import Mathlib namespace OAI noncomputable section open scoped Topology SchwartzMap FourierTransform ContDiff open Set Filter MeasureTheory namespace BiharmonicKernel lemma gaussian_moment_bound (n : ℕ) : ∃ C : ℝ, ∀ x : ℝ, |x|^n * Real.exp (-(x^2/2)) ≤ C := by refine ⟨max 1 ((2:ℝ)^n * (n.factorial : ℝ)), fun x => ?_⟩ by_cases hx : |x| ≤ 1 · apply le_trans _ (le_max_left _ _) have he : Real.exp (-(x^2/2)) ≤ 1 := Real.exp_le_one_iff.mpr (neg_nonpos.mpr (by positivity)) exact (mul_le_of_le_one_left (Real.exp_nonneg _) (pow_le_one₀ (abs_nonneg x) hx)).trans he · have hx1 : 1 ≤ |x| := (lt_of_not_ge hx).le have hxx : |x| ≤ x^2 := by nlinarith [sq_abs x] have hf : (x^2/2)^n ≤ Real.exp (x^2/2)*(n.factorial : ℝ) := (div_le_iff₀ (show (0:ℝ)<(n.factorial : ℝ) by positivity)).mp (Real.pow_div_factorial_le_exp _ (by positivity) n) have hfm : (x^2/2)^n * Real.exp (-(x^2/2)) ≤ (n.factorial : ℝ) := by calc _ ≤ (Real.exp (x^2/2)*(n.factorial : ℝ))*Real.exp (-(x^2/2)) := mul_le_mul_of_nonneg_right hf (Real.exp_nonneg _) _ = (n.factorial : ℝ) := by rw [mul_right_comm, ← Real.exp_add, add_neg_cancel, Real.exp_zero, one_mul] calc _ ≤ (x^2)^n * Real.exp (-(x^2/2)) := by gcongr _ = (2:ℝ)^n *((x^2/2)^n * Real.exp (-(x^2/2))) := by rw [← mul_assoc, ← mul_pow] congr 2 ring _ ≤ (2:ℝ)^n * (n.factorial : ℝ) := mul_le_mul_of_nonneg_left hfm (by positivity) _ ≤ _ := le_max_right _ _ lemma polynomial_gaussian_bound (p : Polynomial ℝ) (k : ℕ) : ∃ C : ℝ, ∀ x : ℝ, |x|^k * |p.eval x| * Real.exp (-(x^2/2)) ≤ C := by induction p using Polynomial.induction_on' with | add p q hp hq => obtain ⟨Cp,hp⟩ := hp obtain ⟨Cq,hq⟩ := hq refine ⟨Cp+Cq, fun x => ?_⟩ calc _ ≤ |x|^k*(|p.eval x|+|q.eval x|)*Real.exp (-(x^2/2)) := by rw [Polynomial.eval_add] gcongr exact abs_add_le _ _ _ = |x|^k*|p.eval x| *Real.exp (-(x^2/2))+ |x|^k*|q.eval x| *Real.exp (-(x^2/2)) := by ring _ ≤ Cp+Cq := add_le_add (hp x) (hq x) | monomial n a => obtain ⟨C,hC⟩ := gaussian_moment_bound (k+n) refine ⟨|a| *C, fun x => ?_⟩ calc _ = |a| *(|x|^(k+n)*Real.exp (-(x^2/2))) := by simp only [Polynomial.eval_monomial, abs_mul, abs_pow, pow_add] ring _ ≤ |a| *C := mul_le_mul_of_nonneg_left (hC x) (abs_nonneg _) def gaussian : SchwartzMap ℝ ℝ where toFun x := Real.exp (-(x^2/2)) smooth' := by fun_prop decay' k n := by obtain ⟨C,hC⟩ := polynomial_gaussian_bound ((Polynomial.hermite n).map (algebraMap ℤ ℝ)) k refine ⟨C,fun x => ?_⟩ rw [norm_iteratedFDeriv_eq_norm_iteratedDeriv, iteratedDeriv_eq_iterate, Polynomial.deriv_gaussian_eq_hermite_mul_gaussian] simpa only [norm_mul, norm_pow, norm_neg, norm_one, one_pow, one_mul, Real.norm_eq_abs, abs_of_pos (Real.exp_pos _), Polynomial.eval_map, Polynomial.aeval_def, mul_assoc] using hC x variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] omit [InnerProductSpace ℝ V] in lemma norm_le_quadratic (q : ℝ) (hq : 0 < q) : ∃ (k : ℕ) (C : ℝ), ∀ x : V, ‖x‖ ≤ C * (1+‖q*‖x‖^2‖)^k := by refine ⟨1,1+q⁻¹,fun x => ?_⟩ rw [pow_one, Real.norm_eq_abs, abs_of_nonneg (by positivity)] have he : (1+q⁻¹)*(1+q*‖x‖^2) = 1+‖x‖^2+q*‖x‖^2+q⁻¹ := by field_simp ring rw [he] nlinarith [sq_nonneg (‖x‖-1/2), inv_pos.mpr hq, mul_nonneg hq.le (sq_nonneg ‖x‖)] def multiplier (b : ℝ) (hb : 0 < b) : SchwartzMap V ℂ := SchwartzMap.postcompCLM Complex.ofRealCLM (SchwartzMap.compCLM ℝ (g := fun x : V => Real.sqrt (2*b)*‖x‖^2) (by fun_prop) (norm_le_quadratic _ (Real.sqrt_pos.mpr (by positivity))) gaussian) @[simp] lemma multiplier_apply (b : ℝ) (hb : 0 < b) (x : V) : multiplier b hb x = (Real.exp (-b*‖x‖^4) : ℂ) := by change (Real.exp (-((Real.sqrt (2*b)*‖x‖^2)^2/2)) : ℂ) = _ congr 2 rw [mul_pow, Real.sq_sqrt (by positivity)] ring variable [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] open Real FourierTransform def heatKernel (θ : ℝ) (hθ : 0 < θ) : SchwartzMap V ℂ := 𝓕⁻ (multiplier (V := V) ((2*Real.pi)^4*θ) (by positivity)) lemma fourier_heatKernel (θ : ℝ) (hθ : 0 < θ) : 𝓕 (heatKernel (V := V) θ hθ) = multiplier ((2*Real.pi)^4*θ) (by positivity) := by exact fourier_fourierInv_eq (multiplier (V := V) ((2*Real.pi)^4*θ) (by positivity)) lemma integral_heatKernel (θ : ℝ) (hθ : 0 < θ) : ∫ x : V, heatKernel θ hθ x = 1 := by have h := congrArg (fun f : SchwartzMap V ℂ => f 0) (fourier_heatKernel (V := V) θ hθ) simpa [SchwartzMap.fourier_coe, Real.fourier_eq, multiplier_apply] using h lemma heatKernel_decay (θ : ℝ) (hθ : 0 < θ) (N j : ℕ) : ∃ C : ℝ, 0 ≤ C ∧ ∀ x : V, ‖iteratedFDeriv ℝ j (heatKernel (V := V) θ hθ) x‖ ≤ C/(1+‖x‖)^N := by let C := 2^N * (Finset.Iic (N,j)).sup (fun m => SchwartzMap.seminorm ℝ m.1 m.2) (heatKernel (V := V) θ hθ) refine ⟨C,by positivity,fun x => ?_⟩ apply (le_div_iff₀ (by positivity)).mpr rw [mul_comm] exact SchwartzMap.one_add_le_sup_seminorm_apply (𝕜 := ℝ) (m := (N,j)) le_rfl le_rfl (heatKernel (V := V) θ hθ) x lemma heatKernel_invariant (θ : ℝ) (hθ : 0 < θ) (A : V ≃ₗᵢ[ℝ] V) (x : V) : heatKernel θ hθ (A x) = heatKernel θ hθ x := by have hm : (fun y : V => multiplier ((2*Real.pi)^4*θ) (by positivity) (A y)) = (multiplier ((2*Real.pi)^4*θ) (by positivity) : V → ℂ) := by ext y simp unfold heatKernel rw [SchwartzMap.fourierInv_coe, Real.fourierInv_eq_fourier_neg, ← map_neg A, ← Real.fourier_comp_linearIsometry, Function.comp_def, hm, ← Real.fourierInv_eq_fourier_neg] open scoped RealInnerProductSpace Real lemma fourierInv_comp_smul (f : V → ℂ) (r : ℝ) (hr : 0 < r) (x : V) : 𝓕⁻ (fun ξ => f (r • ξ)) x = (r ^ Module.finrank ℝ V)⁻¹ • 𝓕⁻ f (r⁻¹ • x) := by rw [Real.fourierInv_eq, Real.fourierInv_eq] convert Measure.integral_comp_smul_of_nonneg volume (fun ξ : V => Real.fourierChar ⟪ξ,r⁻¹ • x⟫ • f ξ) r (hR := hr.le) using 1 congr 1 ext ξ simp only [real_inner_smul_left, real_inner_smul_right] rw [← mul_assoc, inv_mul_cancel₀ hr.ne', one_mul] lemma heatKernel_scaling (θ : ℝ) (hθ : 0 < θ) (r : ℝ) (hr : 0 < r) (x : V) : heatKernel (r^4*θ) (by positivity) x = (r ^ Module.finrank ℝ V)⁻¹ • heatKernel θ hθ (r⁻¹ • x) := by have hm : (multiplier (V := V) ((2*Real.pi)^4*(r^4*θ)) (by positivity) : V → ℂ) = fun ξ => multiplier ((2*Real.pi)^4*θ) (by positivity) (r • ξ) := by ext ξ simp only [multiplier_apply, norm_smul, Real.norm_eq_abs, abs_of_pos hr] congr 2 ring simp only [heatKernel, SchwartzMap.fourierInv_coe, hm] exact fourierInv_comp_smul _ _ hr _ lemma heatKernel_deriv_scaling (θ : ℝ) (hθ : 0 < θ) (r : ℝ) (hr : 0 < r) (j : ℕ) (x : V) : iteratedFDeriv ℝ j (heatKernel (V := V) (r^4*θ) (by positivity)) x = ((r ^ Module.finrank ℝ V)⁻¹ * (r⁻¹)^j) • iteratedFDeriv ℝ j (heatKernel θ hθ) (r⁻¹ • x) := by have he : (heatKernel (V := V) (r^4*θ) (by positivity) : V → ℂ) = fun y => (r ^ Module.finrank ℝ V)⁻¹ • heatKernel θ hθ (r⁻¹ • y) := by ext y exact heatKernel_scaling θ hθ r hr y rw [he, iteratedFDeriv_const_smul_apply'] · rw [iteratedFDeriv_comp_const_smul r⁻¹ ((heatKernel θ hθ).smooth j), smul_smul] · exact (((heatKernel θ hθ).smooth j).comp (by fun_prop)).contDiffAt lemma heatKernel_scaled_decay (N j : ℕ) : ∃ C : ℝ, 0 ≤ C ∧ ∀ r : ℝ, ∀ hr : 0 < r, ∀ x : V, ‖iteratedFDeriv ℝ j (heatKernel (V := V) (r^4) (by positivity)) x‖ ≤ C / r^(Module.finrank ℝ V+j) / (1+‖x‖/r)^N := by obtain ⟨C,hC,hdec⟩ := heatKernel_decay (V := V) 1 zero_lt_one N j refine ⟨C,hC,fun r hr x => ?_⟩ have hid := heatKernel_deriv_scaling (V := V) 1 zero_lt_one r hr j x simp only [mul_one] at hid rw [hid, norm_smul, Real.norm_eq_abs, abs_of_nonneg (by positivity)] have hd := hdec (r⁻¹ • x) simp only [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hr)] at hd calc _ ≤ ((r ^ Module.finrank ℝ V)⁻¹ * r⁻¹^j) * (C/(1+r⁻¹*‖x‖)^N) := by gcongr _ = _ := by rw [pow_add, inv_pow]; field_simp def symbol (x : V) : ℝ := (2*Real.pi)^4 * ‖x‖^4 omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in lemma symbol_nonneg (x : V) : 0 ≤ symbol x := by unfold symbol; positivity omit [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in lemma symbol_temperate : Function.HasTemperateGrowth (symbol (V := V)) := by have he : symbol (V := V) = fun x : V => (2*Real.pi)^4 * (‖x‖^2)^2 := by ext x unfold symbol ring rw [he] fun_prop def kernel (t : ℝ) (x : V) : ℂ := 𝓕⁻ (fun ξ : V => (Real.exp (-t*symbol ξ) : ℂ)) x lemma heatKernel_eq_kernel (t : ℝ) (ht : 0 < t) : (heatKernel (V := V) t ht : V → ℂ) = kernel t := by simp only [heatKernel, SchwartzMap.fourierInv_coe] congr 1 ext ξ simp only [multiplier_apply, symbol] congr 2 ring lemma kernel_hasDerivAt (t : ℝ) (ht : 0 < t) (x : V) : HasDerivAt (fun s => kernel s x) (𝓕⁻ (fun ξ : V => (-symbol ξ) • (Real.exp (-t*symbol ξ) : ℂ)) x) t := by let F := fun s : ℝ => fun ξ : V => Real.fourierChar ⟪ξ,x⟫ • (Real.exp (-s*symbol ξ) : ℂ) let F' := fun s : ℝ => fun ξ : V => Real.fourierChar ⟪ξ,x⟫ • ((-symbol ξ) • (Real.exp (-s*symbol ξ) : ℂ)) let W := SchwartzMap.smulLeftCLM ℂ (symbol (V := V)) (multiplier (V := V) ((2*Real.pi)^4*(t/2)) (by positivity)) have hW (ξ : V) : ‖W ξ‖ = symbol ξ * Real.exp (-(t/2)*symbol ξ) := by simp only [W, SchwartzMap.smulLeftCLM_apply_apply symbol_temperate, multiplier_apply, norm_smul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (symbol_nonneg ξ), abs_of_pos (Real.exp_pos _)] congr 2 unfold symbol ring have htemp : Continuous (symbol (V := V)) := symbol_temperate.1.continuous have hFi : Integrable (F t) := by have hm : (fun ξ => (Real.exp (-t*symbol ξ) : ℂ)) = (multiplier (V := V) ((2*Real.pi)^4*t) (by positivity) : V → ℂ) := by ext ξ rw [multiplier_apply] congr 2 unfold symbol ring have hi : Integrable (multiplier (V := V) ((2*Real.pi)^4*t) (by positivity)) volume := (multiplier (V := V) ((2*Real.pi)^4*t) (by positivity)).integrable convert (Real.fourierIntegral_convergent_iff (-x)).mpr hi using 1 ext ξ rw [← congrFun hm ξ] simp only [F, inner_neg_right, neg_neg] have hder : ∀ ξ : V, ∀ s : ℝ, HasDerivAt (fun u => F u ξ) (F' s ξ) s := by intro ξ s have h : HasDerivAt (fun u : ℝ => (Real.exp (-u*symbol ξ) : ℂ)) ((-symbol ξ) • (Real.exp (-s*symbol ξ) : ℂ)) s := by convert (((hasDerivAt_id s).neg.mul_const (symbol ξ)).exp).ofReal_comp using 1 <;> simp [Complex.real_smul, mul_comm] simpa only [F, F'] using h.fun_const_smul (Real.fourierChar ⟪ξ,x⟫) have hh := hasDerivAt_integral_of_dominated_loc_of_deriv_le (F := F) (F' := F') (bound := fun ξ : V => ‖W ξ‖) (s := Ioi (t/2)) (Ioi_mem_nhds (show t/2 < t by linarith)) (Filter.Eventually.of_forall (fun s => (show Continuous (F s) by dsimp [F]; fun_prop).aestronglyMeasurable)) hFi (show AEStronglyMeasurable (F' t) volume from (show Continuous (F' t) by dsimp [F']; fun_prop).aestronglyMeasurable) (Filter.Eventually.of_forall (fun ξ s hs => by change ‖Real.fourierChar ⟪ξ,x⟫ • ((-symbol ξ) • (Real.exp (-s*symbol ξ) : ℂ))‖ ≤ ‖W ξ‖ simp only [Circle.norm_smul, norm_smul, Complex.norm_real, Real.norm_eq_abs, abs_neg, abs_of_nonneg (symbol_nonneg ξ), abs_of_pos (Real.exp_pos _), hW] apply mul_le_mul_of_nonneg_left _ (symbol_nonneg ξ) apply Real.exp_le_exp.mpr have hs' : t/2 ≤ s := hs.le nlinarith [symbol_nonneg ξ])) W.integrable.norm (Filter.Eventually.of_forall (fun ξ s _ => hder ξ s)) simpa only [kernel, Real.fourierInv_eq, F, F'] using hh.2 open Laplacian lemma biharmonic_eq_fourierMultiplier (f : SchwartzMap V ℂ) : Δ (Δ f) = SchwartzMap.fourierMultiplierCLM ℂ (symbol (V := V)) f := by have hg : Function.HasTemperateGrowth (fun ξ : V => ‖ξ‖^2) := by fun_prop rw [SchwartzMap.laplacian_eq_fourierMultiplierCLM (Δ f), SchwartzMap.laplacian_eq_fourierMultiplierCLM f, map_smul, smul_smul, SchwartzMap.fourierMultiplierCLM_fourierMultiplierCLM_apply hg hg] have he : symbol (V := V) = ((-(2*Real.pi)^2)* (-(2*Real.pi)^2)) • ((fun ξ : V => ‖ξ‖^2) * (fun ξ : V => ‖ξ‖^2)) := by ext ξ simp only [symbol, Pi.smul_apply, Pi.mul_apply, smul_eq_mul] ring rw [he, SchwartzMap.fourierMultiplierCLM_smul (hg.mul hg)] rfl lemma kernel_heat_equation (t : ℝ) (ht : 0 < t) (x : V) : HasDerivAt (fun s => kernel s x) (-((Δ (Δ (heatKernel (V := V) t ht))) x)) t := by have hd := kernel_hasDerivAt t ht x have he : (fun ξ : V => (-symbol ξ) • (Real.exp (-t*symbol ξ) : ℂ)) = ((-SchwartzMap.smulLeftCLM ℂ (symbol (V := V)) (multiplier (V := V) ((2*Real.pi)^4*t) (by positivity)) : SchwartzMap V ℂ) : V → ℂ) := by ext ξ simp only [neg_apply, SchwartzMap.smulLeftCLM_apply_apply symbol_temperate, multiplier_apply, neg_smul] congr 2 congr 1 unfold symbol ring_nf rw [he, ← SchwartzMap.fourierInv_coe, FourierTransform.fourierInv_neg] at hd rw [biharmonic_eq_fourierMultiplier, SchwartzMap.fourierMultiplierCLM_apply, fourier_heatKernel] exact hd lemma kernel_eq_scaled (t : ℝ) (ht : 0 < t) (x : V) : kernel t x = ((Real.sqrt (Real.sqrt t)) ^ Module.finrank ℝ V)⁻¹ • heatKernel 1 zero_lt_one ((Real.sqrt (Real.sqrt t))⁻¹ • x) := by have hr : 0 < Real.sqrt (Real.sqrt t) := Real.sqrt_pos.mpr (Real.sqrt_pos.mpr ht) have he : (Real.sqrt (Real.sqrt t))^4 = t := by calc _ = ((Real.sqrt (Real.sqrt t))^2)^2 := by ring _ = t := by rw [Real.sq_sqrt (Real.sqrt_nonneg t), Real.sq_sqrt ht.le] have hs := heatKernel_scaling (V := V) 1 zero_lt_one _ hr x simp only [mul_one, he] at hs rw [← heatKernel_eq_kernel t ht] exact hs lemma kernel_contDiffAt (q : ℝ × V) (hq : 0 < q.1) : ContDiffAt ℝ ∞ (fun p : ℝ × V => kernel p.1 p.2) q := by have hr : ContDiffAt ℝ ∞ (fun p : ℝ × V => Real.sqrt (Real.sqrt p.1)) q := (contDiffAt_fst.sqrt (ne_of_gt hq)).sqrt (ne_of_gt (Real.sqrt_pos.mpr hq)) have hpos : 0 < Real.sqrt (Real.sqrt q.1) := Real.sqrt_pos.mpr (Real.sqrt_pos.mpr hq) have hs : ContDiffAt ℝ ∞ (fun p : ℝ × V => ((Real.sqrt (Real.sqrt p.1)) ^ Module.finrank ℝ V)⁻¹ • heatKernel 1 zero_lt_one ((Real.sqrt (Real.sqrt p.1))⁻¹ • p.2)) q := ((hr.pow _).inv (pow_ne_zero _ hpos.ne')).smul (((heatKernel (V := V) 1 zero_lt_one).smooth ⊤).contDiffAt.comp q ((hr.inv hpos.ne').smul contDiffAt_snd)) apply hs.congr_of_eventuallyEq filter_upwards [continuous_fst.continuousAt.eventually (Ioi_mem_nhds hq)] with p hp exact kernel_eq_scaled _ hp _ lemma integral_norm_heatKernel (t : ℝ) (ht : 0 < t) : ∫ x : V, ‖heatKernel t ht x‖ = ∫ x : V, ‖heatKernel 1 zero_lt_one x‖ := by let r := Real.sqrt (Real.sqrt t) have hr : 0 < r := Real.sqrt_pos.mpr (Real.sqrt_pos.mpr ht) have he : (fun x : V => ‖heatKernel t ht x‖) = fun x => (r ^ Module.finrank ℝ V)⁻¹ * ‖heatKernel 1 zero_lt_one (r⁻¹ • x)‖ := by ext x rw [congrFun (heatKernel_eq_kernel t ht) x, kernel_eq_scaled t ht x, norm_smul, Real.norm_eq_abs, abs_of_nonneg (by positivity)] rw [he, integral_const_mul, MeasureTheory.Measure.integral_comp_inv_smul_of_nonneg volume (fun x : V => ‖heatKernel 1 zero_lt_one x‖) hr.le] simp [hr.ne'] end BiharmonicKernel open scoped Topology ZeroAtInfty open Set Filter MeasureTheory namespace C0Convolution variable {V : Type*} [NormedAddCommGroup V] def translate (y : V) (f : C₀(V, ℂ)) : C₀(V, ℂ) := f.comp (Homeomorph.subRight y).toCocompactMap @[simp] lemma translate_apply (y : V) (f : C₀(V, ℂ)) (x : V) : translate y f x = f (x-y) := rfl @[simp] lemma translate_zero (f : C₀(V, ℂ)) : translate 0 f = f := by ext x simp @[simp] lemma translate_add (y : V) (f g : C₀(V, ℂ)) : translate y (f+g) = translate y f + translate y g := rfl @[simp] lemma translate_smul (y : V) (c : ℂ) (f : C₀(V, ℂ)) : translate y (c • f) = c • translate y f := rfl lemma norm_le_iff (f : C₀(V, ℂ)) {C : ℝ} (hC : 0 ≤ C) : ‖f‖ ≤ C ↔ ∀ x : V, ‖f x‖ ≤ C := BoundedContinuousFunction.norm_le hC lemma apply_norm_le (f : C₀(V, ℂ)) (x : V) : ‖f x‖ ≤ ‖f‖ := f.toBCF.norm_coe_le_norm x @[simp] lemma norm_translate (y : V) (f : C₀(V, ℂ)) : ‖translate y f‖ = ‖f‖ := by apply le_antisymm · exact (norm_le_iff _ (norm_nonneg _)).mpr (fun x => apply_norm_le f (x-y)) · apply (norm_le_iff _ (norm_nonneg _)).mpr intro x simpa using apply_norm_le (translate y f) (x+y) lemma continuous_translate (f : C₀(V, ℂ)) : Continuous (fun y : V => translate y f) := by apply continuous_iff_continuousAt.mpr intro y apply Metric.continuousAt_iff.mpr intro ε hε obtain ⟨δ,hδ,hf⟩ := Metric.uniformContinuous_iff.mp (ZeroAtInftyContinuousMap.uniformContinuous f) (ε/2) (by positivity) refine ⟨δ,hδ,fun z hz => ?_⟩ apply lt_of_le_of_lt _ (show ε/2 < ε by linarith) change dist (translate z f).toBCF (translate y f).toBCF ≤ ε/2 apply (BoundedContinuousFunction.dist_le (by positivity)).mpr intro x change dist (f (x-z)) (f (x-y)) ≤ ε/2 apply (hf ?_).le simpa only [dist_sub_left] using hz def eval (x : V) : C₀(V, ℂ) →L[ℂ] ℂ := LinearMap.mkContinuous { toFun := fun f => f x map_add' := fun _ _ => rfl map_smul' := fun _ _ => rfl } 1 (fun f => by simpa using apply_norm_le f x) @[simp] lemma eval_apply (x : V) (f : C₀(V, ℂ)) : eval x f = f x := rfl variable [MeasureSpace V] [BorelSpace V] [SecondCountableTopology V] lemma integrable_translated (k : V → ℂ) (hk : Integrable k) (f : C₀(V, ℂ)) : Integrable (fun y : V => k y • translate y f) := by apply (hk.norm.mul_const ‖f‖).mono' · exact hk.aestronglyMeasurable.smul (continuous_translate f).aestronglyMeasurable · filter_upwards with y simp only [norm_smul, norm_translate, le_refl] def convolve (k : V → ℂ) (f : C₀(V, ℂ)) : C₀(V, ℂ) := ∫ y : V, k y • translate y f lemma convolve_apply (k : V → ℂ) (hk : Integrable k) (f : C₀(V, ℂ)) (x : V) : convolve k f x = ∫ y : V, k y * f (x-y) := by have h := (eval x).integral_comp_comm (integrable_translated k hk f) simpa [convolve] using h.symm omit [BorelSpace V] [SecondCountableTopology V] in lemma norm_convolve_le (k : V → ℂ) (f : C₀(V, ℂ)) : ‖convolve k f‖ ≤ (∫ y : V, ‖k y‖) * ‖f‖ := by calc _ ≤ ∫ y : V, ‖k y • translate y f‖ := norm_integral_le_integral_norm _ _ = _ := by simp only [norm_smul, norm_translate, integral_mul_const] lemma convolve_add (k : V → ℂ) (hk : Integrable k) (f g : C₀(V, ℂ)) : convolve k (f+g) = convolve k f + convolve k g := by have he (y : V) : k y • translate y (f+g) = k y • translate y f + k y • translate y g := by ext x exact mul_add (k y) (f (x-y)) (g (x-y)) simp only [convolve, he] exact integral_add (integrable_translated k hk f) (integrable_translated k hk g) omit [BorelSpace V] [SecondCountableTopology V] in lemma convolve_smul (k : V → ℂ) (c : ℂ) (f : C₀(V, ℂ)) : convolve k (c • f) = c • convolve k f := by simp only [convolve, translate_smul, smul_comm (k _) c, integral_smul] def convolveCLM (k : V → ℂ) (hk : Integrable k) : C₀(V, ℂ) →L[ℂ] C₀(V, ℂ) := LinearMap.mkContinuous { toFun := convolve k map_add' := convolve_add k hk map_smul' := convolve_smul k } (∫ y : V, ‖k y‖) (norm_convolve_le k) lemma norm_convolveCLM_le (k : V → ℂ) (hk : Integrable k) : ‖convolveCLM k hk‖ ≤ ∫ y : V, ‖k y‖ := by exact LinearMap.mkContinuous_norm_le _ (integral_nonneg (fun _ => norm_nonneg _)) _ variable [NormedSpace ℝ V] lemma integrable_scaled_translated (k : V → ℂ) (hk : Integrable k) (r : ℝ) (f : C₀(V, ℂ)) : Integrable (fun y : V => k y • translate (r • y) f) := by apply (hk.norm.mul_const ‖f‖).mono' · exact hk.aestronglyMeasurable.smul ((continuous_translate f).comp (continuous_const.smul continuous_id)).aestronglyMeasurable · filter_upwards with y simp only [norm_smul, norm_translate, le_refl] def scaleConvolve (k : V → ℂ) (r : ℝ) (f : C₀(V, ℂ)) : C₀(V, ℂ) := ∫ y : V, k y • translate (r • y) f lemma continuous_scaleConvolve (k : V → ℂ) (hk : Integrable k) (f : C₀(V, ℂ)) : Continuous (fun r : ℝ => scaleConvolve k r f) := by apply continuous_of_dominated (bound := fun y => ‖k y‖ * ‖f‖) · exact fun r => (integrable_scaled_translated k hk r f).aestronglyMeasurable · intro r filter_upwards with y simp only [norm_smul, norm_translate, le_refl] · exact hk.norm.mul_const ‖f‖ · filter_upwards with y exact (continuous_const : Continuous (fun _ : ℝ => k y)).smul ((continuous_translate f).comp (continuous_id.smul (continuous_const : Continuous (fun _ : ℝ => y)))) omit [BorelSpace V] [SecondCountableTopology V] in lemma scaleConvolve_zero (k : V → ℂ) (f : C₀(V, ℂ)) : scaleConvolve k 0 f = (∫ y : V, k y) • f := by simp only [scaleConvolve, zero_smul, translate_zero, integral_smul_const] lemma scaleConvolve_tendsto (k : V → ℂ) (hk : Integrable k) (hmass : ∫ y : V, k y = 1) (f : C₀(V, ℂ)) : Tendsto (fun r : ℝ => scaleConvolve k r f) (𝓝 0) (𝓝 f) := by simpa only [scaleConvolve_zero, hmass, one_smul] using (continuous_scaleConvolve k hk f).tendsto 0 variable [FiniteDimensional ℝ V] [Measure.IsAddHaarMeasure (volume : Measure V)] omit [SecondCountableTopology V] in lemma convolve_scaled (k : V → ℂ) (r : ℝ) (hr : 0 < r) (f : C₀(V, ℂ)) : convolve (fun y : V => (r ^ Module.finrank ℝ V)⁻¹ • k (r⁻¹ • y)) f = scaleConvolve k r f := by have he (y : V) : ((r ^ Module.finrank ℝ V)⁻¹ • k (r⁻¹ • y)) • translate y f = (r ^ Module.finrank ℝ V)⁻¹ • (k (r⁻¹ • y) • translate (r • (r⁻¹ • y)) f) := by rw [smul_inv_smul₀ hr.ne'] ext x exact smul_mul_assoc _ _ _ simp only [convolve, he, integral_smul] rw [Measure.integral_comp_inv_smul_of_nonneg volume (fun y : V => k y • translate (r • y) f) hr.le] rw [inv_smul_smul₀ (pow_ne_zero _ hr.ne')] rfl open scoped Convolution omit [NormedSpace ℝ V] [FiniteDimensional ℝ V] [Measure.IsAddHaarMeasure (volume : Measure V)] in lemma convolve_coe (k : V → ℂ) (hk : Integrable k) (f : C₀(V, ℂ)) : (convolve k f : V → ℂ) = k ⋆[ContinuousLinearMap.mul ℂ ℂ] (f : V → ℂ) := by ext x exact convolve_apply k hk f x lemma convolve_assoc (k l : V → ℂ) (hk : Integrable k) (hl : Integrable l) (hkc : Continuous k) (hlc : Continuous l) (f : C₀(V, ℂ)) : convolve (k ⋆[ContinuousLinearMap.mul ℂ ℂ] l) f = convolve k (convolve l f) := by apply DFunLike.coe_injective rw [convolve_coe _ (hk.integrable_convolution _ hl) f, convolve_coe k hk, convolve_coe l hl] ext x apply convolution_assoc' (ContinuousLinearMap.mul ℂ ℂ) (ContinuousLinearMap.mul ℂ ℂ) (ContinuousLinearMap.mul ℂ ℂ) (ContinuousLinearMap.mul ℂ ℂ) (fun _ _ _ => mul_assoc _ _ _) (hk.ae_convolution_exists _ hl) · filter_upwards with z change Integrable (fun y => l y * f (z-y)) exact (eval z).integrable_comp (integrable_translated l hl f) · apply ((hk.convolution_integrand (ContinuousLinearMap.mul ℂ ℂ) hl).norm.mul_const ‖f‖).mono' · exact (show Continuous (fun p : V × V => k p.2 * (l (p.1-p.2) * f (x-p.1))) by fun_prop).aestronglyMeasurable · filter_upwards with p simp only [Function.uncurry, ContinuousLinearMap.mul_apply', norm_mul] calc _ = (‖k p.2‖ * ‖l (p.1-p.2)‖) * ‖f (x-p.1)‖ := by ring _ ≤ _ := mul_le_mul_of_nonneg_left (apply_norm_le f _) (by positivity) end C0Convolution namespace BiharmonicKernel open scoped FourierTransform ZeroAtInfty variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] lemma heatKernel_semigroup (s t : ℝ) (hs : 0 < s) (ht : 0 < t) : SchwartzMap.convolution (ContinuousLinearMap.mul ℂ ℂ) (heatKernel (V := V) s hs) (heatKernel t ht) = heatKernel (s+t) (by positivity) := by apply (FourierTransform.fourierEquiv ℂ (SchwartzMap V ℂ)).injective ext ξ simp only [FourierTransform.fourierEquiv_apply, SchwartzMap.fourier_convolution, fourier_heatKernel, SchwartzMap.pairing_apply_apply, ContinuousLinearMap.mul_apply', multiplier_apply] rw [← Complex.ofReal_mul, ← Real.exp_add] congr 2 ring def evolution (t : ℝ) (f : C₀(V, ℂ)) : C₀(V, ℂ) := C0Convolution.scaleConvolve (heatKernel 1 zero_lt_one) (Real.sqrt (Real.sqrt t)) f lemma evolution_zero (f : C₀(V, ℂ)) : evolution 0 f = f := by simp only [evolution, Real.sqrt_zero, C0Convolution.scaleConvolve_zero, integral_heatKernel, one_smul] lemma evolution_continuous (f : C₀(V, ℂ)) : Continuous (fun t : ℝ => evolution t f) := by exact (C0Convolution.continuous_scaleConvolve (heatKernel 1 zero_lt_one) (heatKernel (V := V) 1 zero_lt_one).integrable f).comp (Real.continuous_sqrt.comp Real.continuous_sqrt) lemma evolution_initial_trace (f : C₀(V, ℂ)) : Tendsto (fun t : ℝ => evolution t f) (𝓝[>] 0) (𝓝 f) := by simpa only [ContinuousWithinAt, evolution_zero] using ((evolution_continuous f).continuousAt (x := 0)).continuousWithinAt (s := Ioi 0) lemma evolution_eq_convolve (t : ℝ) (ht : 0 < t) (f : C₀(V, ℂ)) : evolution t f = C0Convolution.convolve (heatKernel t ht) f := by have he : (heatKernel (V := V) t ht : V → ℂ) = fun x => ((Real.sqrt (Real.sqrt t)) ^ Module.finrank ℝ V)⁻¹ • heatKernel 1 zero_lt_one ((Real.sqrt (Real.sqrt t))⁻¹ • x) := by ext x rw [congrFun (heatKernel_eq_kernel t ht) x, kernel_eq_scaled t ht x] rw [he, C0Convolution.convolve_scaled _ _ (Real.sqrt_pos.mpr (Real.sqrt_pos.mpr ht))] rfl lemma evolution_apply (t : ℝ) (ht : 0 < t) (f : C₀(V, ℂ)) (x : V) : evolution t f x = ∫ y : V, heatKernel t ht y * f (x-y) := by rw [evolution_eq_convolve t ht f, C0Convolution.convolve_apply _ (heatKernel t ht).integrable] lemma evolution_norm_le (t : ℝ) (ht : 0 < t) (f : C₀(V, ℂ)) : ‖evolution t f‖ ≤ (∫ x : V, ‖heatKernel 1 zero_lt_one x‖) * ‖f‖ := by rw [evolution_eq_convolve t ht f] simpa only [integral_norm_heatKernel] using C0Convolution.norm_convolve_le (heatKernel t ht) f open scoped Convolution lemma evolution_semigroup (s t : ℝ) (hs : 0 < s) (ht : 0 < t) (f : C₀(V, ℂ)) : evolution (s+t) f = evolution s (evolution t f) := by rw [evolution_eq_convolve _ (add_pos hs ht), evolution_eq_convolve s hs, evolution_eq_convolve t ht, ← C0Convolution.convolve_assoc _ _ (heatKernel s hs).integrable (heatKernel t ht).integrable (heatKernel s hs).continuous (heatKernel t ht).continuous] congr 1 ext x have he := congrArg (fun h : SchwartzMap V ℂ => h x) (heatKernel_semigroup s t hs ht) rw [SchwartzMap.convolution_apply] at he exact he.symm end BiharmonicKernel open scoped SchwartzMap Topology ZeroAtInfty Convolution ContDiff BoundedContinuousFunction open Set Filter MeasureTheory namespace SchwartzConvolution section ShiftedMajorant universe u variable {V E F A : Type u} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] [MeasureSpace V] [BorelSpace V] [Measure.IsAddHaarMeasure (volume : Measure V)] [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] lemma shifted_majorant (g : SchwartzMap V E) (x₀ : V) : ∃ b : V → ℝ, Integrable b ∧ ∀ x ∈ Metric.ball x₀ 1, ∀ y, ‖g (x-y)‖ ≤ b y := by let N := Module.finrank ℝ V + 1 let C := 2^N * (Finset.Iic (N,0)).sup (fun i => SchwartzMap.seminorm ℝ i.1 i.2) g have hC : 0 ≤ C := by positivity have hdec (z : V) : (1+‖z‖)^N * ‖g z‖ ≤ C := by simpa only [norm_iteratedFDeriv_zero] using SchwartzMap.one_add_le_sup_seminorm_apply (𝕜 := ℝ) (m := (N,0)) le_rfl le_rfl g z refine ⟨fun y => (2^N*C)/(1+‖x₀-y‖)^N, ?_, ?_⟩ · have hi : Integrable (fun z : V => (1+‖z‖) ^ (-(N : ℝ))) := integrable_one_add_norm (by dsimp [N]; norm_num) simpa only [Real.rpow_neg (by positivity : (0:ℝ) ≤ 1+‖x₀-_‖), Real.rpow_natCast, div_eq_mul_inv] using (hi.comp_sub_left x₀).const_mul (2^N*C) · intro x hx y have hd : ‖x₀-x‖ < 1 := by simpa only [Metric.mem_ball, dist_eq_norm, norm_sub_rev] using hx have hn : ‖x₀-y‖ ≤ ‖x₀-x‖ + ‖x-y‖ := by simpa only [sub_add_sub_cancel] using norm_add_le (x₀-x) (x-y) have hs : 1+‖x₀-y‖ ≤ 2*(1+‖x-y‖) := by nlinarith [norm_nonneg (x-y)] apply (le_div_iff₀ (by positivity)).mpr calc ‖g (x-y)‖ * (1+‖x₀-y‖)^N ≤ ‖g (x-y)‖ * (2*(1+‖x-y‖))^N := by gcongr _ = 2^N * ((1+‖x-y‖)^N * ‖g (x-y)‖) := by rw [mul_pow]; ring _ ≤ _ := mul_le_mul_of_nonneg_left (hdec (x-y)) (by positivity) variable [NormedAddCommGroup A] [NormedSpace ℝ A] lemma integrable_integrand (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (x : V) : Integrable (fun y => L (f y) (g (x-y))) := by refine ((g.integrable.norm.comp_sub_left x).const_mul (‖L‖*‖f‖)).mono' (f.continuous.aestronglyMeasurable.convolution_integrand_snd L g.continuous.aestronglyMeasurable x) (Eventually.of_forall fun y => ?_) exact (L.le_opNorm₂ (f y) (g (x-y))).trans (by gcongr exact f.norm_coe_le_norm y) lemma hasFDerivAt_convolution (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (x₀ : V) : HasFDerivAt (f ⋆[L] g) ((f ⋆[L.precompR V] SchwartzMap.fderivCLM ℝ V E g) x₀) x₀ := by let g' := SchwartzMap.fderivCLM ℝ V E g obtain ⟨b, hb, hbound⟩ := shifted_majorant g' x₀ have hderiv (x y : V) : HasFDerivAt (fun x => g (x-y)) (g' (x-y)) x := by simpa [g', SchwartzMap.fderivCLM_apply, Function.comp_def, Pi.sub_apply] using (g.hasFDerivAt (x-y)).comp x ((hasFDerivAt_id x).sub (hasFDerivAt_const y x)) apply hasFDerivAt_integral_of_dominated_of_fderiv_le (Metric.ball_mem_nhds x₀ zero_lt_one) · exact Eventually.of_forall fun x => (integrable_integrand L f g x).aestronglyMeasurable · exact integrable_integrand L f g x₀ · exact (integrable_integrand (L.precompR V) f g' x₀).aestronglyMeasurable · filter_upwards with y x hx change ‖(L (f y)).comp (g' (x-y))‖ ≤ _ calc ‖(L (f y)).comp (g' (x-y))‖ ≤ ‖L (f y)‖ * ‖g' (x-y)‖ := ContinuousLinearMap.opNorm_comp_le _ _ _ ≤ (‖L‖ * ‖f‖) * ‖g' (x-y)‖ := by gcongr exact (L.le_opNorm (f y)).trans (mul_le_mul_of_nonneg_left (f.norm_coe_le_norm y) (norm_nonneg L)) _ ≤ (‖L‖ * ‖f‖) * b y := mul_le_mul_of_nonneg_left (hbound x hx y) (by positivity) · exact hb.const_mul (‖L‖ * ‖f‖) · filter_upwards with y x _ exact (L (f y)).hasFDerivAt.comp x (hderiv x y) lemma contDiff_convolution_nat (n : ℕ) (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) : ContDiff ℝ n (f ⋆[L] g) := by induction n generalizing E F with | zero => rw [Nat.cast_zero, contDiff_zero] exact continuous_iff_continuousAt.mpr fun x => (hasFDerivAt_convolution L f g x).continuousAt | succ n ih => rw [Nat.cast_add, Nat.cast_one, contDiff_succ_iff_hasFDerivAt] exact ⟨f ⋆[L.precompR V] SchwartzMap.fderivCLM ℝ V E g, ih (L.precompR V) (SchwartzMap.fderivCLM ℝ V E g), hasFDerivAt_convolution L f g⟩ lemma contDiff_convolution (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) : ContDiff ℝ ∞ (f ⋆[L] g) := contDiff_infty.mpr fun n => contDiff_convolution_nat n L f g lemma fderiv_convolution (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) : fderiv ℝ (f ⋆[L] g) = f ⋆[L.precompR V] SchwartzMap.fderivCLM ℝ V E g := by funext x exact (hasFDerivAt_convolution L f g x).fderiv lemma fderiv_convolution_apply (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (x v : V) : fderiv ℝ (f ⋆[L] g) x v = ∫ y, L (f y) (fderiv ℝ g (x-y) v) := by rw [fderiv_convolution, MeasureTheory.convolution_def, ContinuousLinearMap.integral_apply (integrable_integrand (L.precompR V) f (SchwartzMap.fderivCLM ℝ V E g) x)] rfl lemma second_fderiv_convolution_apply (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (x v w : V) : fderiv ℝ (fderiv ℝ (f ⋆[L] g)) x v w = ∫ y, L (f y) (fderiv ℝ (fderiv ℝ g) (x-y) v w) := by rw [fderiv_convolution, fderiv_convolution_apply] rw [ContinuousLinearMap.integral_apply] · rfl · exact integrable_integrand (L.precompR V) f (SchwartzMap.evalCLM ℝ V (V →L[ℝ] E) v (SchwartzMap.fderivCLM ℝ V (V →L[ℝ] E) (SchwartzMap.fderivCLM ℝ V E g))) x end ShiftedMajorant open scoped Laplacian universe u variable {V E F A : Type u} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasureSpace V] [BorelSpace V] [Measure.IsAddHaarMeasure (volume : Measure V)] [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup A] [NormedSpace ℝ A] lemma laplacian_convolution (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) : Δ (f ⋆[L] g) = f ⋆[L] (Δ g : SchwartzMap V E) := by funext x rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis] simp only [iteratedFDeriv_two_apply, Matrix.cons_val_zero, Matrix.cons_val_one, second_fderiv_convolution_apply] rw [MeasureTheory.convolution_def] simp_rw [SchwartzMap.laplacian_apply, InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis, iteratedFDeriv_two_apply, Matrix.cons_val_zero, Matrix.cons_val_one, map_sum] rw [integral_finsetSum] · rfl · intro i _ exact integrable_integrand L f (SchwartzMap.evalCLM ℝ V E ((stdOrthonormalBasis ℝ V) i) (SchwartzMap.evalCLM ℝ V (V →L[ℝ] E) ((stdOrthonormalBasis ℝ V) i) (SchwartzMap.fderivCLM ℝ V (V →L[ℝ] E) (SchwartzMap.fderivCLM ℝ V E g)))) x end SchwartzConvolution open Set Filter MeasureTheory open scoped SchwartzMap Topology namespace SchwartzConvolution variable {V E : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] [MeasureSpace V] [BorelSpace V] [Measure.IsAddHaarMeasure (volume : Measure V)] [NormedAddCommGroup E] [NormedSpace ℝ E] lemma scaled_majorant (g : SchwartzMap V E) (a b : ℝ) (ha : 0 < a) (K : ℕ) : ∃ B : V → ℝ, Integrable B ∧ ∀ c ∈ Icc a b, ∀ x, ‖(c⁻¹)^K • g (c⁻¹ • x)‖ ≤ B x := by let N := Module.finrank ℝ V + 1 let M := max b 1 let C := 2^N * (Finset.Iic (N,0)).sup (fun i => SchwartzMap.seminorm ℝ i.1 i.2) g have hC : 0 ≤ C := by positivity have hM : 0 < M := lt_of_lt_of_le zero_lt_one (le_max_right _ _) have hdec (z : V) : (1+‖z‖)^N * ‖g z‖ ≤ C := by simpa only [norm_iteratedFDeriv_zero] using SchwartzMap.one_add_le_sup_seminorm_apply (𝕜 := ℝ) (m := (N,0)) le_rfl le_rfl g z refine ⟨fun x => ((a⁻¹)^K*(M^N*C))/(1+‖x‖)^N, ?_, ?_⟩ · have hi : Integrable (fun x : V => (1+‖x‖)^(-(N : ℝ))) := integrable_one_add_norm (by dsimp [N]; norm_num) simpa only [Real.rpow_neg (by positivity : (0:ℝ) ≤ 1+‖(_ : V)‖), Real.rpow_natCast, div_eq_mul_inv] using hi.const_mul ((a⁻¹)^K*(M^N*C)) · intro c hc x have hcpos : 0 < c := ha.trans_le hc.1 have hcM : c ≤ M := hc.2.trans (le_max_left _ _) have hnx : ‖x‖ = c * ‖c⁻¹ • x‖ := by rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hcpos), ← mul_assoc, mul_inv_cancel₀ hcpos.ne', one_mul] have hs : 1+‖x‖ ≤ M*(1+‖c⁻¹ • x‖) := by rw [hnx, mul_add, mul_one] exact add_le_add (le_max_right b 1) (mul_le_mul_of_nonneg_right hcM (norm_nonneg _)) have hg : ‖g (c⁻¹ • x)‖ * (1+‖x‖)^N ≤ M^N*C := by calc _ ≤ ‖g (c⁻¹ • x)‖ * (M * (1+‖c⁻¹ • x‖))^N := by gcongr _ = M^N * ((1+‖c⁻¹ • x‖)^N * ‖g (c⁻¹ • x)‖) := by rw [mul_pow]; ring _ ≤ _ := mul_le_mul_of_nonneg_left (hdec _) (by positivity) rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (by positivity)] apply (le_div_iff₀ (by positivity)).mpr calc _ = (c⁻¹)^K * (‖g (c⁻¹ • x)‖ * (1+‖x‖)^N) := by ring _ ≤ (a⁻¹)^K * (‖g (c⁻¹ • x)‖ * (1+‖x‖)^N) := by gcongr exact hc.1 _ ≤ _ := mul_le_mul_of_nonneg_left hg (by positivity) end SchwartzConvolution namespace LaplacianScaling open scoped Laplacian ContDiff variable {V E : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [NormedAddCommGroup E] [NormedSpace ℝ E] lemma laplacian_scaled (f : V → E) (hf : ContDiff ℝ 2 f) (a c : ℝ) : Δ (fun x => a • f (c • x)) = fun x => (a*c^2) • Δ f (c • x) := by have hs : ContDiff ℝ 2 (fun x : V => f (c • x)) := hf.comp (by fun_prop) funext x rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis] simp_rw [iteratedFDeriv_const_smul_apply' hs.contDiffAt, iteratedFDeriv_comp_const_smul c hf] simp only [smul_apply, smul_smul, ← Finset.smul_sum, InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis] end LaplacianScaling namespace BiharmonicKernel section LaplacianKernelScaling open scoped SchwartzMap FourierTransform ZeroAtInfty Convolution Laplacian ContDiff variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] lemma laplacian_kernel_scaling (t : ℝ) (ht : 0 < t) (r : ℝ) (hr : 0 < r) : ((Δ (heatKernel (V := V) (r^4*t) (by positivity)) : SchwartzMap V ℂ) : V → ℂ) = fun x => ((r ^ Module.finrank ℝ V)⁻¹ * (r⁻¹)^2) • (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) (r⁻¹ • x) := by have he : (heatKernel (V := V) (r^4*t) (by positivity) : V → ℂ) = fun x => (r ^ Module.finrank ℝ V)⁻¹ • heatKernel (V := V) t ht (r⁻¹ • x) := by funext x exact heatKernel_scaling t ht r hr x have hs := LaplacianScaling.laplacian_scaled (heatKernel (V := V) t ht) ((heatKernel (V := V) t ht).smooth 2) (r ^ Module.finrank ℝ V)⁻¹ r⁻¹ funext x simpa only [← he, ← SchwartzMap.laplacian_apply] using congrFun hs x lemma biharmonic_kernel_scaling (t : ℝ) (ht : 0 < t) (r : ℝ) (hr : 0 < r) : ((Δ (Δ (heatKernel (V := V) (r^4*t) (by positivity)) : SchwartzMap V ℂ) : SchwartzMap V ℂ) : V → ℂ) = fun x => (r⁻¹)^(Module.finrank ℝ V+4) • (Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ) (r⁻¹ • x) := by have he := laplacian_kernel_scaling (V := V) t ht r hr have hs := LaplacianScaling.laplacian_scaled ((Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : V → ℂ) ((Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ).smooth 2) ((r ^ Module.finrank ℝ V)⁻¹ * (r⁻¹)^2) r⁻¹ funext x have hx := congrFun hs x rw [← he] at hx simp only [← SchwartzMap.laplacian_apply] at hx rw [hx, inv_pow, pow_add] congr 1 ring lemma evolution_coe_convolution (t : ℝ) (ht : 0 < t) (f : C₀(V, ℂ)) : (evolution t f : V → ℂ) = f.toBCF ⋆[ContinuousLinearMap.mul ℝ ℂ] heatKernel (V := V) t ht := by funext x rw [evolution_apply t ht, convolution_def] rw [← integral_sub_left_eq_self (fun y : V => heatKernel (V := V) t ht y * f (x-y)) volume x] simp only [sub_sub_self, ContinuousLinearMap.mul_apply'] congr 1 funext y exact mul_comm _ _ lemma evolution_smooth (t : ℝ) (ht : 0 < t) (f : C₀(V, ℂ)) : ContDiff ℝ ∞ (evolution t f : V → ℂ) := by rw [evolution_coe_convolution t ht] exact SchwartzConvolution.contDiff_convolution (ContinuousLinearMap.mul ℝ ℂ) f.toBCF (heatKernel (V := V) t ht) lemma evolution_biharmonic (t : ℝ) (ht : 0 < t) (f : C₀(V, ℂ)) : Δ (Δ (evolution t f : V → ℂ)) = (f.toBCF ⋆[ContinuousLinearMap.mul ℝ ℂ] (Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ)) := by rw [evolution_coe_convolution t ht, SchwartzConvolution.laplacian_convolution, SchwartzConvolution.laplacian_convolution] lemma kernel_deriv_majorant (t : ℝ) (ht : 0 < t) : ∃ B : V → ℝ, Integrable B ∧ ∀ s ∈ Ioo (t/2) (2*t), ∀ x : V, ‖deriv (fun u => kernel u x) s‖ ≤ B x := by let r : ℝ → ℝ := fun s => Real.sqrt (Real.sqrt s) let g : SchwartzMap V ℂ := Δ (Δ (heatKernel (V := V) 1 zero_lt_one) : SchwartzMap V ℂ) have hr (s : ℝ) (hs : 0 < s) : 0 < r s := Real.sqrt_pos.mpr (Real.sqrt_pos.mpr hs) have hr4 (s : ℝ) (hs : 0 < s) : r s ^ 4 = s := by dsimp [r] calc _ = ((Real.sqrt (Real.sqrt s))^2)^2 := by ring _ = s := by rw [Real.sq_sqrt (Real.sqrt_nonneg s), Real.sq_sqrt hs.le] obtain ⟨B,hB,hbound⟩ := SchwartzConvolution.scaled_majorant g (r (t/2)) (r (2*t)) (hr _ (by positivity)) (Module.finrank ℝ V+4) refine ⟨B,hB,fun s hs x => ?_⟩ have hspos : 0 < s := lt_trans (by positivity) hs.1 rw [(kernel_heat_equation s hspos x).deriv, norm_neg] have he := congrFun (biharmonic_kernel_scaling (V := V) 1 zero_lt_one (r s) (hr s hspos)) x simp only [mul_one, hr4 s hspos] at he rw [he] apply hbound (r s) ⟨?_, ?_⟩ x · exact Real.sqrt_le_sqrt (Real.sqrt_le_sqrt hs.1.le) · exact Real.sqrt_le_sqrt (Real.sqrt_le_sqrt hs.2.le) lemma evolution_hasDerivAt (t : ℝ) (ht : 0 < t) (f : C₀(V, ℂ)) : HasDerivAt (fun s => evolution s f) (-C0Convolution.convolve (Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ) f) t := by let F := fun s : ℝ => fun y : V => kernel s y • C0Convolution.translate y f let F' := fun s : ℝ => fun y : V => deriv (fun u => kernel u y) s • C0Convolution.translate y f have hK (s : ℝ) (hs : 0 < s) : kernel (V := V) s = heatKernel s hs := (heatKernel_eq_kernel s hs).symm have hD (s : ℝ) (hs : 0 < s) : (fun y : V => deriv (fun u => kernel u y) s) = (-(Δ (Δ (heatKernel (V := V) s hs) : SchwartzMap V ℂ) : SchwartzMap V ℂ) : V → ℂ) := by funext y exact (kernel_heat_equation s hs y).deriv have hFi (s : ℝ) (hs : 0 < s) : Integrable (F s) := by dsimp [F] rw [hK s hs] exact C0Convolution.integrable_translated _ (heatKernel s hs).integrable f have hF'i (s : ℝ) (hs : 0 < s) : Integrable (F' s) := by change Integrable (fun y => (fun y : V => deriv (fun u => kernel u y) s) y • _) rw [hD s hs] exact C0Convolution.integrable_translated _ (-(Δ (Δ (heatKernel (V := V) s hs) : SchwartzMap V ℂ) : SchwartzMap V ℂ)).integrable f obtain ⟨B,hB,hbound⟩ := kernel_deriv_majorant (V := V) t ht have hd := hasDerivAt_integral_of_dominated_loc_of_deriv_le (F := F) (F' := F') (bound := fun y => B y * ‖f‖) (s := Ioo (t/2) (2*t)) (Ioo_mem_nhds (by linarith) (by linarith)) (by filter_upwards [Ioi_mem_nhds ht] with s hs exact (hFi s hs).aestronglyMeasurable) (hFi t ht) (hF'i t ht).aestronglyMeasurable (Filter.Eventually.of_forall (fun y s hs => by simp only [F', norm_smul, C0Convolution.norm_translate] exact mul_le_mul_of_nonneg_right (hbound s hs y) (norm_nonneg f))) (hB.mul_const ‖f‖) (Filter.Eventually.of_forall (fun y s hs => by dsimp only [F, F'] exact ((kernel_heat_equation s (lt_trans (by positivity) hs.1) y).differentiableAt.hasDerivAt).smul_const (C0Convolution.translate y f))) have hval : (∫ y, F' t y) = -C0Convolution.convolve (Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ) f := by change (∫ y, (fun y : V => deriv (fun u => kernel u y) t) y • _) = _ rw [hD t ht] change (∫ y, -((Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ) y) • C0Convolution.translate y f) = -(∫ y, (Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ) y • C0Convolution.translate y f) have he (y : V) : -((Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ) y) • C0Convolution.translate y f = -((Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ) y • C0Convolution.translate y f) := neg_smul ((Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ) y) (C0Convolution.translate y f) simp_rw [he] exact integral_neg _ rw [hval] at hd apply hd.2.congr_of_eventuallyEq filter_upwards [Ioi_mem_nhds ht] with s hs rw [evolution_eq_convolve s hs] change C0Convolution.convolve _ _ = ∫ y, kernel s y • _ rw [hK s hs] rfl lemma evolution_heat_equation (t : ℝ) (ht : 0 < t) (f : C₀(V, ℂ)) (x : V) : HasDerivAt (fun s => evolution s f x) (- (Δ (Δ (evolution t f : V → ℂ) : V → ℂ) : V → ℂ) x) t := by have hd := ((C0Convolution.eval x).restrictScalars ℝ).hasFDerivAt.comp_hasDerivAt t (evolution_hasDerivAt t ht f) change HasDerivAt (fun s => evolution s f x) (-C0Convolution.convolve (Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ) f x) t at hd convert hd using 1 congr 1 rw [evolution_biharmonic, convolution_def, C0Convolution.convolve_apply _ (Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable] rw [← integral_sub_left_eq_self (fun y : V => (Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ) y * f (x-y)) volume x] simp only [sub_sub_self, ContinuousLinearMap.mul_apply'] congr 1 funext y exact mul_comm _ _ end LaplacianKernelScaling open scoped SchwartzMap ZeroAtInfty variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] lemma kernel_majorant (a b : ℝ) (ha : 0 < a) : ∃ B : V → ℝ, Integrable B ∧ ∀ t ∈ Icc a b, ∀ x : V, ‖kernel t x‖ ≤ B x := by let r := fun t : ℝ => Real.sqrt (Real.sqrt t) obtain ⟨B,hB,hbound⟩ := SchwartzConvolution.scaled_majorant (heatKernel (V := V) 1 zero_lt_one) (r a) (r b) (Real.sqrt_pos.mpr (Real.sqrt_pos.mpr ha)) (Module.finrank ℝ V) refine ⟨B,hB,fun t ht x => ?_⟩ rw [kernel_eq_scaled t (ha.trans_le ht.1) x, ← inv_pow] exact hbound (r t) ⟨Real.sqrt_le_sqrt (Real.sqrt_le_sqrt ht.1), Real.sqrt_le_sqrt (Real.sqrt_le_sqrt ht.2)⟩ x lemma continuous_kernel_comp {X : Type*} [TopologicalSpace X] (a : X → ℝ) (v : X → V) (ha : Continuous a) (hv : Continuous v) (hpos : ∀ x, 0 < a x) : Continuous (fun x => kernel (a x) (v x)) := by apply continuous_iff_continuousAt.mpr intro x exact ContinuousAt.comp (f := fun u => (a u,v u)) (kernel_contDiffAt (a x,v x) (hpos x)).continuousAt (ha.continuousAt.prodMk hv.continuousAt) end BiharmonicKernel namespace FrozenParametrix open scoped SchwartzMap ZeroAtInfty open BiharmonicKernel variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] def value (a : V → ℝ) (r : ℝ) (f : C₀(V, ℂ)) (x : V) : ℂ := ∫ y : V, kernel (a (x-r • y)) y * f (x-r • y) lemma integrable_value (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (r : ℝ) (f : C₀(V, ℂ)) (x : V) : Integrable (fun y : V => kernel (a (x-r • y)) y * f (x-r • y)) := by obtain ⟨B,hB,hbound⟩ := kernel_majorant (V := V) α β hα apply (hB.mul_const ‖f‖).mono' · exact ((continuous_kernel_comp (fun y : V => a (x-r • y)) id (ha.comp (by fun_prop)) continuous_id (fun y => hα.trans_le (hab _).1)).mul (f.continuous.comp (by fun_prop))).aestronglyMeasurable · filter_upwards with y rw [norm_mul] exact mul_le_mul (hbound _ (hab _) y) (C0Convolution.apply_norm_le f _) (norm_nonneg _) ((norm_nonneg (kernel (a 0) y)).trans (hbound (a 0) (hab 0) y)) lemma continuous_value (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (f : C₀(V, ℂ)) : Continuous (fun p : ℝ × V => value a p.1 f p.2) := by obtain ⟨B,hB,hbound⟩ := kernel_majorant (V := V) α β hα change Continuous (fun p : ℝ × V => ∫ y : V, kernel (a (p.2-p.1 • y)) y * f (p.2-p.1 • y)) apply continuous_of_dominated (bound := fun y => B y * ‖f‖) · intro p exact (integrable_value a ha α β hα hab p.1 f p.2).aestronglyMeasurable · intro p filter_upwards with y rw [norm_mul] exact mul_le_mul (hbound _ (hab _) y) (C0Convolution.apply_norm_le f _) (norm_nonneg _) ((norm_nonneg (kernel (a 0) y)).trans (hbound (a 0) (hab 0) y)) · exact hB.mul_const ‖f‖ · filter_upwards with y exact (continuous_kernel_comp (fun p : ℝ × V => a (p.2-p.1 • y)) (fun _ => y) (ha.comp (by fun_prop)) continuous_const (fun p => hα.trans_le (hab _).1)).mul (f.continuous.comp (by fun_prop)) lemma zero_at_infty_value (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (r : ℝ) (f : C₀(V, ℂ)) : Tendsto (value a r f) (cocompact V) (𝓝 0) := by have : (cocompact V).IsCountablyGenerated := by rw [← comap_dist_left_atTop_eq_cocompact (0 : V)] infer_instance obtain ⟨B,hB,hbound⟩ := kernel_majorant (V := V) α β hα have hlim (y : V) : Tendsto (fun x : V => kernel (a (x-r • y)) y * f (x-r • y)) (cocompact V) (𝓝 0) := by have hf : Tendsto (fun x : V => f (x-r • y)) (cocompact V) (𝓝 0) := zero_at_infty (C0Convolution.translate (r • y) f) apply tendsto_zero_iff_norm_tendsto_zero.mpr apply squeeze_zero (fun x => norm_nonneg _) (fun x => ?_) (by simpa using hf.norm.const_mul (B y)) rw [norm_mul] exact mul_le_mul_of_nonneg_right (hbound _ (hab _) y) (norm_nonneg _) have hi := tendsto_integral_filter_of_dominated_convergence (fun y => B y * ‖f‖) (Filter.Eventually.of_forall (fun x => (integrable_value a ha α β hα hab r f x).aestronglyMeasurable)) (Filter.Eventually.of_forall (fun x => Filter.Eventually.of_forall (fun y => by rw [norm_mul] exact mul_le_mul (hbound _ (hab _) y) (C0Convolution.apply_norm_le f _) (norm_nonneg _) ((norm_nonneg (kernel (a 0) y)).trans (hbound (a 0) (hab 0) y))))) (hB.mul_const ‖f‖) (Filter.Eventually.of_forall hlim) change Tendsto (fun x : V => ∫ y : V, kernel (a (x-r • y)) y * f (x-r • y)) _ _ simpa only [integral_zero] using hi def frozen (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (r : ℝ) (f : C₀(V, ℂ)) : C₀(V, ℂ) where toFun := value a r f continuous_toFun := by change Continuous (fun x : V => value a r f x) exact Continuous.comp (f := fun x : V => (r,x)) (continuous_value a ha α β hα hab f) (continuous_const.prodMk continuous_id) zero_at_infty' := zero_at_infty_value a ha α β hα hab r f end FrozenParametrix open scoped Topology ZeroAtInfty BoundedContinuousFunction open Set Filter MeasureTheory namespace C0Multiplier variable {V : Type*} [TopologicalSpace V] def mul (g : V →ᵇ ℂ) (f : C₀(V,ℂ)) : C₀(V,ℂ) where toFun x := g x * f x continuous_toFun := g.continuous.mul f.continuous zero_at_infty' := by apply tendsto_zero_iff_norm_tendsto_zero.mpr apply squeeze_zero (fun x => norm_nonneg _) (fun x => ?_) (by simpa using (zero_at_infty f).norm.const_mul ‖g‖) rw [norm_mul] exact mul_le_mul_of_nonneg_right (g.norm_coe_le_norm x) (norm_nonneg _) @[simp] lemma mul_apply (g : V →ᵇ ℂ) (f : C₀(V,ℂ)) (x : V) : mul g f x = g x * f x := rfl lemma norm_mul_le (g : V →ᵇ ℂ) (f : C₀(V,ℂ)) : ‖mul g f‖ ≤ ‖g‖ * ‖f‖ := by change ‖(mul g f).toBCF‖ ≤ _ apply (BoundedContinuousFunction.norm_le (by positivity)).mpr intro x change ‖g x * f x‖ ≤ _ rw [norm_mul] exact mul_le_mul (g.norm_coe_le_norm x) (f.toBCF.norm_coe_le_norm x) (norm_nonneg _) (norm_nonneg _) def mulCLM (f : C₀(V,ℂ)) : (V →ᵇ ℂ) →L[ℂ] C₀(V,ℂ) := LinearMap.mkContinuous { toFun := fun g => mul g f map_add' := fun g h => by ext x; dsimp [mul]; ring map_smul' := fun c g => by ext x; dsimp [mul]; ring } ‖f‖ (fun g => by change ‖mul g f‖ ≤ ‖f‖ * ‖g‖ rw [mul_comm] exact norm_mul_le g f) lemma continuous_mul (f : C₀(V,ℂ)) : Continuous (fun g : V →ᵇ ℂ => mul g f) := (mulCLM f).continuous end C0Multiplier namespace C0Convolution open scoped ZeroAtInfty variable {V : Type*} [NormedAddCommGroup V] @[simp] lemma translate_sub (y : V) (f g : C₀(V,ℂ)) : translate y (f-g) = translate y f - translate y g := rfl lemma continuous_translate_comp {X : Type*} [TopologicalSpace X] (y : X → V) (f : X → C₀(V,ℂ)) (hy : Continuous y) (hf : Continuous f) : Continuous (fun x => translate (y x) (f x)) := by apply continuous_iff_continuousAt.mpr intro x have hb := ((continuous_translate (f x)).comp hy).tendsto x apply hb.congr_dist change Tendsto (fun z => dist (translate (y z) (f x)) (translate (y z) (f z))) (𝓝 x) (𝓝 0) have he (z : X) : dist (translate (y z) (f x)) (translate (y z) (f z)) = dist (f x) (f z) := by rw [dist_eq_norm, ← translate_sub, norm_translate, ← dist_eq_norm] simp_rw [he] simpa only [dist_self] using (tendsto_const_nhds (x := f x) (f := 𝓝 x)).dist (hf.tendsto x) end C0Convolution namespace FrozenParametrix open scoped SchwartzMap ZeroAtInfty BoundedContinuousFunction open BiharmonicKernel C0Convolution variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] def kernelFamily (α β : ℝ) (hα : 0 < α) (y : V) : C(Icc α β,ℂ) where toFun t := kernel t.val y continuous_toFun := continuous_kernel_comp Subtype.val (fun _ => y) continuous_subtype_val continuous_const (fun t => hα.trans_le t.property.1) lemma continuous_kernelFamily (α β : ℝ) (hα : 0 < α) : Continuous (kernelFamily (V := V) α β hα) := by apply ContinuousMap.continuous_of_continuous_uncurry exact continuous_kernel_comp (fun p : V × Icc α β => p.2.val) Prod.fst (continuous_subtype_val.comp continuous_snd) continuous_fst (fun p => hα.trans_le p.2.property.1) def sourceMultiplier (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (y : V) : V →ᵇ ℂ := (BoundedContinuousFunction.mkOfCompact (kernelFamily α β hα y)).compContinuous ⟨fun x => ⟨a x,hab x⟩, ha.subtype_mk _⟩ lemma continuous_sourceMultiplier (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) : Continuous (sourceMultiplier a ha α β hα hab) := by exact (BoundedContinuousFunction.continuous_compContinuous ⟨fun x => ⟨a x,hab x⟩, ha.subtype_mk _⟩).comp ((ContinuousMap.isometryEquivBoundedOfCompact (Icc α β) ℂ).continuous.comp (continuous_kernelFamily (V := V) α β hα)) def sourceData (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (f : C₀(V,ℂ)) (y : V) : C₀(V,ℂ) := C0Multiplier.mul (sourceMultiplier a ha α β hα hab y) f lemma continuous_sourceData (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (f : C₀(V,ℂ)) : Continuous (sourceData a ha α β hα hab f) := (C0Multiplier.continuous_mul f).comp (continuous_sourceMultiplier a ha α β hα hab) @[simp] lemma sourceData_apply (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (f : C₀(V,ℂ)) (y x : V) : sourceData a ha α β hα hab f y x = kernel (a x) y * f x := rfl lemma sourceData_norm_le (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (f : C₀(V,ℂ)) (y : V) (B : ℝ) (hB : ∀ t ∈ Icc α β, ‖kernel t y‖ ≤ B) : ‖sourceData a ha α β hα hab f y‖ ≤ B * ‖f‖ := by have hBn : 0 ≤ B := (norm_nonneg (kernel (a 0) y)).trans (hB (a 0) (hab 0)) apply (C0Convolution.norm_le_iff _ (mul_nonneg hBn (norm_nonneg f))).mpr intro x rw [sourceData_apply, norm_mul] exact mul_le_mul (hB (a x) (hab x)) (apply_norm_le f x) (norm_nonneg _) hBn lemma integrable_translated_sourceData (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (r : ℝ) (f : C₀(V,ℂ)) : Integrable (fun y : V => C0Convolution.translate (r • y) (sourceData a ha α β hα hab f y)) := by obtain ⟨B,hB,hbound⟩ := kernel_majorant (V := V) α β hα apply (hB.mul_const ‖f‖).mono' · exact (continuous_translate_comp (fun y : V => r • y) (sourceData a ha α β hα hab f) (by fun_prop) (continuous_sourceData a ha α β hα hab f)).aestronglyMeasurable · filter_upwards with y rw [norm_translate] exact sourceData_norm_le a ha α β hα hab f y (B y) (fun t ht => hbound t ht y) lemma frozen_eq_integral (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (r : ℝ) (f : C₀(V,ℂ)) : frozen a ha α β hα hab r f = ∫ y : V, C0Convolution.translate (r • y) (sourceData a ha α β hα hab f y) := by ext x change value a r f x = (eval x) (∫ y : V, C0Convolution.translate (r • y) (sourceData a ha α β hα hab f y)) rw [← (eval x).integral_comp_comm (integrable_translated_sourceData a ha α β hα hab r f)] rfl lemma continuous_frozen (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (f : C₀(V,ℂ)) : Continuous (fun r : ℝ => frozen a ha α β hα hab r f) := by simp_rw [frozen_eq_integral] obtain ⟨B,hB,hbound⟩ := kernel_majorant (V := V) α β hα apply continuous_of_dominated (bound := fun y => B y * ‖f‖) · intro r exact (integrable_translated_sourceData a ha α β hα hab r f).aestronglyMeasurable · intro r filter_upwards with y rw [norm_translate] exact sourceData_norm_le a ha α β hα hab f y (B y) (fun t ht => hbound t ht y) · exact hB.mul_const ‖f‖ · filter_upwards with y exact (continuous_translate (sourceData a ha α β hα hab f y)).comp (continuous_id.smul (continuous_const : Continuous (fun _ : ℝ => y))) @[simp] lemma frozen_zero (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (f : C₀(V,ℂ)) : frozen a ha α β hα hab 0 f = f := by ext x change (∫ y : V, kernel (a (x-0 • y)) y * f (x-0 • y)) = f x simp only [zero_smul, sub_zero, integral_mul_const] rw [← heatKernel_eq_kernel (a x) (hα.trans_le (hab x).1), integral_heatKernel, one_mul] lemma frozen_tendsto_identity (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (f : C₀(V,ℂ)) : Tendsto (fun r : ℝ => frozen a ha α β hα hab r f) (𝓝 0) (𝓝 f) := by simpa only [frozen_zero] using (continuous_frozen a ha α β hα hab f).tendsto 0 lemma norm_frozen_le (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (r : ℝ) (f : C₀(V,ℂ)) (B : V → ℝ) (hB : Integrable B) (hbound : ∀ t ∈ Icc α β, ∀ y : V, ‖kernel t y‖ ≤ B y) : ‖frozen a ha α β hα hab r f‖ ≤ (∫ y : V, B y) * ‖f‖ := by rw [frozen_eq_integral, ← integral_mul_const] apply norm_integral_le_of_norm_le (hB.mul_const ‖f‖) filter_upwards with y rw [norm_translate] exact sourceData_norm_le a ha α β hα hab f y (B y) (fun t ht => hbound t ht y) lemma frozen_add (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (r : ℝ) (f g : C₀(V,ℂ)) : frozen a ha α β hα hab r (f+g) = frozen a ha α β hα hab r f + frozen a ha α β hα hab r g := by ext x change (∫ y : V, kernel (a (x-r • y)) y * (f (x-r • y)+g (x-r • y))) = (∫ y : V, kernel (a (x-r • y)) y * f (x-r • y)) + (∫ y : V, kernel (a (x-r • y)) y * g (x-r • y)) simp_rw [mul_add] exact integral_add (integrable_value a ha α β hα hab r f x) (integrable_value a ha α β hα hab r g x) lemma frozen_smul (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (r : ℝ) (c : ℂ) (f : C₀(V,ℂ)) : frozen a ha α β hα hab r (c • f) = c • frozen a ha α β hα hab r f := by ext x change (∫ y : V, kernel (a (x-r • y)) y * (c * f (x-r • y))) = c * (∫ y : V, kernel (a (x-r • y)) y * f (x-r • y)) simp_rw [mul_left_comm (kernel _ _) c, integral_const_mul] lemma exists_uniform_norm_bound (α β : ℝ) (hα : 0 < α) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (a : V → ℝ) (ha : Continuous a) (hab : ∀ x, a x ∈ Icc α β) (r : ℝ) (f : C₀(V,ℂ)), ‖frozen a ha α β hα hab r f‖ ≤ C * ‖f‖ := by obtain ⟨B,hB,hbound⟩ := kernel_majorant (V := V) α β hα refine ⟨∫ y : V, |B y|, integral_nonneg (fun y => abs_nonneg _), ?_⟩ intro a ha hab r f exact norm_frozen_le a ha α β hα hab r f (fun y => |B y|) hB.abs (fun t ht y => (hbound t ht y).trans (le_abs_self _)) lemma frozen_apply_unscaled (a : V → ℝ) (ha : Continuous a) (α β : ℝ) (hα : 0 < α) (hab : ∀ x, a x ∈ Icc α β) (r : ℝ) (hr : 0 < r) (f : C₀(V,ℂ)) (x : V) : frozen a ha α β hα hab r f x = ∫ y : V, kernel (r^4*a y) (x-y) * f y := by have hscale (t : ℝ) (ht : 0 < t) (w : V) : kernel (r^4*t) w = (r^Module.finrank ℝ V)⁻¹ • kernel t (r⁻¹ • w) := by rw [← heatKernel_eq_kernel (r^4*t) (by positivity), ← heatKernel_eq_kernel t ht] exact heatKernel_scaling t ht r hr w rw [← integral_sub_left_eq_self (fun y : V => kernel (r^4*a y) (x-y) * f y) volume x] simp only [sub_sub_self] simp_rw [hscale _ (hα.trans_le (hab _).1), smul_mul_assoc] rw [integral_smul] let F := fun y : V => kernel (a (x-r • y)) y * f (x-r • y) have he : (fun y : V => kernel (a (x-y)) (r⁻¹ • y) * f (x-y)) = fun y : V => F (r⁻¹ • y) := by funext y dsimp [F] rw [smul_inv_smul₀ hr.ne'] rw [he, Measure.integral_comp_inv_smul_of_nonneg volume F hr.le, inv_smul_smul₀ (pow_ne_zero _ hr.ne')] rfl end FrozenParametrix open Set Filter MeasureTheory open scoped SchwartzMap Topology namespace SchwartzConvolution variable {V E : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] [MeasureSpace V] [BorelSpace V] [Measure.IsAddHaarMeasure (volume : Measure V)] [NormedAddCommGroup E] [NormedSpace ℝ E] lemma scaled_weighted_majorant (g : SchwartzMap V E) (a b : ℝ) (ha : 0 < a) (K L : ℕ) : ∃ B : V → ℝ, Integrable B ∧ Integrable (fun x => (1+‖x‖)^L * B x) ∧ (∀ x, 0 ≤ B x) ∧ ∀ c ∈ Icc a b, ∀ x, ‖(c⁻¹)^K • g (c⁻¹ • x)‖ ≤ B x := by let N := Module.finrank ℝ V + 1 + L let M := max b 1 let C := 2^N * (Finset.Iic (N,0)).sup (fun i => SchwartzMap.seminorm ℝ i.1 i.2) g have hC : 0 ≤ C := by positivity have hM : 0 < M := lt_of_lt_of_le zero_lt_one (le_max_right _ _) have hdec (z : V) : (1+‖z‖)^N * ‖g z‖ ≤ C := by simpa only [norm_iteratedFDeriv_zero] using SchwartzMap.one_add_le_sup_seminorm_apply (𝕜 := ℝ) (m := (N,0)) le_rfl le_rfl g z refine ⟨fun x => ((a⁻¹)^K*(M^N*C))/(1+‖x‖)^N, ?_, ?_, ?_, ?_⟩ · have hi : Integrable (fun x : V => (1+‖x‖)^(-(N : ℝ))) := integrable_one_add_norm (by dsimp [N]; push_cast; linarith) simpa only [Real.rpow_neg (by positivity : (0:ℝ) ≤ 1+‖(_ : V)‖), Real.rpow_natCast, div_eq_mul_inv] using hi.const_mul ((a⁻¹)^K*(M^N*C)) · have hi : Integrable (fun x : V => (1+‖x‖)^(-(Module.finrank ℝ V + 1 : ℝ))) := integrable_one_add_norm (by linarith) have he (x : V) : (1+‖x‖)^L * (((a⁻¹)^K*(M^N*C))/(1+‖x‖)^N) = ((a⁻¹)^K*(M^N*C)) * (1+‖x‖)^(-(Module.finrank ℝ V + 1 : ℝ)) := by rw [Real.rpow_neg (by positivity), ← Nat.cast_one, ← Nat.cast_add, Real.rpow_natCast] dsimp [N] rw [pow_add] field_simp rw [pow_add] ring simp_rw [he] exact hi.const_mul _ · intro x positivity · intro c hc x have hcpos : 0 < c := ha.trans_le hc.1 have hcM : c ≤ M := hc.2.trans (le_max_left _ _) have hnx : ‖x‖ = c * ‖c⁻¹ • x‖ := by rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hcpos), ← mul_assoc, mul_inv_cancel₀ hcpos.ne', one_mul] have hs : 1+‖x‖ ≤ M*(1+‖c⁻¹ • x‖) := by rw [hnx, mul_add, mul_one] exact add_le_add (le_max_right b 1) (mul_le_mul_of_nonneg_right hcM (norm_nonneg _)) have hg : ‖g (c⁻¹ • x)‖ * (1+‖x‖)^N ≤ M^N*C := by calc _ ≤ ‖g (c⁻¹ • x)‖ * (M * (1+‖c⁻¹ • x‖))^N := by gcongr _ = M^N * ((1+‖c⁻¹ • x‖)^N * ‖g (c⁻¹ • x)‖) := by rw [mul_pow]; ring _ ≤ _ := mul_le_mul_of_nonneg_left (hdec _) (by positivity) rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (by positivity)] apply (le_div_iff₀ (by positivity)).mpr calc _ = (c⁻¹)^K * (‖g (c⁻¹ • x)‖ * (1+‖x‖)^N) := by ring _ ≤ (a⁻¹)^K * (‖g (c⁻¹ • x)‖ * (1+‖x‖)^N) := by gcongr exact hc.1 _ ≤ _ := mul_le_mul_of_nonneg_left hg (by positivity) end SchwartzConvolution namespace BiharmonicKernel open scoped SchwartzMap Laplacian ContDiff Topology variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] def bilapKernel (t : ℝ) (x : V) : ℂ := Δ (Δ (kernel (V := V) t)) x lemma bilapKernel_eq_schwartz (t : ℝ) (ht : 0 < t) (x : V) : bilapKernel t x = (Δ (Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : SchwartzMap V ℂ) x := by have he : ((Δ (heatKernel (V := V) t ht) : SchwartzMap V ℂ) : V → ℂ) = Δ (kernel (V := V) t) := by funext y rw [SchwartzMap.laplacian_apply, heatKernel_eq_kernel t ht] rw [SchwartzMap.laplacian_apply, he] rfl lemma bilapKernel_eq_scaled (t : ℝ) (ht : 0 < t) (x : V) : bilapKernel t x = ((Real.sqrt (Real.sqrt t))⁻¹)^(Module.finrank ℝ V+4) • (Δ (Δ (heatKernel (V := V) 1 zero_lt_one) : SchwartzMap V ℂ) : SchwartzMap V ℂ) ((Real.sqrt (Real.sqrt t))⁻¹ • x) := by let r := Real.sqrt (Real.sqrt t) have hr : 0 < r := Real.sqrt_pos.mpr (Real.sqrt_pos.mpr ht) have he : r^4 = t := by dsimp [r] calc _ = ((Real.sqrt (Real.sqrt t))^2)^2 := by ring _ = t := by rw [Real.sq_sqrt (Real.sqrt_nonneg t), Real.sq_sqrt ht.le] have hs := congrFun (biharmonic_kernel_scaling (V := V) 1 zero_lt_one r hr) x simp only [mul_one, he] at hs rw [bilapKernel_eq_schwartz t ht] exact hs lemma bilapKernel_contDiffAt (q : ℝ × V) (hq : 0 < q.1) : ContDiffAt ℝ ∞ (fun p : ℝ × V => bilapKernel p.1 p.2) q := by have hr : ContDiffAt ℝ ∞ (fun p : ℝ × V => Real.sqrt (Real.sqrt p.1)) q := (contDiffAt_fst.sqrt (ne_of_gt hq)).sqrt (ne_of_gt (Real.sqrt_pos.mpr hq)) have hpos : 0 < Real.sqrt (Real.sqrt q.1) := Real.sqrt_pos.mpr (Real.sqrt_pos.mpr hq) let g : SchwartzMap V ℂ := Δ (Δ (heatKernel (V := V) 1 zero_lt_one) : SchwartzMap V ℂ) have hs : ContDiffAt ℝ ∞ (fun p : ℝ × V => ((Real.sqrt (Real.sqrt p.1))⁻¹)^(Module.finrank ℝ V+4) • g ((Real.sqrt (Real.sqrt p.1))⁻¹ • p.2)) q := ((hr.inv hpos.ne').pow _).smul ((g.smooth ⊤).contDiffAt.comp q ((hr.inv hpos.ne').smul contDiffAt_snd)) apply hs.congr_of_eventuallyEq filter_upwards [continuous_fst.continuousAt.eventually (Ioi_mem_nhds hq)] with p hp exact bilapKernel_eq_scaled _ hp _ lemma continuous_bilapKernel_comp {X : Type*} [TopologicalSpace X] (a : X → ℝ) (v : X → V) (ha : Continuous a) (hv : Continuous v) (hpos : ∀ x, 0 < a x) : Continuous (fun x => bilapKernel (a x) (v x)) := by apply continuous_iff_continuousAt.mpr intro x exact ContinuousAt.comp (f := fun u => (a u,v u)) (bilapKernel_contDiffAt (a x,v x) (hpos x)).continuousAt (ha.continuousAt.prodMk hv.continuousAt) lemma bilapKernel_weighted_majorant (a b : ℝ) (ha : 0 < a) : ∃ B : V → ℝ, Integrable B ∧ Integrable (fun x => (1+‖x‖) * B x) ∧ (∀ x, 0 ≤ B x) ∧ ∀ t ∈ Icc a b, ∀ x : V, ‖bilapKernel t x‖ ≤ B x := by let r := fun t : ℝ => Real.sqrt (Real.sqrt t) obtain ⟨B,hB,hBw,hBn,hbound⟩ := SchwartzConvolution.scaled_weighted_majorant (Δ (Δ (heatKernel (V := V) 1 zero_lt_one) : SchwartzMap V ℂ) : SchwartzMap V ℂ) (r a) (r b) (Real.sqrt_pos.mpr (Real.sqrt_pos.mpr ha)) (Module.finrank ℝ V+4) 1 refine ⟨B,hB,by simpa only [pow_one] using hBw,hBn,fun t ht x => ?_⟩ rw [bilapKernel_eq_scaled t (ha.trans_le ht.1) x] exact hbound (r t) ⟨Real.sqrt_le_sqrt (Real.sqrt_le_sqrt ht.1), Real.sqrt_le_sqrt (Real.sqrt_le_sqrt ht.2)⟩ x end BiharmonicKernel namespace C0Multiplier variable {V : Type*} [TopologicalSpace V] def mulLeftCLM (g : V →ᵇ ℂ) : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ) := LinearMap.mkContinuous { toFun := mul g map_add' := fun f h => by ext x; dsimp [mul]; ring map_smul' := fun c f => by ext x; dsimp [mul]; ring } ‖g‖ (norm_mul_le g) lemma continuous_mul_left (g : V →ᵇ ℂ) : Continuous (mul g) := (mulLeftCLM g).continuous end C0Multiplier namespace FrozenFamily open C0Convolution variable {V P : Type*} [NormedAddCommGroup V] [TopologicalSpace P] [CompactSpace P] def family (g : C(P × V,ℂ)) (y : V) : C(P,ℂ) := ⟨fun t => g (t,y), g.continuous.comp (by fun_prop)⟩ omit [CompactSpace P] in lemma continuous_family (g : C(P × V,ℂ)) : Continuous (family g) := by apply ContinuousMap.continuous_of_continuous_uncurry exact g.continuous.comp (by fun_prop) def multiplier (g : C(P × V,ℂ)) (a : C(V,P)) (y : V) : V →ᵇ ℂ := (BoundedContinuousFunction.mkOfCompact (family g y)).compContinuous a lemma continuous_multiplier (g : C(P × V,ℂ)) (a : C(V,P)) : Continuous (multiplier g a) := (BoundedContinuousFunction.continuous_compContinuous a).comp ((ContinuousMap.isometryEquivBoundedOfCompact P ℂ).continuous.comp (continuous_family g)) def data (g : C(P × V,ℂ)) (a : C(V,P)) (f : C₀(V,ℂ)) (y : V) : C₀(V,ℂ) := C0Multiplier.mul (multiplier g a y) f @[simp] lemma data_apply (g : C(P × V,ℂ)) (a : C(V,P)) (f : C₀(V,ℂ)) (y x : V) : data g a f y x = g (a x,y) * f x := rfl lemma continuous_data (g : C(P × V,ℂ)) (a : C(V,P)) (f : C₀(V,ℂ)) : Continuous (data g a f) := (C0Multiplier.continuous_mul f).comp (continuous_multiplier g a) lemma norm_data_le (g : C(P × V,ℂ)) (a : C(V,P)) (f : C₀(V,ℂ)) (y : V) (B : ℝ) (hB : 0 ≤ B) (hbound : ∀ p, ‖g (p,y)‖ ≤ B) : ‖data g a f y‖ ≤ B * ‖f‖ := by apply (norm_le_iff _ (mul_nonneg hB (norm_nonneg f))).mpr intro x rw [data_apply, norm_mul] exact mul_le_mul (hbound _) (apply_norm_le f x) (norm_nonneg _) hB variable [NormedSpace ℝ V] def residualData (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (f : C₀(V,ℂ)) (y : V) : C₀(V,ℂ) := C0Multiplier.mul c₁ (C0Convolution.translate (r • y) (data g a f y)) - C0Convolution.translate (r • y) (data g a (C0Multiplier.mul c₀ f) y) lemma residualData_apply (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (f : C₀(V,ℂ)) (y x : V) : residualData g a c₀ c₁ r f y x = (c₁ x - c₀ (x-r • y)) * g (a (x-r • y),y) * f (x-r • y) := by change c₁ x * (g (a (x-r • y),y) * f (x-r • y)) - g (a (x-r • y),y) * (c₀ (x-r • y) * f (x-r • y)) = _ ring lemma continuous_residualData (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (f : C₀(V,ℂ)) : Continuous (residualData g a c₀ c₁ r f) := ((C0Multiplier.continuous_mul_left c₁).comp (continuous_translate_comp (fun y : V => r • y) (data g a f) (by fun_prop) (continuous_data g a f))).sub (continuous_translate_comp (fun y : V => r • y) (data g a (C0Multiplier.mul c₀ f)) (by fun_prop) (continuous_data g a (C0Multiplier.mul c₀ f))) lemma norm_residualData_le (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (hr : 0 ≤ r) (f : C₀(V,ℂ)) (y : V) (B L δ : ℝ) (hB : 0 ≤ B) (hL : 0 ≤ L) (hδ : 0 ≤ δ) (hbound : ∀ p, ‖g (p,y)‖ ≤ B) (hLip : ∀ x z, ‖c₁ x-c₀ z‖ ≤ L*(‖x-z‖+δ)) : ‖residualData g a c₀ c₁ r f y‖ ≤ L*(r*‖y‖+δ)*B*‖f‖ := by apply (norm_le_iff _ (by positivity)).mpr intro x rw [residualData_apply, norm_mul, norm_mul] have hc : ‖c₁ x-c₀ (x-r • y)‖ ≤ L*(r*‖y‖+δ) := by simpa only [sub_sub_cancel, norm_smul, Real.norm_eq_abs, abs_of_nonneg hr] using hLip x (x-r • y) exact mul_le_mul (mul_le_mul hc (hbound _) (norm_nonneg _) (by positivity)) (apply_norm_le f _) (norm_nonneg _) (by positivity) variable [MeasureSpace V] [BorelSpace V] [SecondCountableTopology V] lemma integrable_residualData (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (f : C₀(V,ℂ)) (B : V → ℝ) (hB : Integrable B) (hBn : ∀ y, 0 ≤ B y) (hbound : ∀ p y, ‖g (p,y)‖ ≤ B y) : Integrable (residualData g a c₀ c₁ r f) := by apply (hB.const_mul ((‖c₁‖+‖c₀‖)*‖f‖)).mono' · exact (continuous_residualData g a c₀ c₁ r f).aestronglyMeasurable · filter_upwards with y have hBy := hBn y apply (norm_le_iff _ (by positivity)).mpr intro x rw [residualData_apply, norm_mul, norm_mul] have hc : ‖c₁ x-c₀ (x-r • y)‖ ≤ ‖c₁‖+‖c₀‖ := (norm_sub_le _ _).trans (add_le_add (c₁.norm_coe_le_norm _) (c₀.norm_coe_le_norm _)) calc _ ≤ (‖c₁‖+‖c₀‖)*B y*‖f‖ := mul_le_mul (mul_le_mul hc (hbound _ y) (norm_nonneg _) (by positivity)) (apply_norm_le f _) (norm_nonneg _) (by positivity) _ = _ := by ring def residual (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (f : C₀(V,ℂ)) : C₀(V,ℂ) := ∫ y : V, residualData g a c₀ c₁ r f y lemma residual_apply (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (f : C₀(V,ℂ)) (B : V → ℝ) (hB : Integrable B) (hBn : ∀ y, 0 ≤ B y) (hbound : ∀ p y, ‖g (p,y)‖ ≤ B y) (x : V) : residual g a c₀ c₁ r f x = ∫ y : V, (c₁ x-c₀ (x-r • y))*g (a (x-r • y),y)*f (x-r • y) := by change (eval x) (∫ y : V, residualData g a c₀ c₁ r f y) = _ rw [← (eval x).integral_comp_comm (integrable_residualData g a c₀ c₁ r f B hB hBn hbound)] simp only [eval_apply, residualData_apply] omit [BorelSpace V] [SecondCountableTopology V] in lemma norm_residual_le (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (hr : 0 ≤ r) (f : C₀(V,ℂ)) (B : V → ℝ) (hB : Integrable B) (hBw : Integrable (fun y => ‖y‖*B y)) (hBn : ∀ y, 0 ≤ B y) (hbound : ∀ p y, ‖g (p,y)‖ ≤ B y) (L δ : ℝ) (hL : 0 ≤ L) (hδ : 0 ≤ δ) (hLip : ∀ x z, ‖c₁ x-c₀ z‖ ≤ L*(‖x-z‖+δ)) : ‖residual g a c₀ c₁ r f‖ ≤ L*(r*(∫ y : V, ‖y‖*B y)+δ*(∫ y : V, B y))*‖f‖ := by have hi : Integrable (fun y : V => L*(r*‖y‖+δ)*B y*‖f‖) := by convert ((hBw.const_mul r).add (hB.const_mul δ)).const_mul L |>.mul_const ‖f‖ using 1 ext y simp only [Pi.add_apply] ring calc _ ≤ ∫ y : V, L*(r*‖y‖+δ)*B y*‖f‖ := by apply norm_integral_le_of_norm_le hi filter_upwards with y exact norm_residualData_le g a c₀ c₁ r hr f y (B y) L δ (hBn y) hL hδ (fun p => hbound p y) hLip _ = _ := by have he (y : V) : L*(r*‖y‖+δ)*B y*‖f‖ = L*(r*(‖y‖*B y)+δ*B y)*‖f‖ := by ring simp_rw [he] rw [integral_mul_const, integral_const_mul, integral_add (hBw.const_mul r) (hB.const_mul δ), integral_const_mul, integral_const_mul] end FrozenFamily namespace BiharmonicKernel open scoped SchwartzMap Laplacian variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] lemma bilapKernel_scaling (t : ℝ) (ht : 0 < t) (r : ℝ) (hr : 0 < r) (x : V) : bilapKernel (r^4*t) x = (r⁻¹)^(Module.finrank ℝ V+4) • bilapKernel t (r⁻¹ • x) := by rw [bilapKernel_eq_schwartz _ (by positivity), bilapKernel_eq_schwartz _ ht] exact congrFun (biharmonic_kernel_scaling t ht r hr) x end BiharmonicKernel namespace FrozenParametrix open scoped ZeroAtInfty BoundedContinuousFunction open BiharmonicKernel C0Convolution variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] def bilapFamily (α β : ℝ) (hα : 0 < α) : C(Icc α β × V,ℂ) := ⟨fun p => bilapKernel p.1.val p.2, continuous_bilapKernel_comp (fun p : Icc α β × V => p.1.val) Prod.snd (continuous_subtype_val.comp continuous_fst) continuous_snd (fun p => hα.trans_le p.1.property.1)⟩ def coefficient {α β : ℝ} (a : C(V,Icc α β)) : V →ᵇ ℂ := (BoundedContinuousFunction.mkOfCompact (⟨fun t : Icc α β => (t.val : ℂ), by fun_prop⟩ : C(Icc α β,ℂ))).compContinuous a omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in @[simp] lemma coefficient_apply {α β : ℝ} (a : C(V,Icc α β)) (x : V) : coefficient a x = ((a x).val : ℂ) := rfl def principalResidual {α β : ℝ} (hα : 0 < α) (a₀ a₁ : C(V,Icc α β)) (r : ℝ) (f : C₀(V,ℂ)) : C₀(V,ℂ) := (r⁻¹)^4 • FrozenFamily.residual (bilapFamily α β hα) a₀ (coefficient a₀) (coefficient a₁) r f lemma principalResidual_apply_scaled {α β : ℝ} (hα : 0 < α) (a₀ a₁ : C(V,Icc α β)) (r : ℝ) (f : C₀(V,ℂ)) (x : V) : principalResidual hα a₀ a₁ r f x = (r⁻¹)^4 • ∫ y : V, (((a₁ x).val-(a₀ (x-r • y)).val : ℝ) : ℂ) * bilapKernel (a₀ (x-r • y)).val y * f (x-r • y) := by obtain ⟨B,hB,_,hBn,hbound⟩ := bilapKernel_weighted_majorant (V := V) α β hα change (r⁻¹)^4 • FrozenFamily.residual (bilapFamily α β hα) a₀ (coefficient a₀) (coefficient a₁) r f x = _ rw [FrozenFamily.residual_apply _ _ _ _ _ _ B hB hBn (fun p y => hbound p.val p.property y)] simp only [coefficient_apply, Complex.ofReal_sub] rfl lemma principalResidual_apply_unscaled {α β : ℝ} (hα : 0 < α) (a₀ a₁ : C(V,Icc α β)) (r : ℝ) (hr : 0 < r) (f : C₀(V,ℂ)) (x : V) : principalResidual hα a₀ a₁ r f x = ∫ y : V, (((a₁ x).val-(a₀ y).val : ℝ) : ℂ) * bilapKernel (r^4*(a₀ y).val) (x-y) * f y := by rw [principalResidual_apply_scaled] rw [← integral_sub_left_eq_self (fun y : V => (((a₁ x).val-(a₀ y).val : ℝ) : ℂ) * bilapKernel (r^4*(a₀ y).val) (x-y) * f y) volume x] simp only [sub_sub_self] simp_rw [bilapKernel_scaling _ (hα.trans_le (a₀ _).property.1) r hr, mul_smul_comm, smul_mul_assoc] rw [integral_smul] let F := fun y : V => (((a₁ x).val-(a₀ (x-r • y)).val : ℝ) : ℂ) * bilapKernel (a₀ (x-r • y)).val y * f (x-r • y) have he : (fun y : V => (((a₁ x).val-(a₀ (x-y)).val : ℝ) : ℂ) * bilapKernel (a₀ (x-y)).val (r⁻¹ • y) * f (x-y)) = fun y : V => F (r⁻¹ • y) := by funext y dsimp [F] rw [smul_inv_smul₀ hr.ne'] rw [he, Measure.integral_comp_inv_smul_of_nonneg volume F hr.le, smul_smul] congr 1 rw [pow_add, mul_right_comm, ← mul_pow, inv_mul_cancel₀ hr.ne', one_pow, one_mul] lemma norm_principalResidual_le {α β : ℝ} (hα : 0 < α) (a₀ a₁ : C(V,Icc α β)) (r : ℝ) (hr : 0 ≤ r) (f : C₀(V,ℂ)) (B : V → ℝ) (hB : Integrable B) (hBw : Integrable (fun y => ‖y‖*B y)) (hBn : ∀ y, 0 ≤ B y) (hbound : ∀ t ∈ Icc α β, ∀ y : V, ‖bilapKernel t y‖ ≤ B y) (L : ℝ) (hL : 0 ≤ L) (hLip : ∀ x z, |(a₁ x).val-(a₀ z).val| ≤ L*(‖x-z‖+r^4)) : ‖principalResidual hα a₀ a₁ r f‖ ≤ (r⁻¹)^4 * (L*(r*(∫ y : V, ‖y‖*B y)+r^4*(∫ y : V, B y))*‖f‖) := by rw [principalResidual, norm_smul, Real.norm_eq_abs, abs_of_nonneg (by positivity)] apply mul_le_mul_of_nonneg_left _ (by positivity) apply FrozenFamily.norm_residual_le _ _ _ _ r hr f B hB hBw hBn (fun p y => hbound p.val p.property y) L (r^4) hL (by positivity) intro x z simpa only [coefficient_apply, ← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs] using hLip x z lemma exists_uniform_principalResidual_bound (α β : ℝ) (hα : 0 < α) (L : ℝ) (hL : 0 ≤ L) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (a₀ a₁ : C(V,Icc α β)) (r : ℝ), 0 < r → r ≤ 1 → (∀ x z, |(a₁ x).val-(a₀ z).val| ≤ L*(‖x-z‖+r^4)) → ∀ f : C₀(V,ℂ), ‖principalResidual hα a₀ a₁ r f‖ ≤ C*(r⁻¹)^3*‖f‖ := by obtain ⟨B,hB,hBw,hBn,hbound⟩ := bilapKernel_weighted_majorant (V := V) α β hα have hBw' : Integrable (fun y : V => ‖y‖*B y) := by have hi : Integrable (fun y : V => (1+‖y‖)*B y-B y) := hBw.sub hB have he : (fun y : V => (1+‖y‖)*B y-B y) = (fun y : V => ‖y‖*B y) := by funext y ring rwa [he] at hi let M₀ := ∫ y : V, B y let M₁ := ∫ y : V, ‖y‖*B y have hM₀ : 0 ≤ M₀ := integral_nonneg hBn have hM₁ : 0 ≤ M₁ := integral_nonneg (fun y => mul_nonneg (norm_nonneg _) (hBn y)) refine ⟨L*(M₁+M₀),by positivity,fun a₀ a₁ r hr hr1 hLip f => ?_⟩ apply (norm_principalResidual_le hα a₀ a₁ r hr.le f B hB hBw' hBn hbound L hL hLip).trans have hrinv : 1 ≤ (r⁻¹)^3 := one_le_pow₀ (by exact (one_le_inv₀ hr).mpr hr1) have he : (r⁻¹)^4 * (L*(r*M₁+r^4*M₀)*‖f‖) = L*((r⁻¹)^3*M₁+M₀)*‖f‖ := by field_simp change (r⁻¹)^4 * (L*(r*M₁+r^4*M₀)*‖f‖) ≤ _ rw [he] calc _ ≤ L*((r⁻¹)^3*M₁+(r⁻¹)^3*M₀)*‖f‖ := by gcongr; exact le_mul_of_one_le_left hM₀ hrinv _ = _ := by ring end FrozenParametrix open Set Filter MeasureTheory open scoped Topology ZeroAtInfty BoundedContinuousFunction namespace CompactComposition variable {V P F : Type*} [TopologicalSpace V] [PseudoMetricSpace P] [CompactSpace P] [PseudoMetricSpace F] def compose (g : C(P,F)) (a : V →ᵇ P) : V →ᵇ F := (BoundedContinuousFunction.mkOfCompact g).compContinuous a.toContinuousMap @[simp] lemma compose_apply (g : C(P,F)) (a : V →ᵇ P) (x : V) : compose g a x = g (a x) := rfl lemma continuous_compose : Continuous (fun p : C(P,F) × (V →ᵇ P) => compose p.1 p.2) := by apply continuous_iff_continuousAt.mpr rintro ⟨g₀,a₀⟩ apply Metric.continuousAt_iff.mpr intro ε hε obtain ⟨δ,hδ,hg⟩ := Metric.uniformContinuous_iff.mp (CompactSpace.uniformContinuous_of_continuous g₀.continuous) (ε/4) (by positivity) refine ⟨min (ε/4) δ, lt_min (by positivity) hδ, ?_⟩ rintro ⟨g,a⟩ hp change max (dist g g₀) (dist a a₀) < min (ε/4) δ at hp have hgdist : dist g g₀ < ε/4 := (le_max_left _ _).trans_lt (lt_of_lt_of_le hp (min_le_left _ _)) have hadist : dist a a₀ < δ := (le_max_right _ _).trans_lt (lt_of_lt_of_le hp (min_le_right _ _)) apply lt_of_le_of_lt ((BoundedContinuousFunction.dist_le (show (0 : ℝ) ≤ ε/2 by positivity)).mpr ?_) (by linarith) intro x calc dist (g (a x)) (g₀ (a₀ x)) ≤ dist (g (a x)) (g₀ (a x)) + dist (g₀ (a x)) (g₀ (a₀ x)) := dist_triangle _ _ _ _ ≤ ε/2 := by have h₁ := (ContinuousMap.dist_apply_le_dist (f := g) (g := g₀) (a x)).trans_lt hgdist have h₂ := hg ((BoundedContinuousFunction.dist_coe_le_dist (f := a) (g := a₀) x).trans_lt hadist) linarith end CompactComposition namespace C0Multiplier variable {V : Type*} [TopologicalSpace V] lemma continuous_mul_joint : Continuous (fun p : (V →ᵇ ℂ) × C₀(V,ℂ) => mul p.1 p.2) := by apply ZeroAtInftyContinuousMap.isometry_toBCF.isUniformInducing.isInducing.continuous_iff.mpr change Continuous (fun p : (V →ᵇ ℂ) × C₀(V,ℂ) => p.1*p.2.toBCF) exact continuous_fst.mul (ZeroAtInftyContinuousMap.isometry_toBCF.continuous.comp continuous_snd) end C0Multiplier namespace FrozenFamily open C0Convolution variable {V P : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] [PseudoMetricSpace P] [CompactSpace P] omit [NormedSpace ℝ V] in lemma continuous_multiplier_comp {X : Type*} [TopologicalSpace X] (g : C(P × V,ℂ)) (a : X → V →ᵇ P) (y : X → V) (ha : Continuous a) (hy : Continuous y) : Continuous (fun x => multiplier g (a x).toContinuousMap (y x)) := by exact CompactComposition.continuous_compose.comp (((continuous_family g).comp hy).prodMk ha) omit [NormedSpace ℝ V] in lemma continuous_data_comp {X : Type*} [TopologicalSpace X] (g : C(P × V,ℂ)) (a : X → V →ᵇ P) (f : X → C₀(V,ℂ)) (y : X → V) (ha : Continuous a) (hf : Continuous f) (hy : Continuous y) : Continuous (fun x => data g (a x).toContinuousMap (f x) (y x)) := C0Multiplier.continuous_mul_joint.comp ((continuous_multiplier_comp g a y ha hy).prodMk hf) lemma continuous_residualData_comp {X : Type*} [TopologicalSpace X] (g : C(P × V,ℂ)) (a : X → V →ᵇ P) (c₀ c₁ : X → V →ᵇ ℂ) (r : X → ℝ) (f : X → C₀(V,ℂ)) (y : X → V) (ha : Continuous a) (h₀ : Continuous c₀) (h₁ : Continuous c₁) (hr : Continuous r) (hf : Continuous f) (hy : Continuous y) : Continuous (fun x => residualData g (a x).toContinuousMap (c₀ x) (c₁ x) (r x) (f x) (y x)) := by apply Continuous.sub · exact C0Multiplier.continuous_mul_joint.comp (h₁.prodMk (continuous_translate_comp _ _ (hr.smul hy) (continuous_data_comp g a f y ha hf hy))) · exact continuous_translate_comp _ _ (hr.smul hy) (continuous_data_comp g a (fun x => C0Multiplier.mul (c₀ x) (f x)) y ha (C0Multiplier.continuous_mul_joint.comp (h₀.prodMk hf)) hy) variable [MeasureSpace V] [BorelSpace V] [SecondCountableTopology V] omit [MeasureSpace V] [BorelSpace V] [SecondCountableTopology V] in lemma norm_residualData_le_uniform (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (f : C₀(V,ℂ)) (y : V) (B M : ℝ) (hB : 0 ≤ B) (hM : 0 ≤ M) (hbound : ∀ p, ‖g (p,y)‖ ≤ B) (hc : ‖c₀‖+‖c₁‖ ≤ M) : ‖residualData g a c₀ c₁ r f y‖ ≤ M*‖f‖*B := by apply (norm_le_iff _ (by positivity)).mpr intro x rw [residualData_apply, norm_mul, norm_mul] have hn : ‖c₁ x-c₀ (x-r • y)‖ ≤ M := (norm_sub_le _ _).trans <| (add_le_add (c₁.norm_coe_le_norm _) (c₀.norm_coe_le_norm _)).trans (by linarith) calc _ ≤ M*B*‖f‖ := mul_le_mul (mul_le_mul hn (hbound _) (norm_nonneg _) hM) (apply_norm_le f _) (norm_nonneg _) (by positivity) _ = _ := by ring lemma continuous_residual_comp {X : Type*} [TopologicalSpace X] [FirstCountableTopology X] (g : C(P × V,ℂ)) (a : X → V →ᵇ P) (c₀ c₁ : X → V →ᵇ ℂ) (r : X → ℝ) (f : C₀(V,ℂ)) (ha : Continuous a) (h₀ : Continuous c₀) (h₁ : Continuous c₁) (hr : Continuous r) (B : V → ℝ) (hB : Integrable B) (hBn : ∀ y, 0 ≤ B y) (hbound : ∀ p y, ‖g (p,y)‖ ≤ B y) (M : ℝ) (hM : 0 ≤ M) (hc : ∀ x, ‖c₀ x‖+‖c₁ x‖ ≤ M) : Continuous (fun x => residual g (a x).toContinuousMap (c₀ x) (c₁ x) (r x) f) := by apply continuous_of_dominated (fun x => (integrable_residualData g (a x).toContinuousMap (c₀ x) (c₁ x) (r x) f B hB hBn hbound).aestronglyMeasurable) (fun x => Eventually.of_forall (fun y => norm_residualData_le_uniform g (a x).toContinuousMap (c₀ x) (c₁ x) (r x) f y (B y) M (hBn y) hM (fun p => hbound p y) (hc x))) (hB.const_mul (M*‖f‖)) exact Eventually.of_forall (fun y => continuous_residualData_comp g a c₀ c₁ r (fun _ => f) (fun _ => y) ha h₀ h₁ hr continuous_const continuous_const) lemma residual_add (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (f h : C₀(V,ℂ)) (B : V → ℝ) (hB : Integrable B) (hBn : ∀ y, 0 ≤ B y) (hbound : ∀ p y, ‖g (p,y)‖ ≤ B y) : residual g a c₀ c₁ r (f+h) = residual g a c₀ c₁ r f + residual g a c₀ c₁ r h := by have he : residualData g a c₀ c₁ r (f+h) = fun y => residualData g a c₀ c₁ r f y + residualData g a c₀ c₁ r h y := by funext y ext x simp only [residualData_apply, ZeroAtInftyContinuousMap.add_apply] ring rw [residual, he, integral_add (integrable_residualData g a c₀ c₁ r f B hB hBn hbound) (integrable_residualData g a c₀ c₁ r h B hB hBn hbound)] rfl omit [BorelSpace V] [SecondCountableTopology V] in lemma residual_smul (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (c : ℂ) (f : C₀(V,ℂ)) : residual g a c₀ c₁ r (c • f) = c • residual g a c₀ c₁ r f := by have he : residualData g a c₀ c₁ r (c • f) = fun y => c • residualData g a c₀ c₁ r f y := by funext y ext x simp only [residualData_apply, ZeroAtInftyContinuousMap.smul_apply, smul_eq_mul] ring rw [residual, he, integral_smul] rfl omit [BorelSpace V] [SecondCountableTopology V] in lemma norm_residual_le_uniform (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (f : C₀(V,ℂ)) (B : V → ℝ) (hB : Integrable B) (hBn : ∀ y, 0 ≤ B y) (hbound : ∀ p y, ‖g (p,y)‖ ≤ B y) : ‖residual g a c₀ c₁ r f‖ ≤ ((‖c₀‖+‖c₁‖)*(∫ y, B y))*‖f‖ := by calc _ ≤ ∫ y, (‖c₀‖+‖c₁‖)*‖f‖*B y := norm_integral_le_of_norm_le (hB.const_mul _) (Eventually.of_forall (fun y => norm_residualData_le_uniform g a c₀ c₁ r f y (B y) (‖c₀‖+‖c₁‖) (hBn y) (by positivity) (fun p => hbound p y) le_rfl)) _ = _ := by rw [integral_const_mul]; ring def residualCLM (g : C(P × V,ℂ)) (a : C(V,P)) (c₀ c₁ : V →ᵇ ℂ) (r : ℝ) (B : V → ℝ) (hB : Integrable B) (hBn : ∀ y, 0 ≤ B y) (hbound : ∀ p y, ‖g (p,y)‖ ≤ B y) : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ) := LinearMap.mkContinuous { toFun := residual g a c₀ c₁ r map_add' := fun f h => residual_add g a c₀ c₁ r f h B hB hBn hbound map_smul' := fun c f => residual_smul g a c₀ c₁ r c f } ((‖c₀‖+‖c₁‖)*(∫ y, B y)) (fun f => norm_residual_le_uniform g a c₀ c₁ r f B hB hBn hbound) end FrozenFamily namespace FrozenParametrix open scoped SchwartzMap ZeroAtInfty BoundedContinuousFunction open BiharmonicKernel C0Convolution variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] def frozenOp {α β : ℝ} (hα : 0 < α) (a : C(V,Icc α β)) (r : ℝ) (f : C₀(V,ℂ)) : C₀(V,ℂ) := frozen (fun x => (a x).val) (continuous_subtype_val.comp a.continuous) α β hα (fun x => (a x).property) r f lemma kernel_majorant_nonneg (α β : ℝ) (hα : 0 < α) : ∃ B : V → ℝ, Integrable B ∧ (∀ y, 0 ≤ B y) ∧ ∀ t ∈ Icc α β, ∀ y : V, ‖kernel t y‖ ≤ B y := by obtain ⟨B,hB,hbound⟩ := kernel_majorant (V := V) α β hα exact ⟨fun y => |B y|, hB.norm, fun y => abs_nonneg _, fun t ht y => (hbound t ht y).trans (le_abs_self _)⟩ def kernelMajorant (α β : ℝ) (hα : 0 < α) : V → ℝ := (kernel_majorant_nonneg (V := V) α β hα).choose lemma kernelMajorant_spec (α β : ℝ) (hα : 0 < α) : Integrable (kernelMajorant (V := V) α β hα) ∧ (∀ y, 0 ≤ kernelMajorant (V := V) α β hα y) ∧ ∀ t ∈ Icc α β, ∀ y : V, ‖kernel t y‖ ≤ kernelMajorant α β hα y := (kernel_majorant_nonneg (V := V) α β hα).choose_spec def frozenCLM {α β : ℝ} (hα : 0 < α) (a : C(V,Icc α β)) (r : ℝ) : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ) := LinearMap.mkContinuous { toFun := frozenOp hα a r map_add' := fun f g => frozen_add (fun x => (a x).val) (continuous_subtype_val.comp a.continuous) α β hα (fun x => (a x).property) r f g map_smul' := fun c f => frozen_smul (fun x => (a x).val) (continuous_subtype_val.comp a.continuous) α β hα (fun x => (a x).property) r c f } (∫ y : V, kernelMajorant α β hα y) (fun f => norm_frozen_le (fun x => (a x).val) (continuous_subtype_val.comp a.continuous) α β hα (fun x => (a x).property) r f _ (kernelMajorant_spec α β hα).1 (kernelMajorant_spec α β hα).2.2) @[simp] lemma frozenCLM_apply {α β : ℝ} (hα : 0 < α) (a : C(V,Icc α β)) (r : ℝ) (f : C₀(V,ℂ)) : frozenCLM hα a r f = frozenOp hα a r f := rfl lemma norm_frozenCLM_le {α β : ℝ} (hα : 0 < α) (a : C(V,Icc α β)) (r : ℝ) : ‖frozenCLM hα a r‖ ≤ ∫ y : V, kernelMajorant α β hα y := by apply ContinuousLinearMap.opNorm_le_bound (frozenCLM hα a r) (integral_nonneg (kernelMajorant_spec α β hα).2.1) intro f exact norm_frozen_le (fun x => (a x).val) (continuous_subtype_val.comp a.continuous) α β hα (fun x => (a x).property) r f _ (kernelMajorant_spec α β hα).1 (kernelMajorant_spec α β hα).2.2 def kernelFamilyCM (α β : ℝ) (hα : 0 < α) : C(Icc α β × V,ℂ) := ⟨fun p => kernel p.1.val p.2, continuous_kernel_comp (fun p : Icc α β × V => p.1.val) Prod.snd (continuous_subtype_val.comp continuous_fst) continuous_snd (fun p => hα.trans_le p.1.property.1)⟩ lemma continuous_frozenOp_comp {X : Type*} [TopologicalSpace X] [FirstCountableTopology X] {α β : ℝ} (hα : 0 < α) (a : X → V →ᵇ Icc α β) (r : X → ℝ) (ha : Continuous a) (hr : Continuous r) (f : C₀(V,ℂ)) : Continuous (fun x => frozenOp hα (a x).toContinuousMap (r x) f) := by have he (x : X) : frozenOp hα (a x).toContinuousMap (r x) f = ∫ y : V, C0Convolution.translate (r x • y) (FrozenFamily.data (kernelFamilyCM α β hα) (a x).toContinuousMap f y) := frozen_eq_integral (fun z => (a x z).val) (continuous_subtype_val.comp (a x).continuous) α β hα (fun z => (a x z).property) (r x) f simp_rw [he] apply continuous_of_dominated (bound := fun y => kernelMajorant α β hα y*‖f‖) · intro x exact (integrable_translated_sourceData (fun z => (a x z).val) (continuous_subtype_val.comp (a x).continuous) α β hα (fun z => (a x z).property) (r x) f).aestronglyMeasurable · intro x filter_upwards with y rw [norm_translate] exact FrozenFamily.norm_data_le (kernelFamilyCM α β hα) (a x).toContinuousMap f y (kernelMajorant α β hα y) ((kernelMajorant_spec α β hα).2.1 y) (fun p => (kernelMajorant_spec α β hα).2.2 p.val p.property y) · exact (kernelMajorant_spec α β hα).1.mul_const ‖f‖ · exact Eventually.of_forall (fun y => continuous_translate_comp _ _ (hr.smul continuous_const) (FrozenFamily.continuous_data_comp (kernelFamilyCM α β hα) a (fun _ => f) (fun _ => y) ha continuous_const continuous_const)) omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in lemma continuous_coefficient_comp {X : Type*} [TopologicalSpace X] {α β : ℝ} (a : X → V →ᵇ Icc α β) (ha : Continuous a) : Continuous (fun x => coefficient (a x).toContinuousMap) := by exact CompactComposition.continuous_compose.comp (continuous_const.prodMk ha) omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in lemma norm_coefficient_le {α β : ℝ} (a : C(V,Icc α β)) : ‖coefficient a‖ ≤ max |α| |β| := by apply (BoundedContinuousFunction.norm_le (le_trans (abs_nonneg α) (le_max_left _ _))).mpr intro x simp only [coefficient_apply, Complex.norm_real, Real.norm_eq_abs] apply abs_le.mpr exact ⟨(neg_le_neg (le_max_left |α| |β|)).trans ((neg_abs_le α).trans (a x).property.1), (a x).property.2.trans ((le_abs_self β).trans (le_max_right _ _))⟩ def bilapMajorant (α β : ℝ) (hα : 0 < α) : V → ℝ := (bilapKernel_weighted_majorant (V := V) α β hα).choose lemma bilapMajorant_spec (α β : ℝ) (hα : 0 < α) : Integrable (bilapMajorant (V := V) α β hα) ∧ Integrable (fun y : V => (1+‖y‖)*bilapMajorant α β hα y) ∧ (∀ y, 0 ≤ bilapMajorant (V := V) α β hα y) ∧ ∀ t ∈ Icc α β, ∀ y : V, ‖bilapKernel t y‖ ≤ bilapMajorant α β hα y := (bilapKernel_weighted_majorant (V := V) α β hα).choose_spec def principalCLM {α β : ℝ} (hα : 0 < α) (a₀ a₁ : C(V,Icc α β)) (r : ℝ) : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ) := (r⁻¹)^4 • FrozenFamily.residualCLM (bilapFamily α β hα) a₀ (coefficient a₀) (coefficient a₁) r (bilapMajorant α β hα) (bilapMajorant_spec α β hα).1 (bilapMajorant_spec α β hα).2.2.1 (fun p y => (bilapMajorant_spec α β hα).2.2.2 p.val p.property y) @[simp] lemma principalCLM_apply {α β : ℝ} (hα : 0 < α) (a₀ a₁ : C(V,Icc α β)) (r : ℝ) (f : C₀(V,ℂ)) : principalCLM hα a₀ a₁ r f = principalResidual hα a₀ a₁ r f := rfl lemma continuous_principalCLM_apply {X : Type*} [TopologicalSpace X] [FirstCountableTopology X] {α β : ℝ} (hα : 0 < α) (a₀ a₁ : X → V →ᵇ Icc α β) (r : X → ℝ) (h₀ : Continuous a₀) (h₁ : Continuous a₁) (hr : Continuous r) (hr0 : ∀ x, r x ≠ 0) (f : C₀(V,ℂ)) : Continuous (fun x => principalCLM hα (a₀ x).toContinuousMap (a₁ x).toContinuousMap (r x) f) := by change Continuous (fun x => (r x)⁻¹^4 • FrozenFamily.residual (bilapFamily α β hα) (a₀ x).toContinuousMap (coefficient (a₀ x).toContinuousMap) (coefficient (a₁ x).toContinuousMap) (r x) f) apply ((hr.inv₀ hr0).pow 4).smul apply FrozenFamily.continuous_residual_comp _ a₀ _ _ r f h₀ (continuous_coefficient_comp a₀ h₀) (continuous_coefficient_comp a₁ h₁) hr (bilapMajorant α β hα) (bilapMajorant_spec α β hα).1 (bilapMajorant_spec α β hα).2.2.1 (fun p y => (bilapMajorant_spec α β hα).2.2.2 p.val p.property y) (2*max |α| |β|) (by positivity) intro x linarith [norm_coefficient_le (a₀ x).toContinuousMap, norm_coefficient_le (a₁ x).toContinuousMap] def fullResidualCLM {α β : ℝ} (hα : 0 < α) (a₀ a₁ : C(V,Icc α β)) (b : V →ᵇ ℂ) (r : ℝ) : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ) := principalCLM hα a₀ a₁ r + (C0Multiplier.mulLeftCLM b).comp (frozenCLM hα a₀ r) lemma fullResidualCLM_apply {α β : ℝ} (hα : 0 < α) (a₀ a₁ : C(V,Icc α β)) (b : V →ᵇ ℂ) (r : ℝ) (f : C₀(V,ℂ)) : fullResidualCLM hα a₀ a₁ b r f = principalResidual hα a₀ a₁ r f + C0Multiplier.mul b (frozenOp hα a₀ r f) := rfl lemma continuous_fullResidualCLM_apply {X : Type*} [TopologicalSpace X] [FirstCountableTopology X] {α β : ℝ} (hα : 0 < α) (a₀ a₁ : X → V →ᵇ Icc α β) (b : X → V →ᵇ ℂ) (r : X → ℝ) (h₀ : Continuous a₀) (h₁ : Continuous a₁) (hb : Continuous b) (hr : Continuous r) (hr0 : ∀ x, r x ≠ 0) (f : C₀(V,ℂ)) : Continuous (fun x => fullResidualCLM hα (a₀ x).toContinuousMap (a₁ x).toContinuousMap (b x) (r x) f) := by simp_rw [fullResidualCLM_apply] exact (continuous_principalCLM_apply hα a₀ a₁ r h₀ h₁ hr hr0 f).add (C0Multiplier.continuous_mul_joint.comp (hb.prodMk (continuous_frozenOp_comp hα a₀ r h₀ hr f))) lemma exists_uniform_fullResidual_bound (α β : ℝ) (hα : 0 < α) (L B : ℝ) (hL : 0 ≤ L) (hB : 0 ≤ B) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (a₀ a₁ : C(V,Icc α β)) (b : V →ᵇ ℂ) (r : ℝ), 0 < r → r ≤ 1 → ‖b‖ ≤ B → (∀ x z, |(a₁ x).val-(a₀ z).val| ≤ L*(‖x-z‖+r^4)) → ‖fullResidualCLM hα a₀ a₁ b r‖ ≤ C*(r⁻¹)^3 := by obtain ⟨C,hC,hbound⟩ := exists_uniform_principalResidual_bound (V := V) α β hα L hL let M := ∫ y : V, kernelMajorant α β hα y have hM : 0 ≤ M := integral_nonneg (kernelMajorant_spec α β hα).2.1 refine ⟨C+B*M,by positivity,fun a₀ a₁ b r hr hr1 hb hLip => ?_⟩ apply ContinuousLinearMap.opNorm_le_bound _ (by positivity) intro f rw [fullResidualCLM_apply] have hrinv : 1 ≤ (r⁻¹)^3 := one_le_pow₀ ((one_le_inv₀ hr).mpr hr1) have hK : ‖frozenOp hα a₀ r f‖ ≤ M*‖f‖ := by exact norm_frozen_le (fun x => (a₀ x).val) (continuous_subtype_val.comp a₀.continuous) α β hα (fun x => (a₀ x).property) r f _ (kernelMajorant_spec α β hα).1 (kernelMajorant_spec α β hα).2.2 calc _ ≤ ‖principalResidual hα a₀ a₁ r f‖+‖C0Multiplier.mul b (frozenOp hα a₀ r f)‖ := norm_add_le _ _ _ ≤ C*(r⁻¹)^3*‖f‖+B*(M*‖f‖) := add_le_add (hbound a₀ a₁ r hr hr1 hLip f) ((C0Multiplier.norm_mul_le b _).trans (mul_le_mul hb hK (norm_nonneg _) hB)) _ ≤ C*(r⁻¹)^3*‖f‖+B*(M*‖f‖)*(r⁻¹)^3 := add_le_add le_rfl (le_mul_of_one_le_right (show 0 ≤ B*(M*‖f‖) by positivity) hrinv) _ = _ := by ring end FrozenParametrix open Set Filter MeasureTheory open scoped Topology BoundedContinuousFunction namespace StrongWeightedVolterra abbrev UnitTime := Ioo (0 : ℝ) 1 abbrev Time (T : ℝ) := Ioc (0 : ℝ) T instance : MeasureSpace UnitTime := Measure.Subtype.measureSpace def weight (u : UnitTime) : ℝ := u.val ^ (-3/4 : ℝ) * (1-u.val)^(-3/4 : ℝ) lemma weight_nonneg (u : UnitTime) : 0 ≤ weight u := by exact mul_nonneg (Real.rpow_nonneg u.property.1.le _) (Real.rpow_nonneg (by linarith [u.property.2]) _) lemma continuous_weight : Continuous weight := by apply Continuous.mul · exact continuous_subtype_val.rpow_const (fun u => Or.inl u.property.1.ne') · exact (continuous_const.sub continuous_subtype_val).rpow_const (fun u => Or.inl (sub_pos.mpr u.property.2).ne') lemma integrable_weight : Integrable weight := by have hb := (Complex.betaIntegral_convergent (u := (1/4 : ℂ)) (v := (1/4 : ℂ)) (by norm_num) (by norm_num)).norm.1 have hi := hb.mono_set Ioo_subset_Ioc_self have he : EqOn (fun x : ℝ => ‖(x : ℂ)^((1/4 : ℂ)-1) * (1-(x : ℂ))^((1/4 : ℂ)-1)‖) (fun x : ℝ => x^(-3/4 : ℝ)*(1-x)^(-3/4 : ℝ)) (Ioo 0 1) := by intro x hx dsimp only rw [norm_mul] have hq : (1/4 : ℂ)-1 = ((-3/4 : ℝ) : ℂ) := by norm_num rw [hq, show (1 : ℂ)-(x : ℂ) = ((1-x : ℝ) : ℂ) by push_cast; rfl, Complex.norm_cpow_real, Complex.norm_cpow_real, Complex.norm_real, Complex.norm_real, Real.norm_eq_abs, Real.norm_eq_abs, abs_of_pos hx.1, abs_of_pos (sub_pos.mpr hx.2)] have hw := hi.congr_fun he measurableSet_Ioo exact (integrableOn_iff_comap_subtypeVal measurableSet_Ioo).mp hw variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] variable {T : ℝ} def lag (t : Time T) (u : UnitTime) : Time T := ⟨t.val*u.val, mul_pos t.property.1 u.property.1, (mul_le_of_le_one_right t.property.1.le u.property.2.le).trans t.property.2⟩ lemma continuous_lag : Continuous (fun p : Time T × UnitTime => lag p.1 p.2) := by exact (continuous_subtype_val.comp continuous_fst |>.mul (continuous_subtype_val.comp continuous_snd)).subtype_mk _ structure Kernel (T : ℝ) (E : Type*) [NormedAddCommGroup E] [NormedSpace ℂ E] where toFun : Time T × UnitTime → E →L[ℂ] E continuous_apply : Continuous (fun p : (Time T × UnitTime) × E => toFun p.1 p.2) instance : CoeFun (Kernel T E) (fun _ => Time T × UnitTime → E →L[ℂ] E) := ⟨Kernel.toFun⟩ lemma continuous_apply_of_strong {X : Type*} [TopologicalSpace X] (A : X → E →L[ℂ] E) (hA : ∀ f, Continuous (fun x => A x f)) (C : ℝ) (_hC : 0 ≤ C) (hbound : ∀ x, ‖A x‖ ≤ C) : Continuous (fun p : X × E => A p.1 p.2) := by apply continuous_iff_continuousAt.mpr rintro ⟨x,f⟩ have hb := ((hA f).comp continuous_fst).tendsto (x,f) apply hb.congr_dist apply squeeze_zero (fun p => dist_nonneg) (fun p => ?_) (show Tendsto (fun p : X × E => C*dist f p.2) (𝓝 (x,f)) (𝓝 0) from by have hd : Continuous (fun p : X × E => C*dist f p.2) := by fun_prop simpa using hd.tendsto (x,f)) dsimp only [Function.comp_apply] rw [dist_eq_norm, ← map_sub, dist_eq_norm] exact (A p.1).le_opNorm _ |>.trans (mul_le_mul_of_nonneg_right (hbound _) (norm_nonneg _)) def Kernel.ofStrong (A : Time T × UnitTime → E →L[ℂ] E) (hA : ∀ f, Continuous (fun p => A p f)) (C : ℝ) (hC : 0 ≤ C) (hbound : ∀ p, ‖A p‖ ≤ C) : Kernel T E := ⟨A, continuous_apply_of_strong A hA C hC hbound⟩ variable (N : Kernel T E) (C : ℝ) (hC : 0 ≤ C) (hN : ∀ p, ‖N p‖ ≤ C) def data (j : Time T →ᵇ E) (t : Time T) (u : UnitTime) : E := weight u • N (t,u) (j (lag t u)) lemma continuous_data (j : Time T →ᵇ E) : Continuous (fun p : Time T × UnitTime => data N j p.1 p.2) := by exact (continuous_weight.comp continuous_snd).smul (N.continuous_apply.comp (continuous_id.prodMk (j.continuous.comp continuous_lag))) include hC hN lemma norm_data_le (j : Time T →ᵇ E) (t : Time T) (u : UnitTime) : ‖data N j t u‖ ≤ weight u * (C*‖j‖) := by rw [data, norm_smul, Real.norm_eq_abs, abs_of_nonneg (weight_nonneg u)] apply mul_le_mul_of_nonneg_left _ (weight_nonneg u) exact (N (t,u)).le_opNorm _ |>.trans (mul_le_mul (hN _) (j.norm_coe_le_norm _) (norm_nonneg _) hC) lemma integrable_data (j : Time T →ᵇ E) (t : Time T) : Integrable (data N j t) := by exact (integrable_weight.mul_const (C*‖j‖)).mono' ((continuous_data N j).comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable (Eventually.of_forall (norm_data_le N C hC hN j t)) lemma continuous_integral_data (j : Time T →ᵇ E) : Continuous (fun t => ∫ u : UnitTime, data N j t u) := by apply continuous_of_dominated (fun t => (integrable_data N C hC hN j t).aestronglyMeasurable) (fun t => Eventually.of_forall (norm_data_le N C hC hN j t)) (integrable_weight.mul_const (C*‖j‖)) exact Eventually.of_forall (fun u => (continuous_data N j).comp (continuous_id.prodMk continuous_const)) lemma norm_integral_data_le (j : Time T →ᵇ E) (t : Time T) : ‖∫ u : UnitTime, data N j t u‖ ≤ (∫ u : UnitTime, weight u)*C*‖j‖ := by calc _ ≤ ∫ u : UnitTime, weight u*(C*‖j‖) := norm_integral_le_of_norm_le (integrable_weight.mul_const _) <| Eventually.of_forall (norm_data_le N C hC hN j t) _ = _ := by rw [integral_mul_const]; ring omit hC hN in def value (j : Time T →ᵇ E) (t : Time T) : E := t.val^(1/4 : ℝ) • ∫ u : UnitTime, data N j t u lemma continuous_value (j : Time T →ᵇ E) : Continuous (value N j) := (continuous_subtype_val.rpow_const (fun t => Or.inl t.property.1.ne')).smul (continuous_integral_data N C hC hN j) lemma norm_value_le (hT : 0 ≤ T) (j : Time T →ᵇ E) (t : Time T) : ‖value N j t‖ ≤ (T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*C)*‖j‖ := by rw [value, norm_smul, Real.norm_eq_abs, abs_of_nonneg (Real.rpow_nonneg t.property.1.le _)] calc _ ≤ T^(1/4 : ℝ) * ((∫ u : UnitTime, weight u)*C*‖j‖) := mul_le_mul (Real.rpow_le_rpow t.property.1.le t.property.2 (by norm_num)) (norm_integral_data_le N C hC hN j t) (norm_nonneg _) (Real.rpow_nonneg hT _) _ = _ := by ring def operator (hT : 0 ≤ T) (j : Time T →ᵇ E) : Time T →ᵇ E := .mkOfBound ⟨value N j, continuous_value N C hC hN j⟩ (2*((T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*C)*‖j‖)) (by intro x y exact (dist_le_norm_add_norm _ _).trans <| by dsimp linarith [norm_value_le N C hC hN hT j x, norm_value_le N C hC hN hT j y]) @[simp] lemma operator_apply (hT : 0 ≤ T) (j : Time T →ᵇ E) (t : Time T) : operator N C hC hN hT j t = value N j t := rfl lemma norm_operator_le (hT : 0 ≤ T) (j : Time T →ᵇ E) : ‖operator N C hC hN hT j‖ ≤ (T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*C)*‖j‖ := by apply (BoundedContinuousFunction.norm_le (by apply mul_nonneg _ (norm_nonneg _) exact mul_nonneg (mul_nonneg (Real.rpow_nonneg hT _) (integral_nonneg weight_nonneg)) hC)).mpr exact norm_value_le N C hC hN hT j lemma operator_add (hT : 0 ≤ T) (j k : Time T →ᵇ E) : operator N C hC hN hT (j+k) = operator N C hC hN hT j + operator N C hC hN hT k := by ext t simp only [operator_apply, BoundedContinuousFunction.add_apply, value] have hd : (fun u => data N (j+k) t u) = (fun u => data N j t u+data N k t u) := by funext u simp [data, map_add, smul_add] rw [hd, integral_add (integrable_data N C hC hN j t) (integrable_data N C hC hN k t), smul_add] lemma operator_smul (hT : 0 ≤ T) (c : ℂ) (j : Time T →ᵇ E) : operator N C hC hN hT (c • j) = c • operator N C hC hN hT j := by ext t simp only [operator_apply, BoundedContinuousFunction.smul_apply, value] have hd : (fun u => data N (c • j) t u) = (fun u => c • data N j t u) := by funext u simp only [data, BoundedContinuousFunction.smul_apply, map_smul] exact smul_comm _ _ _ rw [hd, integral_smul, smul_comm] def operatorCLM (hT : 0 ≤ T) : (Time T →ᵇ E) →L[ℂ] (Time T →ᵇ E) := LinearMap.mkContinuous { toFun := operator N C hC hN hT map_add' := operator_add N C hC hN hT map_smul' := operator_smul N C hC hN hT } (T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*C) (norm_operator_le N C hC hN hT) lemma norm_operatorCLM_le (hT : 0 ≤ T) : ‖operatorCLM N C hC hN hT‖ ≤ T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*C := by apply ContinuousLinearMap.opNorm_le_bound _ (mul_nonneg (mul_nonneg (Real.rpow_nonneg hT _) (integral_nonneg weight_nonneg)) hC) exact norm_operator_le N C hC hN hT theorem exists_unique_solution [CompleteSpace E] (hT : 0 ≤ T) (hsmall : T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*C < 1) (g : Time T →ᵇ E) : ∃! j : Time T →ᵇ E, j + operator N C hC hN hT j = g := by let A := operatorCLM N C hC hN hT have hA : ‖A‖ < 1 := (norm_operatorCLM_le N C hC hN hT).trans_lt hsmall let f : (Time T →ᵇ E) → (Time T →ᵇ E) := fun j => g-A j have hf : ContractingWith ‖A‖₊ f := by refine ⟨hA, LipschitzWith.of_dist_le_mul (fun j k => ?_)⟩ rw [dist_eq_norm, dist_eq_norm] change ‖(g-A j)-(g-A k)‖ ≤ ‖A‖*‖j-k‖ rw [sub_sub_sub_cancel_left, ← A.map_sub, norm_sub_rev j k] exact A.le_opNorm (k-j) refine ⟨hf.fixedPoint, ?_, fun j hj => ?_⟩ · have hx := hf.fixedPoint_isFixedPt change g-A hf.fixedPoint = hf.fixedPoint at hx change hf.fixedPoint + A hf.fixedPoint = g exact (eq_sub_iff_add_eq.mp hx.symm) · apply hf.fixedPoint_unique change g-A j = j change j + A j = g at hj exact (eq_sub_iff_add_eq.mpr hj).symm lemma norm_solution_le (hT : 0 ≤ T) (hsmall : T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*C < 1) (g j : Time T →ᵇ E) (hj : j+operator N C hC hN hT j=g) : ‖j‖ ≤ ‖g‖/(1-T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*C) := by have hpos := sub_pos.mpr hsmall apply (le_div_iff₀ hpos).mpr have hb := norm_operator_le N C hC hN hT j have hjg : j = g-operator N C hC hN hT j := eq_sub_iff_add_eq.mpr hj have hn := norm_sub_le g (operator N C hC hN hT j) rw [← hjg] at hn nlinarith omit hC hN in def unweighted (j : Time T →ᵇ E) (r : ℝ) : E := if hr : r ∈ Time T then r^(-3/4 : ℝ) • j ⟨r,hr⟩ else 0 omit hC hN in lemma unweighted_lag (j : Time T →ᵇ E) (t : Time T) (u : UnitTime) : unweighted j (t.val*u.val) = (t.val*u.val)^(-3/4 : ℝ) • j (lag t u) := by have hr : t.val*u.val ∈ Time T := (lag t u).property rw [unweighted, dite_eq_left hr] rfl omit hC hN in def normalizedArg (t : Time T) (r : ℝ) (hr : r ∈ Ioo 0 t.val) : UnitTime := ⟨r/t.val, div_pos hr.1 t.property.1, (div_lt_one t.property.1).mpr hr.2⟩ omit hC hN in def kernel (t : Time T) (r : ℝ) : E →L[ℂ] E := if hr : r ∈ Ioo 0 t.val then (t.val-r)^(-3/4 : ℝ) • N (t,normalizedArg t r hr) else 0 omit hC hN in lemma normalizedArg_lag (t : Time T) (u : UnitTime) (hr : t.val*u.val ∈ Ioo 0 t.val) : normalizedArg t (t.val*u.val) hr = u := by apply Subtype.ext change t.val*u.val/t.val = u.val field_simp [t.property.1.ne'] omit hC hN in lemma kernel_lag (t : Time T) (u : UnitTime) : kernel N t (t.val*u.val) = (t.val*(1-u.val))^(-3/4 : ℝ) • N (t,u) := by have hr : t.val*u.val ∈ Ioo 0 t.val := ⟨mul_pos t.property.1 u.property.1, mul_lt_of_lt_one_right t.property.1 u.property.2⟩ rw [kernel, dite_eq_left hr, normalizedArg_lag] congr 2 ring omit hC hN in lemma rescaling_scalar (t u : ℝ) (ht : 0 < t) (hu : 0 < u) (hu1 : u < 1) : t^(3/4 : ℝ)*t*((t*(1-u))^(-3/4 : ℝ)*(t*u)^(-3/4 : ℝ)) = t^(1/4 : ℝ)*(u^(-3/4 : ℝ)*(1-u)^(-3/4 : ℝ)) := by rw [Real.mul_rpow ht.le (sub_pos.mpr hu1).le, Real.mul_rpow ht.le hu.le] have hprod : t^(3/4 : ℝ)*t*t^(-3/4 : ℝ)*t^(-3/4 : ℝ) = t^(1/4 : ℝ) := by calc _ = t^(3/4 : ℝ)*t^(1 : ℝ)*t^(-3/4 : ℝ)*t^(-3/4 : ℝ) := by rw [Real.rpow_one] _ = t^((3/4:ℝ)+1+(-3/4)+(-3/4)) := by rw [Real.rpow_add ht, Real.rpow_add ht, Real.rpow_add ht] _ = _ := by norm_num calc _ = (t^(3/4 : ℝ)*t*t^(-3/4 : ℝ)*t^(-3/4 : ℝ))* (u^(-3/4 : ℝ)*(1-u)^(-3/4 : ℝ)) := by ring _ = _ := by rw [hprod] omit hC hN in lemma value_eq_volterra (j : Time T →ᵇ E) (t : Time T) : value N j t = t.val^(3/4 : ℝ) • ∫ r in 0..t.val, kernel N t r (unweighted j r) := by let F : ℝ → E := fun r => kernel N t r (unweighted j r) have hc := intervalIntegral.smul_integral_comp_mul_left (a := 0) (b := 1) F t.val simp only [mul_zero, mul_one] at hc change value N j t = t.val^(3/4 : ℝ) • ∫ r in 0..t.val, F r rw [← hc, intervalIntegral.integral_of_le (by norm_num : (0:ℝ) ≤ 1), integral_Ioc_eq_integral_Ioo, ← integral_subtype measurableSet_Ioo] rw [← integral_smul, ← integral_smul] unfold value rw [← integral_smul] apply integral_congr_ae apply Eventually.of_forall intro u dsimp [F] rw [kernel_lag, unweighted_lag, smul_apply, ContinuousLinearMap.map_smul_of_tower] simp only [data, smul_smul] congr 1 simpa only [weight, mul_assoc] using (rescaling_scalar t.val u.val t.property.1 u.property.1 u.property.2).symm omit hC hN in lemma unweighted_at_time (j : Time T →ᵇ E) (t : Time T) : unweighted j t.val = t.val^(-3/4 : ℝ) • j t := by rw [unweighted, dite_eq_left t.property] omit hC hN in lemma kernel_unweighted_lag (j : Time T →ᵇ E) (t : Time T) (u : UnitTime) : kernel N t (t.val*u.val) (unweighted j (t.val*u.val)) = (t.val^(-3/4 : ℝ)*t.val^(-3/4 : ℝ)) • data N j t u := by rw [kernel_lag, unweighted_lag, smul_apply, ContinuousLinearMap.map_smul_of_tower] simp only [data, smul_smul] congr 1 rw [Real.mul_rpow t.property.1.le (sub_pos.mpr u.property.2).le, Real.mul_rpow t.property.1.le u.property.1.le, weight] ring lemma intervalIntegrable_kernel_unweighted (j : Time T →ᵇ E) (t : Time T) : IntervalIntegrable (fun r => kernel N t r (unweighted j r)) volume 0 t.val := by have hi : Integrable (fun u : UnitTime => kernel N t (t.val*u.val) (unweighted j (t.val*u.val))) := by simp_rw [kernel_unweighted_lag] exact (integrable_data N C hC hN j t).smul _ have hs : IntegrableOn (fun u : ℝ => kernel N t (t.val*u) (unweighted j (t.val*u))) (Ioo 0 1) := (integrableOn_iff_comap_subtypeVal measurableSet_Ioo).mpr hi have hii := (intervalIntegrable_iff_integrableOn_Ioo_of_le (by norm_num : (0:ℝ) ≤ 1)).mpr hs apply (IntervalIntegrable.comp_mul_left_iff (a := 0) (b := t.val) t.property.1.ne').mp simpa only [zero_div, div_self t.property.1.ne'] using hii lemma unweighted_solves (hT : 0 ≤ T) (g j : Time T →ᵇ E) (hj : j+operator N C hC hN hT j=g) (t : Time T) : unweighted j t.val + ∫ r in 0..t.val, kernel N t r (unweighted j r) = unweighted g t.val := by have ht : t.val^(-3/4 : ℝ)*t.val^(3/4 : ℝ)=1 := by rw [← Real.rpow_add t.property.1] norm_num have he : j t+value N j t=g t := congrArg (fun k : Time T →ᵇ E => k t) hj rw [unweighted_at_time, unweighted_at_time] calc _ = t.val^(-3/4 : ℝ) • (j t+value N j t) := by rw [smul_add, value_eq_volterra, smul_smul, ht, one_smul] _ = _ := by rw [he] theorem exists_unweighted_solution [CompleteSpace E] (hT : 0 ≤ T) (hsmall : T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*C < 1) (g : Time T →ᵇ E) : ∃ j : Time T →ᵇ E, (∀ t : Time T, IntervalIntegrable (fun r => kernel N t r (unweighted j r)) volume 0 t.val) ∧ (∀ t : Time T, unweighted j t.val + ∫ r in 0..t.val, kernel N t r (unweighted j r) = unweighted g t.val) ∧ (∀ t : Time T, ‖unweighted j t.val‖ ≤ t.val^(-3/4 : ℝ) * (‖g‖/(1-T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*C))) := by obtain ⟨j,hj,_⟩ := exists_unique_solution N C hC hN hT hsmall g refine ⟨j, intervalIntegrable_kernel_unweighted N C hC hN j, unweighted_solves N C hC hN hT g j hj, fun t => ?_⟩ rw [unweighted_at_time, norm_smul, Real.norm_eq_abs, abs_of_nonneg (Real.rpow_nonneg t.property.1.le _)] exact mul_le_mul_of_nonneg_left ((j.norm_coe_le_norm t).trans (norm_solution_le N C hC hN hT hsmall g j hj)) (Real.rpow_nonneg t.property.1.le _) end StrongWeightedVolterra open Set Filter open scoped Topology namespace ParabolicRadius def radius (q : ℝ) : ℝ := q^(1/4 : ℝ) lemma pos {q : ℝ} (hq : 0 < q) : 0 < radius q := Real.rpow_pos_of_pos hq _ lemma nonneg {q : ℝ} (hq : 0 ≤ q) : 0 ≤ radius q := Real.rpow_nonneg hq _ lemma pow_four {q : ℝ} (hq : 0 ≤ q) : radius q ^ 4 = q := by rw [radius, ← Real.rpow_natCast, ← Real.rpow_mul hq] norm_num lemma pow_three {q : ℝ} (hq : 0 ≤ q) : radius q ^ 3 = q^(3/4 : ℝ) := by rw [radius, ← Real.rpow_natCast, ← Real.rpow_mul hq] norm_num lemma inv_pow_three {q : ℝ} (hq : 0 ≤ q) : (radius q)⁻¹ ^ 3 = q^(-3/4 : ℝ) := by rw [inv_pow, pow_three hq, ← Real.rpow_neg hq] congr 1 norm_num lemma le_one {q : ℝ} (hq : 0 ≤ q) (hq1 : q ≤ 1) : radius q ≤ 1 := Real.rpow_le_one hq hq1 (by positivity) lemma continuous_comp {X : Type*} [TopologicalSpace X] (q : X → ℝ) (hq : Continuous q) : Continuous (fun x => radius (q x)) := hq.rpow_const (fun _ => Or.inr (by positivity)) lemma normalized_cancel {q : ℝ} (hq : 0 < q) : q^(-3/4 : ℝ)*radius q^3=1 := by rw [pow_three hq.le, ← Real.rpow_add hq] norm_num end ParabolicRadius namespace ParametrixVolterra section Slab open scoped ZeroAtInfty BoundedContinuousFunction open FrozenParametrix C0Convolution open StrongWeightedVolterra variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {T α β : ℝ} abbrev Slab (T : ℝ) := Icc (0 : ℝ) T def now (t : Time T) : Slab T := ⟨t.val,t.property.1.le,t.property.2⟩ def before (p : Time T × UnitTime) : Slab T := now (lag p.1 p.2) def elapsed (p : Time T × UnitTime) : ℝ := p.1.val*(1-p.2.val) lemma continuous_now : Continuous (now : Time T → Slab T) := continuous_subtype_val.subtype_mk _ lemma continuous_before : Continuous (before : Time T × UnitTime → Slab T) := continuous_now.comp continuous_lag lemma continuous_elapsed : Continuous (elapsed : Time T × UnitTime → ℝ) := by unfold elapsed fun_prop lemma elapsed_pos (p : Time T × UnitTime) : 0 < elapsed p := mul_pos p.1.property.1 (sub_pos.mpr p.2.property.2) lemma elapsed_le (p : Time T × UnitTime) : elapsed p ≤ T := by have hu : 1-p.2.val ≤ 1 := by linarith [p.2.property.1] exact (mul_le_of_le_one_right p.1.property.1.le hu).trans p.1.property.2 lemma elapsed_eq_sub (p : Time T × UnitTime) : elapsed p = (now p.1).val-(before p).val := by dsimp [elapsed, now, before, lag] ring variable (hα : 0 < α) (A : C(Slab T, V →ᵇ Icc α β)) (b : C(Slab T, V →ᵇ ℂ)) def actualResidual (t : Time T) (s : Slab T) : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ) := fullResidualCLM hα (A s).toContinuousMap (A (now t)).toContinuousMap (b (now t)) (ParabolicRadius.radius (t.val-s.val)) def normalized (p : Time T × UnitTime) : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ) := ParabolicRadius.radius (elapsed p)^3 • actualResidual hα A b p.1 (before p) lemma continuous_actualResidual_before (f : C₀(V,ℂ)) : Continuous (fun p : Time T × UnitTime => actualResidual hα A b p.1 (before p) f) := by have he (p : Time T × UnitTime) : p.1.val-(before p).val=elapsed p := (elapsed_eq_sub p).symm unfold actualResidual simp_rw [he] apply continuous_fullResidualCLM_apply hα (fun p => A (before p)) (fun p => A (now p.1)) (fun p => b (now p.1)) (fun p => ParabolicRadius.radius (elapsed p)) (A.continuous.comp continuous_before) (A.continuous.comp (continuous_now.comp continuous_fst)) (b.continuous.comp (continuous_now.comp continuous_fst)) (ParabolicRadius.continuous_comp _ continuous_elapsed) exact fun p => (ParabolicRadius.pos (elapsed_pos p)).ne' lemma continuous_normalized_apply (f : C₀(V,ℂ)) : Continuous (fun p : Time T × UnitTime => normalized hα A b p f) := by change Continuous (fun p => ParabolicRadius.radius (elapsed p)^3 • actualResidual hα A b p.1 (before p) f) exact ((ParabolicRadius.continuous_comp _ continuous_elapsed).pow 3).smul (continuous_actualResidual_before hα A b f) lemma exists_normalized_bound (hT : T ≤ 1) (L B : ℝ) (hL : 0 ≤ L) (hB : 0 ≤ B) (hb : ∀ s, ‖b s‖ ≤ B) (hLip : ∀ s r : Slab T, ∀ x z : V, |(A s x).val-(A r z).val| ≤ L*(‖x-z‖+|s.val-r.val|)) : ∃ C : ℝ, 0 ≤ C ∧ ∀ p : Time T × UnitTime, ‖normalized hα A b p‖ ≤ C := by obtain ⟨C,hC,hbound⟩ := exists_uniform_fullResidual_bound (V := V) α β hα L B hL hB refine ⟨C,hC,fun p => ?_⟩ let r := ParabolicRadius.radius (elapsed p) have hr : 0 < r := ParabolicRadius.pos (elapsed_pos p) have hr1 : r ≤ 1 := ParabolicRadius.le_one (elapsed_pos p).le ((elapsed_le p).trans hT) have hgap : |(now p.1).val-(before p).val| = r^4 := by rw [← elapsed_eq_sub, abs_of_pos (elapsed_pos p), ParabolicRadius.pow_four (elapsed_pos p).le] have hR : ‖actualResidual hα A b p.1 (before p)‖ ≤ C*(r⁻¹)^3 := by unfold actualResidual rw [show p.1.val-(before p).val = elapsed p from (elapsed_eq_sub p).symm] apply hbound _ _ _ r hr hr1 (hb _) intro x z simpa only [hgap, BoundedContinuousFunction.coe_toContinuousMap] using hLip (now p.1) (before p) x z apply ContinuousLinearMap.opNorm_le_bound _ hC intro f change ‖r^3 • actualResidual hα A b p.1 (before p) f‖ ≤ C*‖f‖ rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (pow_nonneg hr.le _)] calc _ ≤ r^3*(C*(r⁻¹)^3*‖f‖) := mul_le_mul_of_nonneg_left ((actualResidual hα A b p.1 (before p)).le_opNorm f |>.trans (mul_le_mul_of_nonneg_right hR (norm_nonneg f))) (pow_nonneg hr.le _) _ = C*‖f‖ := by field_simp theorem exists_correction (hT₀ : 0 ≤ T) (hT : T ≤ 1) (L B : ℝ) (hL : 0 ≤ L) (hB : 0 ≤ B) (hb : ∀ s, ‖b s‖ ≤ B) (hLip : ∀ s r : Slab T, ∀ x z : V, |(A s x).val-(A r z).val| ≤ L*(‖x-z‖+|s.val-r.val|)) : ∃ C : ℝ, 0 ≤ C ∧ (T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*C < 1 → ∀ g : Time T →ᵇ C₀(V,ℂ), ∃ j : Time T →ᵇ C₀(V,ℂ), ∀ t : Time T, j t + t.val^(1/4 : ℝ) • ∫ u : UnitTime, weight u • normalized hα A b (t,u) (j (lag t u)) = g t) := by obtain ⟨C,hC,hbound⟩ := exists_normalized_bound hα A b hT L B hL hB hb hLip refine ⟨C,hC,fun hsmall g => ?_⟩ let N : Kernel T C₀(V,ℂ) := Kernel.ofStrong (normalized hα A b) (continuous_normalized_apply hα A b) C hC hbound obtain ⟨j,hj,_⟩ := exists_unique_solution N C hC hbound hT₀ hsmall g refine ⟨j,fun t => ?_⟩ exact congrArg (fun f : Time T →ᵇ C₀(V,ℂ) => f t) hj lemma exists_actualResidual_bound (hT : T ≤ 1) (L B : ℝ) (hL : 0 ≤ L) (hB : 0 ≤ B) (hb : ∀ s, ‖b s‖ ≤ B) (hLip : ∀ s r : Slab T, ∀ x z : V, |(A s x).val-(A r z).val| ≤ L*(‖x-z‖+|s.val-r.val|)) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (t : Time T) (s : Slab T), s.val ?_⟩ have hq : 0 < t.val-s.val := sub_pos.mpr hst have hq1 : t.val-s.val ≤ 1 := by linarith [t.property.2, s.property.1] have he : |(now t).val-s.val| = ParabolicRadius.radius (t.val-s.val)^4 := by rw [ParabolicRadius.pow_four hq.le] exact abs_of_pos hq unfold actualResidual rw [← ParabolicRadius.inv_pow_three hq.le] apply hbound _ _ _ _ (ParabolicRadius.pos hq) (ParabolicRadius.le_one hq.le hq1) (hb _) intro x z simpa only [he, BoundedContinuousFunction.coe_toContinuousMap] using hLip (now t) s x z def initial (hT₀ : 0 ≤ T) : Slab T := ⟨0,le_rfl,hT₀⟩ lemma continuous_initialResidual (hT₀ : 0 ≤ T) (d : C₀(V,ℂ)) : Continuous (fun t : Time T => actualResidual hα A b t (initial hT₀) d) := by unfold actualResidual initial simp only [sub_zero] apply continuous_fullResidualCLM_apply hα (fun _ => A ⟨0,le_rfl,hT₀⟩) (fun t => A (now t)) (fun t => b (now t)) (fun t => ParabolicRadius.radius t.val) continuous_const (A.continuous.comp continuous_now) (b.continuous.comp continuous_now) (ParabolicRadius.continuous_comp _ continuous_subtype_val) exact fun t => (ParabolicRadius.pos t.property.1).ne' def sourceRHS (hT₀ : 0 ≤ T) (d : C₀(V,ℂ)) (l : Time T →ᵇ C₀(V,ℂ)) (t : Time T) : C₀(V,ℂ) := t.val^(3/4 : ℝ) • (l t-actualResidual hα A b t (initial hT₀) d) lemma continuous_sourceRHS (hT₀ : 0 ≤ T) (d : C₀(V,ℂ)) (l : Time T →ᵇ C₀(V,ℂ)) : Continuous (sourceRHS hα A b hT₀ d l) := (continuous_subtype_val.rpow_const (fun t => Or.inl t.property.1.ne')).smul (l.continuous.sub (continuous_initialResidual hα A b hT₀ d)) lemma norm_sourceRHS_le (hT₀ : 0 ≤ T) (hT : T ≤ 1) (C : ℝ) (_hC : 0 ≤ C) (hR : ∀ t : Time T, ‖actualResidual hα A b t (initial hT₀)‖ ≤ C*t.val^(-3/4 : ℝ)) (d : C₀(V,ℂ)) (l : Time T →ᵇ C₀(V,ℂ)) (t : Time T) : ‖sourceRHS hα A b hT₀ d l t‖ ≤ ‖l‖+C*‖d‖ := by have ht : 0 ≤ t.val^(3/4 : ℝ) := Real.rpow_nonneg t.property.1.le _ have ht1 : t.val^(3/4 : ℝ) ≤ 1 := Real.rpow_le_one t.property.1.le (t.property.2.trans hT) (by positivity) have hc : t.val^(3/4 : ℝ)*t.val^(-3/4 : ℝ)=1 := by rw [← Real.rpow_add t.property.1] norm_num rw [sourceRHS, norm_smul, Real.norm_eq_abs, abs_of_nonneg ht] calc _ ≤ t.val^(3/4 : ℝ)*(‖l‖+C*t.val^(-3/4 : ℝ)*‖d‖) := mul_le_mul_of_nonneg_left ((norm_sub_le _ _).trans <| add_le_add (l.norm_coe_le_norm t) ((actualResidual hα A b t (initial hT₀)).le_opNorm d |>.trans (mul_le_mul_of_nonneg_right (hR t) (norm_nonneg d)))) ht _ = t.val^(3/4 : ℝ)*‖l‖+C*‖d‖*(t.val^(3/4 : ℝ)*t.val^(-3/4 : ℝ)) := by ring _ = t.val^(3/4 : ℝ)*‖l‖+C*‖d‖ := by rw [hc,mul_one] _ ≤ ‖l‖+C*‖d‖ := add_le_add (mul_le_of_le_one_left (norm_nonneg l) ht1) le_rfl def weightedSource (hT₀ : 0 ≤ T) (hT : T ≤ 1) (C : ℝ) (hC : 0 ≤ C) (hR : ∀ t : Time T, ‖actualResidual hα A b t (initial hT₀)‖ ≤ C*t.val^(-3/4 : ℝ)) (d : C₀(V,ℂ)) (l : Time T →ᵇ C₀(V,ℂ)) : Time T →ᵇ C₀(V,ℂ) := .mkOfBound ⟨sourceRHS hα A b hT₀ d l, continuous_sourceRHS hα A b hT₀ d l⟩ (2*(‖l‖+C*‖d‖)) (by intro t s exact (dist_le_norm_add_norm _ _).trans <| by dsimp linarith [norm_sourceRHS_le hα A b hT₀ hT C hC hR d l t, norm_sourceRHS_le hα A b hT₀ hT C hC hR d l s]) lemma unweighted_weightedSource (hT₀ : 0 ≤ T) (hT : T ≤ 1) (C : ℝ) (hC : 0 ≤ C) (hR : ∀ t : Time T, ‖actualResidual hα A b t (initial hT₀)‖ ≤ C*t.val^(-3/4 : ℝ)) (d : C₀(V,ℂ)) (l : Time T →ᵇ C₀(V,ℂ)) (t : Time T) : unweighted (weightedSource hα A b hT₀ hT C hC hR d l) t.val = l t-actualResidual hα A b t (initial hT₀) d := by rw [unweighted_at_time] change t.val^(-3/4 : ℝ) • (t.val^(3/4 : ℝ) • (l t-actualResidual hα A b t (initial hT₀) d)) = _ rw [smul_smul, ← Real.rpow_add t.property.1] norm_num def residualKernel (hT₀ : 0 ≤ T) (t : Time T) (r : ℝ) : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ) := if r ∈ Ioo 0 t.val then actualResidual hα A b t (projIcc 0 T hT₀ r) else 0 lemma elapsed_normalizedArg (t : Time T) (r : ℝ) (hr : r ∈ Ioo 0 t.val) : elapsed (t,normalizedArg t r hr) = t.val-r := by dsimp [elapsed, normalizedArg] field_simp [t.property.1.ne'] lemma before_normalizedArg (hT₀ : 0 ≤ T) (t : Time T) (r : ℝ) (hr : r ∈ Ioo 0 t.val) : before (t,normalizedArg t r hr) = projIcc 0 T hT₀ r := by have hmem : r ∈ Icc 0 T := ⟨hr.1.le,hr.2.le.trans t.property.2⟩ rw [projIcc_of_mem hT₀ hmem] apply Subtype.ext dsimp [before,now,lag,normalizedArg] field_simp [t.property.1.ne'] lemma kernel_eq_residualKernel (hT₀ : 0 ≤ T) (C : ℝ) (hC : 0 ≤ C) (hN : ∀ p : Time T × UnitTime, ‖normalized hα A b p‖ ≤ C) (t : Time T) (r : ℝ) : kernel (Kernel.ofStrong (normalized hα A b) (continuous_normalized_apply hα A b) C hC hN) t r = residualKernel hα A b hT₀ t r := by classical by_cases hr : r ∈ Ioo 0 t.val · rw [kernel,dite_eq_left hr,residualKernel,ite_eq_left hr] change (t.val-r)^(-3/4 : ℝ) • (ParabolicRadius.radius (elapsed (t,normalizedArg t r hr))^3 • actualResidual hα A b t (before (t,normalizedArg t r hr))) = _ rw [elapsed_normalizedArg,before_normalizedArg hT₀] apply ContinuousLinearMap.ext intro f change (t.val-r)^(-3/4 : ℝ) • (ParabolicRadius.radius (t.val-r)^3 • actualResidual hα A b t (projIcc 0 T hT₀ r) f) = _ rw [smul_smul,ParabolicRadius.normalized_cancel (sub_pos.mpr hr.2),one_smul] · simp only [kernel,dite_eq_right hr,residualKernel,ite_eq_right hr] end Slab open scoped ZeroAtInfty BoundedContinuousFunction open FrozenParametrix C0Convolution StrongWeightedVolterra variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {T α β : ℝ} theorem exists_parametrix_correction (hα : 0 < α) (A : C(Slab T, V →ᵇ Icc α β)) (b : C(Slab T, V →ᵇ ℂ)) (hT₀ : 0 ≤ T) (hT : T ≤ 1) (L B : ℝ) (hL : 0 ≤ L) (hB : 0 ≤ B) (hb : ∀ s, ‖b s‖ ≤ B) (hLip : ∀ s r : Slab T, ∀ x z : V, |(A s x).val-(A r z).val| ≤ L*(‖x-z‖+|s.val-r.val|)) : ∃ C D : ℝ, 0 ≤ C ∧ 0 ≤ D ∧ (T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*D < 1 → ∀ (d : C₀(V,ℂ)) (l : Time T →ᵇ C₀(V,ℂ)), ∃ j : Time T →ᵇ C₀(V,ℂ), (∀ t : Time T, IntervalIntegrable (fun r => residualKernel hα A b hT₀ t r (unweighted j r)) volume 0 t.val) ∧ (∀ t : Time T, unweighted j t.val + ∫ r in 0..t.val, residualKernel hα A b hT₀ t r (unweighted j r) = l t-actualResidual hα A b t (initial hT₀) d) ∧ (∀ t : Time T, ‖unweighted j t.val‖ ≤ t.val^(-3/4 : ℝ) * ((‖l‖+C*‖d‖)/(1-T^(1/4 : ℝ)*(∫ u : UnitTime, weight u)*D)))) := by obtain ⟨C,hC,hR⟩ := exists_actualResidual_bound hα A b hT L B hL hB hb hLip obtain ⟨D,hD,hN⟩ := exists_normalized_bound hα A b hT L B hL hB hb hLip have hR₀ : ∀ t : Time T, ‖actualResidual hα A b t (initial hT₀)‖ ≤ C*t.val^(-3/4 : ℝ) := by intro t simpa only [initial, sub_zero] using hR t (initial hT₀) t.property.1 refine ⟨C,D,hC,hD,fun hsmall d l => ?_⟩ let N : Kernel T C₀(V,ℂ) := Kernel.ofStrong (normalized hα A b) (continuous_normalized_apply hα A b) D hD hN let g := weightedSource hα A b hT₀ hT C hC hR₀ d l obtain ⟨j,hi,hj,hbound⟩ := exists_unweighted_solution N D hD hN hT₀ hsmall g have he : ∀ t r, kernel N t r = residualKernel hα A b hT₀ t r := kernel_eq_residualKernel hα A b hT₀ D hD hN simp_rw [he] at hi hj have hg : ∀ t : Time T, unweighted g t.val = l t-actualResidual hα A b t (initial hT₀) d := unweighted_weightedSource hα A b hT₀ hT C hC hR₀ d l simp_rw [hg] at hj refine ⟨j,hi,hj,fun t => (hbound t).trans ?_⟩ have hgnorm : ‖g‖ ≤ ‖l‖+C*‖d‖ := by apply (BoundedContinuousFunction.norm_le (by positivity)).mpr exact norm_sourceRHS_le hα A b hT₀ hT C hC hR₀ d l exact mul_le_mul_of_nonneg_left (div_le_div_of_nonneg_right hgnorm (by linarith)) (Real.rpow_nonneg t.property.1.le _) end ParametrixVolterra namespace MildVolterra open StrongWeightedVolterra def massWeight (u : UnitTime) : ℝ := u.val^(-3/4 : ℝ) lemma massWeight_nonneg (u : UnitTime) : 0 ≤ massWeight u := Real.rpow_nonneg u.property.1.le _ lemma continuous_massWeight : Continuous massWeight := continuous_subtype_val.rpow_const (fun u => Or.inl u.property.1.ne') lemma integrable_massWeight : Integrable massWeight := by have hi := (intervalIntegral.intervalIntegrable_rpow' (a := 0) (b := 1) (r := (-3/4 : ℝ)) (by norm_num)).1.mono_set Ioo_subset_Ioc_self exact (integrableOn_iff_comap_subtypeVal measurableSet_Ioo).mp hi variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] variable {T : ℝ} variable (K : Kernel T E) (C : ℝ) (hC : 0 ≤ C) (hK : ∀ p, ‖K p‖ ≤ C) def data (j : Time T →ᵇ E) (t : Time T) (u : UnitTime) : E := massWeight u • K (t,u) (j (lag t u)) lemma continuous_data (j : Time T →ᵇ E) : Continuous (fun p : Time T × UnitTime => data K j p.1 p.2) := by exact (continuous_massWeight.comp continuous_snd).smul (K.continuous_apply.comp (continuous_id.prodMk (j.continuous.comp continuous_lag))) include hC hK lemma norm_data_le (j : Time T →ᵇ E) (t : Time T) (u : UnitTime) : ‖data K j t u‖ ≤ massWeight u*(C*‖j‖) := by rw [data,norm_smul,Real.norm_eq_abs,abs_of_nonneg (massWeight_nonneg u)] apply mul_le_mul_of_nonneg_left _ (massWeight_nonneg u) exact (K (t,u)).le_opNorm _ |>.trans (mul_le_mul (hK _) (j.norm_coe_le_norm _) (norm_nonneg _) hC) lemma integrable_data (j : Time T →ᵇ E) (t : Time T) : Integrable (data K j t) := (integrable_massWeight.mul_const (C*‖j‖)).mono' ((continuous_data K j).comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable (Eventually.of_forall (norm_data_le K C hC hK j t)) lemma continuous_integral_data (j : Time T →ᵇ E) : Continuous (fun t => ∫ u : UnitTime, data K j t u) := by apply continuous_of_dominated (fun t => (integrable_data K C hC hK j t).aestronglyMeasurable) (fun t => Eventually.of_forall (norm_data_le K C hC hK j t)) (integrable_massWeight.mul_const (C*‖j‖)) exact Eventually.of_forall (fun u => (continuous_data K j).comp (continuous_id.prodMk continuous_const)) lemma norm_integral_data_le (j : Time T →ᵇ E) (t : Time T) : ‖∫ u : UnitTime, data K j t u‖ ≤ (∫ u : UnitTime, massWeight u)*C*‖j‖ := by calc _ ≤ ∫ u : UnitTime, massWeight u*(C*‖j‖) := norm_integral_le_of_norm_le (integrable_massWeight.mul_const _) <| Eventually.of_forall (norm_data_le K C hC hK j t) _ = _ := by rw [integral_mul_const]; ring omit hC hK in def value (j : Time T →ᵇ E) (t : Time T) : E := t.val^(1/4 : ℝ) • ∫ u : UnitTime, data K j t u lemma continuous_value (j : Time T →ᵇ E) : Continuous (value K j) := (continuous_subtype_val.rpow_const (fun t => Or.inl t.property.1.ne')).smul (continuous_integral_data K C hC hK j) lemma norm_value_le (j : Time T →ᵇ E) (t : Time T) : ‖value K j t‖ ≤ t.val^(1/4 : ℝ)*((∫ u : UnitTime, massWeight u)*C*‖j‖) := by rw [value,norm_smul,Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg t.property.1.le _)] exact mul_le_mul_of_nonneg_left (norm_integral_data_le K C hC hK j t) (Real.rpow_nonneg t.property.1.le _) lemma tendsto_value_zero (j : Time T →ᵇ E) : Tendsto (value K j) (comap Subtype.val (𝓝[>] (0 : ℝ))) (𝓝 0) := by apply tendsto_zero_iff_norm_tendsto_zero.mpr apply squeeze_zero (fun t => norm_nonneg _) (norm_value_le K C hC hK j) have ht : Tendsto (fun t : Time T => t.val) (comap Subtype.val (𝓝[>] (0 : ℝ))) (𝓝 0) := tendsto_comap.mono_right nhdsWithin_le_nhds have hp := (Real.continuous_rpow_const (by norm_num : (0:ℝ) ≤ 1/4)).continuousAt.tendsto.comp ht simpa using hp.mul_const ((∫ u : UnitTime, massWeight u)*C*‖j‖) omit hC hK in def rawKernel (t : Time T) (r : ℝ) : E →L[ℂ] E := if hr : r ∈ Ioo 0 t.val then K (t,normalizedArg t r hr) else 0 omit hC hK in lemma rawKernel_lag (t : Time T) (u : UnitTime) : rawKernel K t (t.val*u.val) = K (t,u) := by have hr : t.val*u.val ∈ Ioo 0 t.val := ⟨mul_pos t.property.1 u.property.1, mul_lt_of_lt_one_right t.property.1 u.property.2⟩ rw [rawKernel,dite_eq_left hr,normalizedArg_lag] omit hC hK in lemma rescaling_scalar (t u : ℝ) (ht : 0 < t) (hu : 0 < u) : t*(t*u)^(-3/4 : ℝ) = t^(1/4 : ℝ)*u^(-3/4 : ℝ) := by rw [Real.mul_rpow ht.le hu.le] have he : t*t^(-3/4 : ℝ) = t^(1/4 : ℝ) := by calc _ = t^(1 : ℝ)*t^(-3/4 : ℝ) := by rw [Real.rpow_one] _ = t^((1:ℝ)+(-3/4)) := (Real.rpow_add ht _ _).symm _ = _ := by norm_num rw [← mul_assoc,he] omit hC hK in lemma value_eq_integral (j : Time T →ᵇ E) (t : Time T) : value K j t = ∫ r in 0..t.val, rawKernel K t r (unweighted j r) := by let F : ℝ → E := fun r => rawKernel K t r (unweighted j r) have hc := intervalIntegral.smul_integral_comp_mul_left (a := 0) (b := 1) F t.val simp only [mul_zero,mul_one] at hc change value K j t = ∫ r in 0..t.val, F r rw [← hc,intervalIntegral.integral_of_le (by norm_num : (0:ℝ) ≤ 1), integral_Ioc_eq_integral_Ioo,← integral_subtype measurableSet_Ioo,← integral_smul] unfold value rw [← integral_smul] apply integral_congr_ae apply Eventually.of_forall intro u dsimp [F] rw [rawKernel_lag,unweighted_lag,ContinuousLinearMap.map_smul_of_tower] simp only [data,smul_smul,massWeight] rw [rescaling_scalar t.val u.val t.property.1 u.property.1] omit hC hK in lemma rawKernel_unweighted_lag (j : Time T →ᵇ E) (t : Time T) (u : UnitTime) : rawKernel K t (t.val*u.val) (unweighted j (t.val*u.val)) = t.val^(-3/4 : ℝ) • data K j t u := by rw [rawKernel_lag,unweighted_lag,ContinuousLinearMap.map_smul_of_tower] simp only [data,smul_smul,massWeight] rw [Real.mul_rpow t.property.1.le u.property.1.le] lemma intervalIntegrable_rawKernel_unweighted (j : Time T →ᵇ E) (t : Time T) : IntervalIntegrable (fun r => rawKernel K t r (unweighted j r)) volume 0 t.val := by have hi : Integrable (fun u : UnitTime => rawKernel K t (t.val*u.val) (unweighted j (t.val*u.val))) := by simp_rw [rawKernel_unweighted_lag] exact (integrable_data K C hC hK j t).smul _ have hs : IntegrableOn (fun u : ℝ => rawKernel K t (t.val*u) (unweighted j (t.val*u))) (Ioo 0 1) := (integrableOn_iff_comap_subtypeVal measurableSet_Ioo).mpr hi have hii := (intervalIntegrable_iff_integrableOn_Ioo_of_le (by norm_num : (0:ℝ) ≤ 1)).mpr hs apply (IntervalIntegrable.comp_mul_left_iff (a := 0) (b := t.val) t.property.1.ne').mp simpa only [zero_div,div_self t.property.1.ne'] using hii end MildVolterra namespace ParametrixVolterra open scoped ZeroAtInfty BoundedContinuousFunction open FrozenParametrix C0Convolution StrongWeightedVolterra variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {T α β : ℝ} variable (hα : 0 < α) (A : C(Slab T, V →ᵇ Icc α β)) def actualFrozen (t : Time T) (s : Slab T) : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ) := frozenCLM hα (A s).toContinuousMap (ParabolicRadius.radius (t.val-s.val)) lemma continuous_actualFrozen_before (d : C₀(V,ℂ)) : Continuous (fun p : Time T × UnitTime => actualFrozen hα A p.1 (before p) d) := by unfold actualFrozen simp_rw [show ∀ p : Time T × UnitTime, p.1.val-(before p).val=elapsed p from fun p => (elapsed_eq_sub p).symm] exact continuous_frozenOp_comp hα (fun p => A (before p)) (fun p => ParabolicRadius.radius (elapsed p)) (A.continuous.comp continuous_before) (ParabolicRadius.continuous_comp _ continuous_elapsed) d def frozenBound : ℝ := ∫ y : V, kernelMajorant α β hα y lemma frozenBound_nonneg : 0 ≤ frozenBound (V := V) (β := β) hα := integral_nonneg (kernelMajorant_spec α β hα).2.1 lemma norm_actualFrozen_le (t : Time T) (s : Slab T) : ‖actualFrozen hα A t s‖ ≤ frozenBound (V := V) (β := β) hα := norm_frozenCLM_le hα _ _ def mildKernel : Kernel T C₀(V,ℂ) := Kernel.ofStrong (fun p => actualFrozen hα A p.1 (before p)) (continuous_actualFrozen_before hα A) (frozenBound (V := V) (β := β) hα) (frozenBound_nonneg hα) (fun p => norm_actualFrozen_le hα A p.1 (before p)) lemma norm_mildKernel_le (p : Time T × UnitTime) : ‖mildKernel hα A p‖ ≤ frozenBound (V := V) (β := β) hα := norm_actualFrozen_le hα A p.1 (before p) lemma continuous_frozen_initial (hT₀ : 0 ≤ T) (d : C₀(V,ℂ)) : Continuous (fun t : Time T => actualFrozen hα A t (initial hT₀) d) := by change Continuous (fun t : Time T => frozenOp hα (A (initial hT₀)).toContinuousMap (ParabolicRadius.radius (t.val-0)) d) simp only [sub_zero] exact continuous_frozenOp_comp hα (fun _ : Time T => A (initial hT₀)) (fun t : Time T => ParabolicRadius.radius t.val) continuous_const (ParabolicRadius.continuous_comp _ continuous_subtype_val) d lemma tendsto_frozen_initial (hT₀ : 0 ≤ T) (d : C₀(V,ℂ)) : Tendsto (fun t : Time T => actualFrozen hα A t (initial hT₀) d) (comap Subtype.val (𝓝[>] (0 : ℝ))) (𝓝 d) := by have ht : Tendsto (fun t : Time T => t.val) (comap Subtype.val (𝓝[>] (0 : ℝ))) (𝓝 0) := tendsto_comap.mono_right nhdsWithin_le_nhds have hf : Continuous (fun q : ℝ => frozenOp hα (A (initial hT₀)).toContinuousMap (ParabolicRadius.radius q) d) := continuous_frozenOp_comp hα (fun _ => A (initial hT₀)) ParabolicRadius.radius continuous_const (ParabolicRadius.continuous_comp _ continuous_id) d have hz : frozenOp hα (A (initial hT₀)).toContinuousMap (ParabolicRadius.radius 0) d = d := by rw [show ParabolicRadius.radius 0 = 0 from Real.zero_rpow (by norm_num)] exact frozen_zero _ _ _ _ _ _ d change Tendsto (fun t : Time T => frozenOp hα (A (initial hT₀)).toContinuousMap (ParabolicRadius.radius (t.val-0)) d) _ (𝓝 d) simp only [sub_zero] have hout := (hf.tendsto 0).comp ht rw [hz] at hout exact hout def mild (hT₀ : 0 ≤ T) (d : C₀(V,ℂ)) (j : Time T →ᵇ C₀(V,ℂ)) (t : Time T) : C₀(V,ℂ) := actualFrozen hα A t (initial hT₀) d + MildVolterra.value (mildKernel hα A) j t lemma continuous_mild (hT₀ : 0 ≤ T) (d : C₀(V,ℂ)) (j : Time T →ᵇ C₀(V,ℂ)) : Continuous (mild hα A hT₀ d j) := (continuous_frozen_initial hα A hT₀ d).add (MildVolterra.continuous_value (mildKernel hα A) (frozenBound (V := V) (β := β) hα) (frozenBound_nonneg hα) (norm_mildKernel_le hα A) j) lemma tendsto_mild_initial (hT₀ : 0 ≤ T) (d : C₀(V,ℂ)) (j : Time T →ᵇ C₀(V,ℂ)) : Tendsto (mild hα A hT₀ d j) (comap Subtype.val (𝓝[>] (0 : ℝ))) (𝓝 d) := by have ht := (tendsto_frozen_initial hα A hT₀ d).add (MildVolterra.tendsto_value_zero (mildKernel hα A) (frozenBound (V := V) (β := β) hα) (frozenBound_nonneg hα) (norm_mildKernel_le hα A) j) unfold mild simpa only [add_zero] using ht def frozenRaw (hT₀ : 0 ≤ T) (t : Time T) (r : ℝ) : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ) := if r ∈ Ioo 0 t.val then actualFrozen hα A t (projIcc 0 T hT₀ r) else 0 lemma rawKernel_eq_frozenRaw (hT₀ : 0 ≤ T) (t : Time T) (r : ℝ) : MildVolterra.rawKernel (mildKernel hα A) t r = frozenRaw hα A hT₀ t r := by classical by_cases hr : r ∈ Ioo 0 t.val · rw [MildVolterra.rawKernel,dite_eq_left hr,frozenRaw,ite_eq_left hr] change actualFrozen hα A t (before (t,normalizedArg t r hr)) = _ rw [before_normalizedArg hT₀] · simp only [MildVolterra.rawKernel,dite_eq_right hr,frozenRaw,ite_eq_right hr] lemma intervalIntegrable_frozenRaw (hT₀ : 0 ≤ T) (j : Time T →ᵇ C₀(V,ℂ)) (t : Time T) : IntervalIntegrable (fun r => frozenRaw hα A hT₀ t r (unweighted j r)) volume 0 t.val := by simpa only [rawKernel_eq_frozenRaw hα A hT₀] using MildVolterra.intervalIntegrable_rawKernel_unweighted (mildKernel hα A) (frozenBound (V := V) (β := β) hα) (frozenBound_nonneg hα) (norm_mildKernel_le hα A) j t lemma mild_eq_integral (hT₀ : 0 ≤ T) (d : C₀(V,ℂ)) (j : Time T →ᵇ C₀(V,ℂ)) (t : Time T) : mild hα A hT₀ d j t = actualFrozen hα A t (initial hT₀) d + ∫ r in 0..t.val, frozenRaw hα A hT₀ t r (unweighted j r) := by rw [mild,MildVolterra.value_eq_integral] simp_rw [rawKernel_eq_frozenRaw hα A hT₀] end ParametrixVolterra open Set Filter MeasureTheory open scoped SchwartzMap Topology ContDiff BoundedContinuousFunction namespace ScaledSchwartzConvolution section ScaledDecay universe u variable {V E : Type u} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] [MeasureSpace V] [BorelSpace V] [Measure.IsAddHaarMeasure (volume : Measure V)] [NormedAddCommGroup E] [NormedSpace ℝ E] omit [FiniteDimensional ℝ V] [MeasureSpace V] [BorelSpace V] [Measure.IsAddHaarMeasure (volume : Measure V)] in lemma scaled_decay (g : SchwartzMap V E) (a b : ℝ) (ha : 0 < a) (K N : ℕ) : ∃ C : ℝ, 0 ≤ C ∧ ∀ c ∈ Icc a b, ∀ z, (1+‖z‖)^N * ‖(c⁻¹)^K • g (c⁻¹ • z)‖ ≤ C := by let M := max b 1 let D := 2^N * (Finset.Iic (N,0)).sup (fun i => SchwartzMap.seminorm ℝ i.1 i.2) g have hD : 0 ≤ D := by positivity have hM : 0 < M := lt_of_lt_of_le zero_lt_one (le_max_right _ _) have hdec (z : V) : (1+‖z‖)^N * ‖g z‖ ≤ D := by simpa only [norm_iteratedFDeriv_zero] using SchwartzMap.one_add_le_sup_seminorm_apply (𝕜 := ℝ) (m := (N,0)) le_rfl le_rfl g z refine ⟨(a⁻¹)^K*(M^N*D), by positivity, ?_⟩ intro c hc z have hcpos : 0 < c := ha.trans_le hc.1 have hcM : c ≤ M := hc.2.trans (le_max_left _ _) have hn : ‖z‖ = c * ‖c⁻¹ • z‖ := by rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hcpos), ← mul_assoc, mul_inv_cancel₀ hcpos.ne', one_mul] have hs : 1+‖z‖ ≤ M*(1+‖c⁻¹ • z‖) := by rw [hn, mul_add, mul_one] exact add_le_add (le_max_right b 1) (mul_le_mul_of_nonneg_right hcM (norm_nonneg _)) have hg : (1+‖z‖)^N * ‖g (c⁻¹ • z)‖ ≤ M^N*D := by calc _ ≤ (M*(1+‖c⁻¹ • z‖))^N * ‖g (c⁻¹ • z)‖ := by gcongr _ = M^N * ((1+‖c⁻¹ • z‖)^N * ‖g (c⁻¹ • z)‖) := by rw [mul_pow]; ring _ ≤ _ := mul_le_mul_of_nonneg_left (hdec _) (by positivity) rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (by positivity)] calc _ = (c⁻¹)^K * ((1+‖z‖)^N * ‖g (c⁻¹ • z)‖) := by ring _ ≤ (a⁻¹)^K * ((1+‖z‖)^N * ‖g (c⁻¹ • z)‖) := by gcongr; exact hc.1 _ ≤ _ := mul_le_mul_of_nonneg_left hg (by positivity) lemma shifted_majorant (g : SchwartzMap V E) (a b : ℝ) (ha : 0 < a) (K : ℕ) (x₀ : V) : ∃ B : V → ℝ, Integrable B ∧ ∀ c ∈ Icc a b, ∀ x ∈ Metric.ball x₀ 1, ∀ y, ‖(c⁻¹)^K • g (c⁻¹ • (x-y))‖ ≤ B y := by let N := Module.finrank ℝ V + 1 obtain ⟨C,hC,hdec⟩ := scaled_decay g a b ha K N refine ⟨fun y => (2^N*C)/(1+‖x₀-y‖)^N, ?_, ?_⟩ · have hi : Integrable (fun z : V => (1+‖z‖)^(-(N : ℝ))) := integrable_one_add_norm (by dsimp [N]; norm_num) simpa only [Real.rpow_neg (by positivity : (0:ℝ) ≤ 1+‖x₀-_‖), Real.rpow_natCast, div_eq_mul_inv] using (hi.comp_sub_left x₀).const_mul (2^N*C) · intro c hc x hx y have hd : ‖x₀-x‖ < 1 := by simpa only [Metric.mem_ball,dist_eq_norm,norm_sub_rev] using hx have hn : ‖x₀-y‖ ≤ ‖x₀-x‖ + ‖x-y‖ := by simpa only [sub_add_sub_cancel] using norm_add_le (x₀-x) (x-y) have hs : 1+‖x₀-y‖ ≤ 2*(1+‖x-y‖) := by nlinarith [norm_nonneg (x-y)] apply (le_div_iff₀ (by positivity)).mpr calc _ ≤ ‖(c⁻¹)^K • g (c⁻¹ • (x-y))‖ * (2*(1+‖x-y‖))^N := by gcongr _ = 2^N * ((1+‖x-y‖)^N * ‖(c⁻¹)^K • g (c⁻¹ • (x-y))‖) := by rw [mul_pow]; ring _ ≤ _ := mul_le_mul_of_nonneg_left (hdec c hc _) (by positivity) variable {A F : Type u} [NormedAddCommGroup A] [NormedSpace ℝ A] [NormedAddCommGroup F] [NormedSpace ℝ F] variable {a b : ℝ} (ha : 0 < a) (c : V → ℝ) (hc : Continuous c) (hcb : ∀ y, c y ∈ Icc a b) def value (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (K : ℕ) (x : V) : F := ∫ y, L (f y) ((c y)⁻¹^K • g ((c y)⁻¹ • (x-y))) include ha hc hcb lemma integrable_integrand (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (K : ℕ) (x : V) : Integrable (fun y => L (f y) ((c y)⁻¹^K • g ((c y)⁻¹ • (x-y)))) := by obtain ⟨B,hB,hbound⟩ := shifted_majorant g a b ha K x have hci : Continuous (fun y => (c y)⁻¹) := hc.inv₀ (fun y => (ha.trans_le (hcb y).1).ne') refine (hB.const_mul (‖L‖*‖f‖)).mono' ?_ (Eventually.of_forall fun y => ?_) · exact ((L.continuous.comp f.continuous).clm_apply ((hci.pow K).smul (g.continuous.comp (hci.smul (continuous_const.sub continuous_id))))).aestronglyMeasurable · exact (L.le_opNorm₂ _ _).trans <| mul_le_mul (mul_le_mul_of_nonneg_left (f.norm_coe_le_norm y) (norm_nonneg L)) (hbound (c y) (hcb y) x (Metric.mem_ball_self zero_lt_one) y) (norm_nonneg _) (by positivity) lemma hasFDerivAt_value (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (K : ℕ) (x₀ : V) : HasFDerivAt (value c L f g K) (value c (L.precompR V) f (SchwartzMap.fderivCLM ℝ V E g) (K+1) x₀) x₀ := by let g' := SchwartzMap.fderivCLM ℝ V E g obtain ⟨B,hB,hbound⟩ := shifted_majorant g' a b ha (K+1) x₀ have hd (x y : V) : HasFDerivAt (fun x => (c y)⁻¹^K • g ((c y)⁻¹ • (x-y))) ((c y)⁻¹^(K+1) • g' ((c y)⁻¹ • (x-y))) x := by have h := ((g.hasFDerivAt ((c y)⁻¹ • (x-y))).comp x (((hasFDerivAt_id x).sub (hasFDerivAt_const y x)).const_smul (c y)⁻¹)).const_smul ((c y)⁻¹^K) convert h using 1 · rfl · ext v simp [g',SchwartzMap.fderivCLM_apply,pow_succ,smul_smul] apply hasFDerivAt_integral_of_dominated_of_fderiv_le (Metric.ball_mem_nhds x₀ zero_lt_one) · exact Eventually.of_forall fun x => (integrable_integrand ha c hc hcb L f g K x).aestronglyMeasurable · exact integrable_integrand ha c hc hcb L f g K x₀ · exact (integrable_integrand ha c hc hcb (L.precompR V) f g' (K+1) x₀).aestronglyMeasurable · filter_upwards with y x hx change ‖(L (f y)).comp ((c y)⁻¹^(K+1) • g' ((c y)⁻¹ • (x-y)))‖ ≤ _ calc _ ≤ ‖L (f y)‖ * ‖(c y)⁻¹^(K+1) • g' ((c y)⁻¹ • (x-y))‖ := ContinuousLinearMap.opNorm_comp_le _ _ _ ≤ (‖L‖*‖f‖) * ‖(c y)⁻¹^(K+1) • g' ((c y)⁻¹ • (x-y))‖ := by gcongr exact (L.le_opNorm _).trans (mul_le_mul_of_nonneg_left (f.norm_coe_le_norm y) (norm_nonneg L)) _ ≤ (‖L‖*‖f‖)*B y := mul_le_mul_of_nonneg_left (hbound (c y) (hcb y) x hx y) (by positivity) · exact hB.const_mul (‖L‖*‖f‖) · filter_upwards with y x _ exact (L (f y)).hasFDerivAt.comp x (hd x y) lemma contDiff_value_nat (n : ℕ) (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (K : ℕ) : ContDiff ℝ n (value c L f g K) := by induction n generalizing E F K with | zero => rw [Nat.cast_zero,contDiff_zero] exact continuous_iff_continuousAt.mpr (fun x => (hasFDerivAt_value ha c hc hcb L f g K x).continuousAt) | succ n ih => rw [Nat.cast_add,Nat.cast_one,contDiff_succ_iff_hasFDerivAt] exact ⟨value c (L.precompR V) f (SchwartzMap.fderivCLM ℝ V E g) (K+1), ih (L.precompR V) (SchwartzMap.fderivCLM ℝ V E g) (K+1), hasFDerivAt_value ha c hc hcb L f g K⟩ lemma contDiff_value (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (K : ℕ) : ContDiff ℝ ∞ (value c L f g K) := contDiff_infty.mpr (fun n => contDiff_value_nat ha c hc hcb n L f g K) lemma fderiv_value (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (K : ℕ) : fderiv ℝ (value c L f g K) = value c (L.precompR V) f (SchwartzMap.fderivCLM ℝ V E g) (K+1) := by funext x exact (hasFDerivAt_value ha c hc hcb L f g K x).fderiv lemma fderiv_value_apply (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (K : ℕ) (x v : V) : fderiv ℝ (value c L f g K) x v = ∫ y, L (f y) ((c y)⁻¹^(K+1) • fderiv ℝ g ((c y)⁻¹ • (x-y)) v) := by rw [fderiv_value ha c hc hcb L f g K, value, ContinuousLinearMap.integral_apply (integrable_integrand ha c hc hcb (L.precompR V) f (SchwartzMap.fderivCLM ℝ V E g) (K+1) x)] rfl lemma second_fderiv_value_apply (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (K : ℕ) (x v w : V) : fderiv ℝ (fderiv ℝ (value c L f g K)) x v w = ∫ y, L (f y) ((c y)⁻¹^(K+2) • fderiv ℝ (fderiv ℝ g) ((c y)⁻¹ • (x-y)) v w) := by rw [fderiv_value ha c hc hcb L f g K, fderiv_value_apply ha c hc hcb] rw [ContinuousLinearMap.integral_apply] · rfl · exact integrable_integrand ha c hc hcb (L.precompR V) f (SchwartzMap.evalCLM ℝ V (V →L[ℝ] E) v (SchwartzMap.fderivCLM ℝ V (V →L[ℝ] E) (SchwartzMap.fderivCLM ℝ V E g))) (K+2) x end ScaledDecay open scoped Laplacian universe u variable {V E F A : Type u} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasureSpace V] [BorelSpace V] [Measure.IsAddHaarMeasure (volume : Measure V)] [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup A] [NormedSpace ℝ A] variable {a b : ℝ} (ha : 0 < a) (c : V → ℝ) (hc : Continuous c) (hcb : ∀ y, c y ∈ Icc a b) include ha hc hcb lemma laplacian_value (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (K : ℕ) : Δ (value c L f g K) = value c L f (Δ g : SchwartzMap V E) (K+2) := by funext x rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis] simp only [iteratedFDeriv_two_apply, Matrix.cons_val_zero, Matrix.cons_val_one, second_fderiv_value_apply ha c hc hcb] rw [value] simp_rw [SchwartzMap.laplacian_apply, InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis, iteratedFDeriv_two_apply, Matrix.cons_val_zero, Matrix.cons_val_one, Finset.smul_sum, map_sum] rw [integral_finsetSum] · rfl · intro i _ exact integrable_integrand ha c hc hcb L f (SchwartzMap.evalCLM ℝ V E ((stdOrthonormalBasis ℝ V) i) (SchwartzMap.evalCLM ℝ V (V →L[ℝ] E) ((stdOrthonormalBasis ℝ V) i) (SchwartzMap.fderivCLM ℝ V (V →L[ℝ] E) (SchwartzMap.fderivCLM ℝ V E g)))) (K+2) x lemma bilaplacian_value (L : A →L[ℝ] E →L[ℝ] F) (f : V →ᵇ A) (g : SchwartzMap V E) (K : ℕ) : Δ (Δ (value c L f g K)) = value c L f (Δ (Δ g : SchwartzMap V E) : SchwartzMap V E) (K+4) := by rw [laplacian_value ha c hc hcb, laplacian_value ha c hc hcb] end ScaledSchwartzConvolution namespace FrozenParametrix open scoped SchwartzMap Laplacian ZeroAtInfty BoundedContinuousFunction open BiharmonicKernel variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {α β : ℝ} def sourceScale (a : C(V,Icc α β)) (q : ℝ) (y : V) : ℝ := Real.sqrt (Real.sqrt (q*(a y).val)) omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in lemma continuous_sourceScale (a : C(V,Icc α β)) (q : ℝ) : Continuous (sourceScale a q) := Real.continuous_sqrt.comp (Real.continuous_sqrt.comp (continuous_const.mul (continuous_subtype_val.comp a.continuous))) omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in lemma sourceScale_bounds (a : C(V,Icc α β)) (q : ℝ) (hq : 0 ≤ q) (y : V) : sourceScale a q y ∈ Icc (Real.sqrt (Real.sqrt (q*α))) (Real.sqrt (Real.sqrt (q*β))) := by exact ⟨Real.sqrt_le_sqrt (Real.sqrt_le_sqrt (mul_le_mul_of_nonneg_left (a y).property.1 hq)), Real.sqrt_le_sqrt (Real.sqrt_le_sqrt (mul_le_mul_of_nonneg_left (a y).property.2 hq))⟩ def timeValue (a : C(V,Icc α β)) (q : ℝ) (f : C₀(V,ℂ)) (x : V) : ℂ := ∫ y, f y * kernel (q*(a y).val) (x-y) lemma timeValue_eq_scaled (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) : timeValue a q f = ScaledSchwartzConvolution.value (sourceScale a q) (ContinuousLinearMap.mul ℝ ℂ) f.toBCF (heatKernel 1 zero_lt_one) (Module.finrank ℝ V) := by funext x unfold timeValue ScaledSchwartzConvolution.value congr 1 funext y rw [kernel_eq_scaled _ (mul_pos hq (hα.trans_le (a y).property.1)), ← inv_pow] rfl lemma timeValue_eq_frozenOp (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) : timeValue a q f = frozenOp hα a (ParabolicRadius.radius q) f := by funext x rw [frozenOp, frozen_apply_unscaled _ _ _ _ _ _ _ (ParabolicRadius.pos hq)] rw [ParabolicRadius.pow_four hq.le] unfold timeValue congr 1 funext y exact mul_comm _ _ lemma contDiff_timeValue (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) : ContDiff ℝ ∞ (timeValue a q f) := by rw [timeValue_eq_scaled hα a q hq f] exact ScaledSchwartzConvolution.contDiff_value (Real.sqrt_pos.mpr (Real.sqrt_pos.mpr (mul_pos hq hα))) (sourceScale a q) (continuous_sourceScale a q) (sourceScale_bounds a q hq.le) (ContinuousLinearMap.mul ℝ ℂ) f.toBCF (heatKernel 1 zero_lt_one) (Module.finrank ℝ V) lemma bilaplacian_timeValue (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (x : V) : Δ (Δ (timeValue a q f)) x = ∫ y, f y * bilapKernel (q*(a y).val) (x-y) := by rw [timeValue_eq_scaled hα a q hq f, ScaledSchwartzConvolution.bilaplacian_value (Real.sqrt_pos.mpr (Real.sqrt_pos.mpr (mul_pos hq hα))) (sourceScale a q) (continuous_sourceScale a q) (sourceScale_bounds a q hq.le)] unfold ScaledSchwartzConvolution.value congr 1 funext y rw [bilapKernel_eq_scaled _ (mul_pos hq (hα.trans_le (a y).property.1))] rfl lemma integrable_timeValue_integrand (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (x : V) : Integrable (fun y => f y * kernel (q*(a y).val) (x-y)) := by have he (y : V) : kernel (q*(a y).val) (x-y) = (sourceScale a q y)⁻¹^(Module.finrank ℝ V) • heatKernel (V := V) 1 zero_lt_one ((sourceScale a q y)⁻¹ • (x-y)) := by rw [kernel_eq_scaled _ (mul_pos hq (hα.trans_le (a y).property.1)), ← inv_pow] rfl simp_rw [he] exact ScaledSchwartzConvolution.integrable_integrand (Real.sqrt_pos.mpr (Real.sqrt_pos.mpr (mul_pos hq hα))) (sourceScale a q) (continuous_sourceScale a q) (sourceScale_bounds a q hq.le) (ContinuousLinearMap.mul ℝ ℂ) f.toBCF (heatKernel 1 zero_lt_one) (Module.finrank ℝ V) x lemma integrable_bilapValue_integrand (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (x : V) : Integrable (fun y => f y * bilapKernel (q*(a y).val) (x-y)) := by simp_rw [bilapKernel_eq_scaled _ (mul_pos hq (hα.trans_le (a _).property.1))] exact ScaledSchwartzConvolution.integrable_integrand (Real.sqrt_pos.mpr (Real.sqrt_pos.mpr (mul_pos hq hα))) (sourceScale a q) (continuous_sourceScale a q) (sourceScale_bounds a q hq.le) (ContinuousLinearMap.mul ℝ ℂ) f.toBCF (Δ (Δ (heatKernel 1 zero_lt_one) : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Module.finrank ℝ V+4) x end FrozenParametrix namespace BiharmonicKernel open scoped SchwartzMap Laplacian variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] lemma bilapKernel_shifted_majorant (a b : ℝ) (ha : 0 < a) (x₀ : V) : ∃ B : V → ℝ, Integrable B ∧ ∀ t ∈ Icc a b, ∀ x ∈ Metric.ball x₀ 1, ∀ y, ‖bilapKernel t (x-y)‖ ≤ B y := by obtain ⟨B,hB,hbound⟩ := ScaledSchwartzConvolution.shifted_majorant (Δ (Δ (heatKernel (V := V) 1 zero_lt_one) : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Real.sqrt (Real.sqrt a)) (Real.sqrt (Real.sqrt b)) (Real.sqrt_pos.mpr (Real.sqrt_pos.mpr ha)) (Module.finrank ℝ V+4) x₀ refine ⟨B,hB,fun t ht x hx y => ?_⟩ rw [bilapKernel_eq_scaled t (ha.trans_le ht.1)] exact hbound _ ⟨Real.sqrt_le_sqrt (Real.sqrt_le_sqrt ht.1), Real.sqrt_le_sqrt (Real.sqrt_le_sqrt ht.2)⟩ x hx y lemma hasDerivAt_kernel_bilap (t : ℝ) (ht : 0 < t) (x : V) : HasDerivAt (fun s => kernel s x) (-bilapKernel t x) t := by rw [bilapKernel_eq_schwartz t ht] exact kernel_heat_equation t ht x end BiharmonicKernel namespace FrozenParametrix open scoped SchwartzMap Laplacian ZeroAtInfty BoundedContinuousFunction open BiharmonicKernel variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {α β : ℝ} lemma hasDerivAt_timeValue (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (x : V) : HasDerivAt (fun s => timeValue a s f x) (-(∫ y, ((a y).val : ℂ)*f y*bilapKernel (q*(a y).val) (x-y))) q := by let F := fun s : ℝ => fun y : V => f y * kernel (s*(a y).val) (x-y) let F' := fun s : ℝ => fun y : V => f y * ((a y).val • (-bilapKernel (s*(a y).val) (x-y))) have hay (y : V) : 0 < (a y).val := hα.trans_le (a y).property.1 have hβ : 0 < β := (hay 0).trans_le (a 0).property.2 have hFi (s : ℝ) (hs : 0 < s) : Integrable (F s) := integrable_timeValue_integrand hα a s hs f x have hF'cont (s : ℝ) (hs : 0 < s) : Continuous (F' s) := f.continuous.mul ((continuous_subtype_val.comp a.continuous).smul (continuous_bilapKernel_comp (fun y : V => s*(a y).val) (fun y => x-y) (continuous_const.mul (continuous_subtype_val.comp a.continuous)) (continuous_const.sub continuous_id) (fun y => mul_pos hs (hay y))).neg) obtain ⟨B,hB,hbound⟩ := bilapKernel_shifted_majorant (V := V) ((q/2)*α) ((2*q)*β) (by positivity) x have hspos (s : ℝ) (hs : s ∈ Ioo (q/2) (2*q)) : 0 < s := lt_trans (by positivity) hs.1 have hsb (s : ℝ) (hs : s ∈ Ioo (q/2) (2*q)) (y : V) : s*(a y).val ∈ Icc ((q/2)*α) ((2*q)*β) := ⟨mul_le_mul hs.1.le (a y).property.1 hα.le (hspos s hs).le, mul_le_mul hs.2.le (a y).property.2 (hay y).le (by positivity)⟩ have hd := hasDerivAt_integral_of_dominated_loc_of_deriv_le (F := F) (F' := F') (bound := fun y => (‖f‖*β)*B y) (s := Ioo (q/2) (2*q)) (Ioo_mem_nhds (by linarith) (by linarith)) (by filter_upwards [Ioi_mem_nhds hq] with s hs; exact (hFi s hs).aestronglyMeasurable) (hFi q hq) (hF'cont q hq).aestronglyMeasurable (Eventually.of_forall (fun y s hs => by simp only [F',norm_mul,norm_smul,Real.norm_eq_abs,abs_of_pos (hay y),norm_neg] rw [← mul_assoc] exact mul_le_mul (mul_le_mul (f.toBCF.norm_coe_le_norm y) (a y).property.2 (hay y).le (norm_nonneg f)) (hbound _ (hsb s hs y) x (Metric.mem_ball_self zero_lt_one) y) (norm_nonneg _) (by positivity))) (hB.const_mul (‖f‖*β)) (Eventually.of_forall (fun y s hs => by simpa [F,F',Function.comp_def,Complex.real_smul,mul_assoc] using ((hasDerivAt_kernel_bilap (s*(a y).val) (mul_pos (hspos s hs) (hay y)) (x-y)).scomp s ((hasDerivAt_id s).mul_const (a y).val)).const_mul (f y))) have he : (∫ y, F' q y) = -(∫ y, ((a y).val : ℂ)*f y*bilapKernel (q*(a y).val) (x-y)) := by rw [← integral_neg] congr 1 funext y dsimp [F'] ring rw [he] at hd exact hd.2 lemma frozen_pde_residual (hα : 0 < α) (a₀ a₁ : C(V,Icc α β)) (b : V →ᵇ ℂ) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (x : V) : deriv (fun s => timeValue a₀ s f x) q + ((a₁ x).val : ℂ) * Δ (Δ (timeValue a₀ q f)) x + b x * timeValue a₀ q f x = fullResidualCLM hα a₀ a₁ b (ParabolicRadius.radius q) f x := by rw [(hasDerivAt_timeValue hα a₀ q hq f x).deriv, bilaplacian_timeValue hα a₀ q hq f x, fullResidualCLM_apply] change _ = principalResidual hα a₀ a₁ (ParabolicRadius.radius q) f x + b x * frozenOp hα a₀ (ParabolicRadius.radius q) f x rw [← timeValue_eq_frozenOp hα a₀ q hq f, principalResidual_apply_unscaled hα a₀ a₁ _ (ParabolicRadius.pos hq), ParabolicRadius.pow_four hq.le] have hi₀ := integrable_bilapValue_integrand hα a₀ q hq f x have hi₁ := integrable_bilapValue_integrand hα a₀ q hq (C0Multiplier.mul (coefficient a₀) f) x change Integrable (fun y => (((a₀ y).val : ℂ)*f y)*bilapKernel (q*(a₀ y).val) (x-y)) at hi₁ have he : (fun y => (((a₁ x).val-(a₀ y).val : ℝ) : ℂ) * bilapKernel (q*(a₀ y).val) (x-y) * f y) = fun y => ((a₁ x).val : ℂ)*(f y*bilapKernel (q*(a₀ y).val) (x-y)) - (((a₀ y).val : ℂ)*f y)*bilapKernel (q*(a₀ y).val) (x-y) := by funext y rw [Complex.ofReal_sub] ring rw [he, integral_sub (hi₀.const_mul _) hi₁, integral_const_mul] ring end FrozenParametrix open scoped Topology SchwartzMap ZeroAtInfty BoundedContinuousFunction open Set Filter MeasureTheory namespace KernelPairing section IntegrableProdOfTranslationBound variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] lemma integrable_prod_of_translation_bound (F : V × V → ℂ) (hF : Continuous F) (φ B : V → ℝ) (hφ : Integrable φ) (hB : Integrable B) (C : ℝ) (hbound : ∀ x y, ‖F (x,y)‖ ≤ φ x * C * B (x-y)) : Integrable F (volume.prod volume) := by have hBi (x : V) : Integrable (fun y => B (x-y)) := (integrable_comp_sub_left B x).mpr hB have hFi (x : V) : Integrable (fun y => F (x,y)) := by apply ((hBi x).const_mul (φ x * C)).mono' · exact (hF.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable · exact Eventually.of_forall (hbound x) apply (integrable_prod_iff hF.aestronglyMeasurable).mpr refine ⟨Eventually.of_forall hFi, ?_⟩ apply ((hφ.mul_const C).mul_const (∫ y, B y)).mono' · exact hF.aestronglyMeasurable.norm.integral_prod_right' · filter_upwards with x rw [Real.norm_eq_abs, abs_of_nonneg (integral_nonneg (fun y => norm_nonneg _))] calc _ ≤ ∫ y, φ x*C*B (x-y) := integral_mono (hFi x).norm ((hBi x).const_mul _) (hbound x) _ = φ x*C*(∫ y, B y) := by rw [integral_const_mul, integral_sub_left_eq_self] lemma integrable_pair (φ : V → ℂ) (hφ : Integrable φ) (f : V →ᵇ ℂ) : Integrable (fun x => φ x * f x) := by apply (hφ.norm.mul_const ‖f‖).mono' · exact hφ.aestronglyMeasurable.mul f.continuous.aestronglyMeasurable · filter_upwards with x rw [norm_mul] exact mul_le_mul_of_nonneg_left (f.norm_coe_le_norm x) (norm_nonneg _) lemma norm_pair_le (φ : V → ℂ) (hφ : Integrable φ) (f : V →ᵇ ℂ) : ‖∫ x, φ x*f x‖ ≤ (∫ x, ‖φ x‖)*‖f‖ := by apply (norm_integral_le_integral_norm _).trans rw [← integral_mul_const] apply integral_mono (integrable_pair φ hφ f).norm (hφ.norm.mul_const _) intro x dsimp only rw [norm_mul] exact mul_le_mul_of_nonneg_left (f.norm_coe_le_norm x) (norm_nonneg _) def pairCLM (φ : V → ℂ) (hφ : Integrable φ) : C₀(V,ℂ) →L[ℂ] ℂ := LinearMap.mkContinuous { toFun := fun f => ∫ x, φ x * f x map_add' := by intro f g change (∫ x, φ x*(f x+g x)) = _ simp_rw [mul_add] exact integral_add (integrable_pair φ hφ f.toBCF) (integrable_pair φ hφ g.toBCF) map_smul' := by intro c f change (∫ x, φ x*(c*f x)) = c*(∫ x, φ x*f x) simp_rw [mul_left_comm (φ _) c] exact integral_const_mul _ _ } (∫ x, ‖φ x‖) (fun f => norm_pair_le φ hφ f.toBCF) @[simp] lemma pairCLM_apply (φ : V → ℂ) (hφ : Integrable φ) (f : C₀(V,ℂ)) : pairCLM φ hφ f = ∫ x, φ x * f x := rfl end IntegrableProdOfTranslationBound open scoped Laplacian variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] omit [MeasurableSpace V] [BorelSpace V] in lemma laplacian_translate (f : V → ℂ) (y : V) : Δ (fun x => f (x-y)) = fun x => Δ f (x-y) := by simp_rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis, iteratedFDeriv_comp_sub] omit [MeasurableSpace V] [BorelSpace V] in lemma schwartz_laplacian_translate (f : SchwartzMap V ℂ) (y : V) : Δ (f.compSubConstCLM ℂ y : SchwartzMap V ℂ) = (Δ f : SchwartzMap V ℂ).compSubConstCLM ℂ y := by ext x rw [SchwartzMap.laplacian_apply, SchwartzMap.compSubConstCLM_apply, SchwartzMap.laplacian_apply] exact congrFun (laplacian_translate (f : V → ℂ) y) x lemma schwartz_bilaplacian_pair (φ g : SchwartzMap V ℂ) : (∫ x, φ x*(Δ (Δ g : SchwartzMap V ℂ) : SchwartzMap V ℂ) x) = ∫ x, (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) x*g x := by rw [SchwartzMap.integral_mul_laplacian_right_eq_left, SchwartzMap.integral_mul_laplacian_right_eq_left] lemma bilaplacian_pair_kernel (φ : SchwartzMap V ℂ) (f : V → ℂ) (K : V → SchwartzMap V ℂ) (h₀ : Integrable (fun p : V × V => (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) p.1 * (f p.2*K p.2 (p.1-p.2))) (volume.prod volume)) (h₄ : Integrable (fun p : V × V => φ p.1*(f p.2*(Δ (Δ (K p.2) : SchwartzMap V ℂ) : SchwartzMap V ℂ) (p.1-p.2))) (volume.prod volume)) : (∫ x, φ x*(∫ y, f y*(Δ (Δ (K y) : SchwartzMap V ℂ) : SchwartzMap V ℂ) (x-y))) = ∫ x, (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) x*(∫ y, f y*K y (x-y)) := by simp_rw [← integral_const_mul] conv_lhs => rw [integral_integral_swap h₄] conv_rhs => rw [integral_integral_swap h₀] apply integral_congr_ae filter_upwards with y have he := schwartz_bilaplacian_pair φ ((K y).compSubConstCLM ℂ y) simp only [schwartz_laplacian_translate, SchwartzMap.compSubConstCLM_apply] at he calc _ = f y*(∫ x, φ x*(Δ (Δ (K y) : SchwartzMap V ℂ) : SchwartzMap V ℂ) (x-y)) := by rw [← integral_const_mul] congr 1 funext x ring _ = f y*(∫ x, (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) x*K y (x-y)) := by rw [he] _ = _ := by rw [← integral_const_mul] congr 1 funext x ring end KernelPairing namespace FrozenParametrix section IntegrableKernelPairProd open scoped SchwartzMap Laplacian ZeroAtInfty BoundedContinuousFunction open BiharmonicKernel variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {α β : ℝ} lemma integrable_kernel_pair_prod (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (φ : SchwartzMap V ℂ) : Integrable (fun p : V × V => φ p.1 * (f p.2*kernel (q*(a p.2).val) (p.1-p.2))) (volume.prod volume) := by obtain ⟨B,hB,hbound⟩ := kernel_majorant (V := V) (q*α) (q*β) (mul_pos hq hα) apply KernelPairing.integrable_prod_of_translation_bound _ (φ.continuous.comp continuous_fst |>.mul ((f.continuous.comp continuous_snd).mul (continuous_kernel_comp (fun p : V × V => q*(a p.2).val) (fun p => p.1-p.2) (continuous_const.mul (continuous_subtype_val.comp (a.continuous.comp continuous_snd))) (continuous_fst.sub continuous_snd) (fun p => mul_pos hq (hα.trans_le (a p.2).property.1))))) (fun x => ‖φ x‖) B φ.integrable.norm hB ‖f‖ intro x y change ‖φ x * (f y * kernel (q*(a y).val) (x-y))‖ ≤ ‖φ x‖*‖f‖*B (x-y) rw [norm_mul,norm_mul,mul_assoc] apply mul_le_mul_of_nonneg_left _ (norm_nonneg _) exact mul_le_mul (f.toBCF.norm_coe_le_norm y) (hbound _ ⟨mul_le_mul_of_nonneg_left (a y).property.1 hq.le, mul_le_mul_of_nonneg_left (a y).property.2 hq.le⟩ (x-y)) (norm_nonneg _) (norm_nonneg _) lemma integrable_bilap_pair_prod (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (φ : SchwartzMap V ℂ) : Integrable (fun p : V × V => φ p.1 * (f p.2*bilapKernel (q*(a p.2).val) (p.1-p.2))) (volume.prod volume) := by obtain ⟨B,hB,_,_,hbound⟩ := bilapKernel_weighted_majorant (V := V) (q*α) (q*β) (mul_pos hq hα) apply KernelPairing.integrable_prod_of_translation_bound _ (φ.continuous.comp continuous_fst |>.mul ((f.continuous.comp continuous_snd).mul (continuous_bilapKernel_comp (fun p : V × V => q*(a p.2).val) (fun p => p.1-p.2) (continuous_const.mul (continuous_subtype_val.comp (a.continuous.comp continuous_snd))) (continuous_fst.sub continuous_snd) (fun p => mul_pos hq (hα.trans_le (a p.2).property.1))))) (fun x => ‖φ x‖) B φ.integrable.norm hB ‖f‖ intro x y change ‖φ x * (f y * bilapKernel (q*(a y).val) (x-y))‖ ≤ ‖φ x‖*‖f‖*B (x-y) rw [norm_mul,norm_mul,mul_assoc] apply mul_le_mul_of_nonneg_left _ (norm_nonneg _) exact mul_le_mul (f.toBCF.norm_coe_le_norm y) (hbound _ ⟨mul_le_mul_of_nonneg_left (a y).property.1 hq.le, mul_le_mul_of_nonneg_left (a y).property.2 hq.le⟩ (x-y)) (norm_nonneg _) (norm_nonneg _) lemma pair_bilaplacian_timeValue (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (φ : SchwartzMap V ℂ) : (∫ x, φ x * Δ (Δ (timeValue a q f)) x) = ∫ x, (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) x * timeValue a q f x := by let K : V → SchwartzMap V ℂ := fun y : V => heatKernel (q*(a y).val) (mul_pos hq (hα.trans_le (a y).property.1)) have he₀ (y x : V) : K y x = kernel (q*(a y).val) x := by exact congrFun (heatKernel_eq_kernel _ _) x have he₄ (y x : V) : (Δ (Δ (K y) : SchwartzMap V ℂ) : SchwartzMap V ℂ) x = bilapKernel (q*(a y).val) x := (bilapKernel_eq_schwartz _ _ x).symm have h₀ := integrable_kernel_pair_prod hα a q hq f (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) have h₄ := integrable_bilap_pair_prod hα a q hq f φ simp_rw [bilaplacian_timeValue hα a q hq f] unfold timeValue have hp := KernelPairing.bilaplacian_pair_kernel φ f K (by simpa only [he₀] using h₀) (by simpa only [he₄] using h₄) simpa only [he₀,he₄] using hp lemma norm_bilap_integral_le (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (B : V → ℝ) (hB : Integrable B) (hbound : ∀ y z, ‖bilapKernel (q*(a y).val) z‖ ≤ B z) (x : V) : ‖∫ y, f y*bilapKernel (q*(a y).val) (x-y)‖ ≤ ‖f‖*(∫ y, B y) := by apply (norm_integral_le_integral_norm _).trans calc _ ≤ ∫ y, ‖f‖*B (x-y) := by apply integral_mono (integrable_bilapValue_integrand hα a q hq f x).norm ((hB.comp_sub_left x).const_mul _) intro y dsimp only rw [norm_mul] exact mul_le_mul (f.toBCF.norm_coe_le_norm y) (hbound y (x-y)) (norm_nonneg _) (norm_nonneg _) _ = _ := by rw [integral_const_mul, integral_sub_left_eq_self] lemma hasDerivAt_pair_timeValue (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (φ : SchwartzMap V ℂ) : HasDerivAt (fun s => ∫ x, φ x*timeValue a s f x) (-(∫ x, (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) x * timeValue a q (C0Multiplier.mul (coefficient a) f) x)) q := by let f₁ := C0Multiplier.mul (coefficient a) f let F := fun s : ℝ => fun x : V => φ x*timeValue a s f x let F' := fun s : ℝ => fun x : V => -(φ x*(∫ y, f₁ y*bilapKernel (s*(a y).val) (x-y))) have hay (y : V) : 0 < (a y).val := hα.trans_le (a y).property.1 have hβ : 0 < β := (hay 0).trans_le (a 0).property.2 have hFi (s : ℝ) (hs : 0 < s) : Integrable (F s) := by have hh := (integrable_kernel_pair_prod hα a s hs f φ).integral_prod_left simpa only [F,timeValue,integral_const_mul] using hh have hF'i (s : ℝ) (hs : 0 < s) : Integrable (F' s) := by have hh := (integrable_bilap_pair_prod hα a s hs f₁ φ).integral_prod_left.neg change Integrable (fun x => -(∫ y, φ x * (f₁ y * bilapKernel (s*(a y).val) (x-y)))) at hh simpa only [F',integral_const_mul] using hh obtain ⟨B,hB,_,_,hbound⟩ := bilapKernel_weighted_majorant (V := V) ((q/2)*α) ((2*q)*β) (by positivity) have hspos (s : ℝ) (hs : s ∈ Ioo (q/2) (2*q)) : 0 < s := lt_trans (by positivity) hs.1 have hsb (s : ℝ) (hs : s ∈ Ioo (q/2) (2*q)) (y : V) : s*(a y).val ∈ Icc ((q/2)*α) ((2*q)*β) := ⟨mul_le_mul hs.1.le (a y).property.1 hα.le (hspos s hs).le, mul_le_mul hs.2.le (a y).property.2 (hay y).le (by positivity)⟩ have hd := hasDerivAt_integral_of_dominated_loc_of_deriv_le (F := F) (F' := F') (bound := fun x => ‖φ x‖*(‖f₁‖*(∫ y, B y))) (s := Ioo (q/2) (2*q)) (Ioo_mem_nhds (by linarith) (by linarith)) (by filter_upwards [Ioi_mem_nhds hq] with s hs; exact (hFi s hs).aestronglyMeasurable) (hFi q hq) (hF'i q hq).aestronglyMeasurable (Eventually.of_forall (fun x s hs => by simp only [F',norm_neg,norm_mul] exact mul_le_mul_of_nonneg_left (norm_bilap_integral_le hα a s (hspos s hs) f₁ B hB (fun y z => hbound _ (hsb s hs y) z) x) (norm_nonneg _))) (φ.integrable.norm.mul_const _) (Eventually.of_forall (fun x s hs => by convert (hasDerivAt_timeValue hα a s (hspos s hs) f x).const_mul (φ x) using 1 simp only [F',f₁,C0Multiplier.mul_apply,coefficient_apply,mul_neg])) have he : (∫ x, F' q x) = -(∫ x, (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) x * timeValue a q f₁ x) := by unfold F' rw [integral_neg] congr 1 simpa only [bilaplacian_timeValue hα a q hq f₁] using pair_bilaplacian_timeValue hα a q hq f₁ φ rw [he] at hd exact hd.2 lemma integrable_pair_bilaplacian (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (φ : SchwartzMap V ℂ) : Integrable (fun x => φ x * Δ (Δ (timeValue a q f)) x) := by simp_rw [bilaplacian_timeValue hα a q hq f] have hh := (integrable_bilap_pair_prod hα a q hq f φ).integral_prod_left simpa only [integral_const_mul] using hh lemma integrable_pair_timeValue (hα : 0 < α) (a : C(V,Icc α β)) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (φ : V → ℂ) (hφ : Integrable φ) : Integrable (fun x => φ x * timeValue a q f x) := by rw [timeValue_eq_frozenOp hα a q hq f] exact KernelPairing.integrable_pair φ hφ (frozenOp hα a (ParabolicRadius.radius q) f).toBCF lemma weak_frozen_residual (hα : 0 < α) (a₀ a₁ : C(V,Icc α β)) (b : V →ᵇ ℂ) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (φ ψ : SchwartzMap V ℂ) (hψ : ∀ x, ψ x = ((a₁ x).val : ℂ)*φ x) : deriv (fun s => ∫ x, φ x*timeValue a₀ s f x) q + (∫ x, (Δ (Δ ψ : SchwartzMap V ℂ) : SchwartzMap V ℂ) x*timeValue a₀ q f x) + (∫ x, φ x*b x*timeValue a₀ q f x) = ∫ x, φ x*fullResidualCLM hα a₀ a₁ b (ParabolicRadius.radius q) f x := by let f₁ := C0Multiplier.mul (coefficient a₀) f have hD (x : V) : deriv (fun s => timeValue a₀ s f x) q = -Δ (Δ (timeValue a₀ q f₁)) x := by rw [(hasDerivAt_timeValue hα a₀ q hq f x).deriv, bilaplacian_timeValue hα a₀ q hq f₁] rfl have hid : Integrable (fun x => -(φ x * Δ (Δ (timeValue a₀ q f₁)) x)) := (integrable_pair_bilaplacian hα a₀ q hq f₁ φ).neg have his := integrable_pair_bilaplacian hα a₀ q hq f ψ have hiz := integrable_pair_timeValue hα a₀ q hq f (fun x => φ x*b x) (KernelPairing.integrable_pair φ φ.integrable b) have he : (∫ x, -(φ x * Δ (Δ (timeValue a₀ q f₁)) x) + ψ x * Δ (Δ (timeValue a₀ q f)) x + φ x*b x*timeValue a₀ q f x) = ∫ x, φ x*fullResidualCLM hα a₀ a₁ b (ParabolicRadius.radius q) f x := by apply integral_congr_ae filter_upwards with x rw [← frozen_pde_residual hα a₀ a₁ b q hq f x, hD, hψ] ring have hids : Integrable (fun x => -(φ x * Δ (Δ (timeValue a₀ q f₁)) x) + ψ x * Δ (Δ (timeValue a₀ q f)) x) := hid.add his rw [integral_add hids hiz, integral_add hid his, integral_neg, pair_bilaplacian_timeValue hα a₀ q hq f₁ φ, pair_bilaplacian_timeValue hα a₀ q hq f ψ] at he rw [(hasDerivAt_pair_timeValue hα a₀ q hq f φ).deriv] exact he end IntegrableKernelPairProd open scoped SchwartzMap Laplacian ZeroAtInfty BoundedContinuousFunction open BiharmonicKernel KernelPairing variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {α β : ℝ} def c0Time (hα : 0 < α) (a : V →ᵇ Icc α β) (q : ℝ) (f : C₀(V,ℂ)) : C₀(V,ℂ) := frozenOp hα a.toContinuousMap (ParabolicRadius.radius q) f lemma continuous_c0Time (hα : 0 < α) (a : V →ᵇ Icc α β) (f : C₀(V,ℂ)) : Continuous (fun q : ℝ => c0Time hα a q f) := continuous_frozenOp_comp hα (fun _ => a) ParabolicRadius.radius continuous_const (ParabolicRadius.continuous_comp _ continuous_id) f lemma c0Time_zero (hα : 0 < α) (a : V →ᵇ Icc α β) (f : C₀(V,ℂ)) : c0Time hα a 0 f = f := by unfold c0Time ParabolicRadius.radius rw [Real.zero_rpow (by norm_num : (1/4 : ℝ) ≠ 0)] exact frozen_zero _ _ _ _ _ _ f lemma c0Time_apply (hα : 0 < α) (a : V →ᵇ Icc α β) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (x : V) : c0Time hα a q f x = timeValue a.toContinuousMap q f x := (congrFun (timeValue_eq_frozenOp hα a.toContinuousMap q hq f) x).symm lemma hasDerivAt_pair_c0Time (hα : 0 < α) (a : V →ᵇ Icc α β) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (φ : SchwartzMap V ℂ) : HasDerivAt (fun s => pairCLM φ φ.integrable (c0Time hα a s f)) (-pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable (c0Time hα a q (C0Multiplier.mul (coefficient a.toContinuousMap) f))) q := by have he (s : ℝ) (hs : 0 < s) (g : C₀(V,ℂ)) (ψ : SchwartzMap V ℂ) : pairCLM ψ ψ.integrable (c0Time hα a s g) = ∫ x, ψ x*timeValue a.toContinuousMap s g x := by simp only [pairCLM_apply,c0Time_apply hα a s hs g] rw [he q hq] apply (hasDerivAt_pair_timeValue hα a.toContinuousMap q hq f φ).congr_of_eventuallyEq filter_upwards [Ioi_mem_nhds hq] with s hs exact he s hs f φ lemma pair_c0Time_integral (hα : 0 < α) (a : V →ᵇ Icc α β) (q : ℝ) (hq : 0 ≤ q) (f : C₀(V,ℂ)) (φ : SchwartzMap V ℂ) : pairCLM φ φ.integrable (c0Time hα a q f) = pairCLM φ φ.integrable f - ∫ s in (0 : ℝ)..q, pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable (c0Time hα a s (C0Multiplier.mul (coefficient a.toContinuousMap) f)) := by let L := pairCLM φ φ.integrable let M := pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable let f₁ := C0Multiplier.mul (coefficient a.toContinuousMap) f have hc := L.continuous.comp (continuous_c0Time hα a f) have hd := (M.continuous.comp (continuous_c0Time hα a f₁)).neg have hh := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hq hc.continuousOn (fun s hs => hasDerivAt_pair_c0Time hα a s hs.1 f φ) (hd.intervalIntegrable 0 q) change (∫ y in (0 : ℝ)..q, -M (c0Time hα a y f₁)) = L (c0Time hα a q f) - L (c0Time hα a 0 f) at hh rw [c0Time_zero, intervalIntegral.integral_neg] at hh change L (c0Time hα a q f) = L f - ∫ s in (0 : ℝ)..q, M (c0Time hα a s f₁) dsimp only [L] at hh ⊢ linear_combination -hh end FrozenParametrix namespace StrongWeightedVolterra variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] variable {T : ℝ} lemma continuous_unweighted_on_time (j : Time T →ᵇ E) : Continuous (fun t : Time T => unweighted j t.val) := by simp_rw [unweighted_at_time] exact (continuous_subtype_val.rpow_const (fun t => Or.inl t.property.1.ne')).smul j.continuous lemma stronglyMeasurable_unweighted (j : Time T →ᵇ E) : StronglyMeasurable (unweighted j) := by classical unfold unweighted apply StronglyMeasurable.dite (s := Time T) (f := fun t : Time T => t.val^(-3/4 : ℝ) • j t) (g := fun _ => (0 : E)) _ stronglyMeasurable_const measurableSet_Ioc exact ((continuous_subtype_val.rpow_const (fun t : Time T => Or.inl t.property.1.ne')).smul j.continuous).stronglyMeasurable lemma norm_unweighted_le (j : Time T →ᵇ E) (r : ℝ) (hr : 0 < r) : ‖unweighted j r‖ ≤ r^(-3/4 : ℝ)*‖j‖ := by by_cases hrt : r ≤ T · rw [unweighted,dite_eq_left (show r ∈ Ioc 0 T from ⟨hr,hrt⟩),norm_smul, Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg hr.le _)] exact mul_le_mul_of_nonneg_left (j.norm_coe_le_norm _) (Real.rpow_nonneg hr.le _) · rw [unweighted,dite_eq_right (by simpa only [mem_Ioc,not_and,not_le] using fun _ => lt_of_not_ge hrt),norm_zero] positivity lemma intervalIntegrable_unweighted (j : Time T →ᵇ E) (t : ℝ) (ht : 0 ≤ t) : IntervalIntegrable (unweighted j) volume 0 t := by apply (intervalIntegrable_iff_integrableOn_Ioo_of_le ht).mpr have hp : IntegrableOn (fun r : ℝ => r^(-3/4 : ℝ)*‖j‖) (Ioo 0 t) := by rcases eq_or_lt_of_le ht with he | ht' · subst t; simp · exact ((intervalIntegral.integrableOn_Ioo_rpow_iff ht').mpr (by norm_num)).mul_const ‖j‖ apply hp.mono' (stronglyMeasurable_unweighted j).aestronglyMeasurable filter_upwards [ae_restrict_mem measurableSet_Ioo] with r hr exact norm_unweighted_le j r hr.1 end StrongWeightedVolterra namespace VolterraIntegral section IntegralTriangleLeft variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] variable {T : ℝ} lemma integral_triangle_left (F : ℝ → ℝ → E) (hT : 0 ≤ T) (r : ℝ) (hr : r ∈ Icc 0 T) : (∫ s in (0 : ℝ)..T, if s < r then F r s else 0) = ∫ s in (0 : ℝ)..r, F r s := by rw [intervalIntegral.integral_of_le hT, intervalIntegral.integral_of_le hr.1] change (∫ s in Ioc (0 : ℝ) T, (Iio r).indicator (F r) s) = _ rw [integral_indicator measurableSet_Iio, Measure.restrict_restrict measurableSet_Iio] have he : Iio r ∩ Ioc (0 : ℝ) T = Ioo 0 r := by ext x simp only [mem_inter_iff, mem_Iio, mem_Ioc, mem_Ioo] constructor · rintro ⟨hxr,hx0,_⟩; exact ⟨hx0,hxr⟩ · rintro ⟨hx0,hxr⟩; exact ⟨hxr,hx0,hxr.le.trans hr.2⟩ rw [he, integral_Ioc_eq_integral_Ioo] lemma integral_triangle_right (F : ℝ → ℝ → E) (hT : 0 ≤ T) (s : ℝ) (hs : s ∈ Icc 0 T) : (∫ r in (0 : ℝ)..T, if s < r then F r s else 0) = ∫ r in s..T, F r s := by rw [intervalIntegral.integral_of_le hT, intervalIntegral.integral_of_le hs.2] change (∫ r in Ioc (0 : ℝ) T, (Ioi s).indicator (fun r => F r s) r) = _ rw [integral_indicator measurableSet_Ioi, Measure.restrict_restrict measurableSet_Ioi] have he : Ioi s ∩ Ioc (0 : ℝ) T = Ioc s T := by ext x simp only [mem_inter_iff, mem_Ioi, mem_Ioc] constructor · rintro ⟨hsx,_,hxT⟩; exact ⟨hsx,hxT⟩ · rintro ⟨hsx,hxT⟩; exact ⟨hsx,lt_of_le_of_lt hs.1 hsx,hxT⟩ rw [he] lemma integral_triangle_swap (F : ℝ → ℝ → E) (hT : 0 ≤ T) (hF : IntegrableOn (fun p : ℝ × ℝ => if p.2 < p.1 then F p.1 p.2 else 0) (Ioc 0 T ×ˢ Ioc 0 T)) : (∫ r in (0 : ℝ)..T, ∫ s in (0 : ℝ)..r, F r s) = ∫ s in (0 : ℝ)..T, ∫ r in s..T, F r s := by have hh := intervalIntegral_intervalIntegral_swap (F := fun r s => if s < r then F r s else 0) (a := 0) (b := T) (c := 0) (d := T) (by simpa only [uIoc_of_le hT, Function.uncurry_def] using hF) calc _ = ∫ r in (0 : ℝ)..T, ∫ s in (0 : ℝ)..T, if s < r then F r s else 0 := by apply intervalIntegral.integral_congr intro r hr exact (integral_triangle_left F hT r (by simpa only [uIcc_of_le hT] using hr)).symm _ = _ := hh _ = _ := by apply intervalIntegral.integral_congr intro s hs exact integral_triangle_right F hT s (by simpa only [uIcc_of_le hT] using hs) end IntegralTriangleLeft section IntegrableOnTriangleOfBound variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] variable {T : ℝ} omit [NormedSpace ℝ E] in lemma integrableOn_triangle_of_bound (F : ℝ → ℝ → E) (hF : AEStronglyMeasurable (fun p : ℝ × ℝ => if p.2 < p.1 then F p.1 p.2 else 0) (volume.restrict (Ioc 0 T ×ˢ Ioc 0 T))) (B : ℝ → ℝ) (hB : IntegrableOn B (Ioc 0 T)) (hB0 : ∀ s ∈ Ioc 0 T, 0 ≤ B s) (hbound : ∀ r ∈ Ioc 0 T, ∀ s ∈ Ioc 0 T, s < r → ‖F r s‖ ≤ B s) : IntegrableOn (fun p : ℝ × ℝ => if p.2 < p.1 then F p.1 p.2 else 0) (Ioc 0 T ×ˢ Ioc 0 T) := by have hb : IntegrableOn (fun p : ℝ × ℝ => B p.2) (Ioc 0 T ×ˢ Ioc 0 T) := by have hh := hB.comp_snd (volume.restrict (Ioc (0 : ℝ) T)) simpa only [Measure.prod_restrict, Measure.volume_eq_prod, IntegrableOn] using hh refine hb.mono' hF ?_ filter_upwards [ae_restrict_mem (measurableSet_Ioc.prod measurableSet_Ioc)] with p hp by_cases hlt : p.2 < p.1 · simpa only [ite_eq_left hlt] using hbound p.1 hp.1 p.2 hp.2 hlt · simpa only [ite_eq_right hlt, norm_zero] using hB0 p.2 hp.2 lemma duhamel_integral_identity (A F : ℝ → ℝ → E) (d : ℝ → E) (hT : 0 ≤ T) (hA : IntervalIntegrable (A T) volume 0 T) (hd : IntervalIntegrable d volume 0 T) (hF : IntegrableOn (fun p : ℝ × ℝ => if p.2 < p.1 then F p.1 p.2 else 0) (Ioc 0 T ×ˢ Ioc 0 T)) (hFTC : ∀ s ∈ Ioo 0 T, A T s - d s = ∫ r in s..T, F r s) : (∫ s in (0 : ℝ)..T, A T s) = (∫ s in (0 : ℝ)..T, d s) + ∫ r in (0 : ℝ)..T, ∫ s in (0 : ℝ)..r, F r s := by rw [integral_triangle_swap F hT hF] have he : (∫ s in (0 : ℝ)..T, A T s - d s) = ∫ s in (0 : ℝ)..T, ∫ r in s..T, F r s := by exact intervalIntegral.integral_congr_Ioo_of_le hT hFTC rw [intervalIntegral.integral_sub hA hd] at he exact sub_eq_iff_eq_add'.mp he end IntegrableOnTriangleOfBound variable {E : Type*} [NormedAddCommGroup E] variable {T : ℝ} def sourceExtend (F : ℝ × Ioc (0 : ℝ) T → E) (r s : ℝ) : E := if hs : s ∈ Ioc (0 : ℝ) T then F (r,⟨s,hs⟩) else 0 lemma sourceExtend_apply (F : ℝ × Ioc (0 : ℝ) T → E) (r s : ℝ) (hs : s ∈ Ioc (0 : ℝ) T) : sourceExtend F r s = F (r,⟨s,hs⟩) := by rw [sourceExtend,dite_eq_left hs] lemma stronglyMeasurable_sourceExtend (F : ℝ × Ioc (0 : ℝ) T → E) (hF : Continuous F) : StronglyMeasurable (fun p : ℝ × ℝ => sourceExtend F p.1 p.2) := by classical let D : Set (ℝ × ℝ) := {p | p.2 ∈ Ioc (0 : ℝ) T} let G : D → ℝ × Ioc (0 : ℝ) T := fun p => (p.val.1,⟨p.val.2,p.property⟩) have hG : Continuous G := by exact (continuous_fst.comp continuous_subtype_val).prodMk ((continuous_snd.comp continuous_subtype_val).subtype_mk _) have hD : MeasurableSet D := measurableSet_Ioc.preimage measurable_snd exact StronglyMeasurable.dite (s := D) (f := F ∘ G) (g := fun _ => 0) (hF.comp hG).stronglyMeasurable stronglyMeasurable_const hD lemma integrableOn_sourceExtend (F : ℝ × Ioc (0 : ℝ) T → E) (hF : Continuous F) (B : ℝ → ℝ) (hB : IntegrableOn B (Ioc 0 T)) (hbound : ∀ (r : ℝ) (s : Ioc (0 : ℝ) T), ‖F (r,s)‖ ≤ B s.val) (r : ℝ) : IntegrableOn (sourceExtend F r) (Ioc 0 T) := by apply hB.mono' · exact ((stronglyMeasurable_sourceExtend F hF).comp_measurable (measurable_const.prodMk measurable_id)).aestronglyMeasurable · filter_upwards [ae_restrict_mem measurableSet_Ioc] with s hs rw [sourceExtend_apply F r s hs] exact hbound r ⟨s,hs⟩ lemma integrableOn_sourceExtend_triangle (F : ℝ × Ioc (0 : ℝ) T → E) (hF : Continuous F) (B : ℝ → ℝ) (hB : IntegrableOn B (Ioc 0 T)) (hbound : ∀ (r : ℝ) (s : Ioc (0 : ℝ) T), ‖F (r,s)‖ ≤ B s.val) : IntegrableOn (fun p : ℝ × ℝ => if p.2 < p.1 then sourceExtend F p.1 p.2 else 0) (Ioc 0 T ×ˢ Ioc 0 T) := by apply integrableOn_triangle_of_bound (sourceExtend F) (((stronglyMeasurable_sourceExtend F hF).ite (measurableSet_lt measurable_snd measurable_fst) stronglyMeasurable_const).aestronglyMeasurable) B hB · intro s hs exact (norm_nonneg _).trans (hbound 0 ⟨s,hs⟩) · intro r _ s hs _ rw [sourceExtend_apply F r s hs] exact hbound r ⟨s,hs⟩ end VolterraIntegral namespace FrozenParametrix open scoped ZeroAtInfty BoundedContinuousFunction open StrongWeightedVolterra ParametrixVolterra variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {α β : ℝ} lemma continuous_c0Time_comp {X : Type*} [TopologicalSpace X] [FirstCountableTopology X] (hα : 0 < α) (a : X → V →ᵇ Icc α β) (q : X → ℝ) (f : X → C₀(V,ℂ)) (ha : Continuous a) (hq : Continuous q) (hf : Continuous f) : Continuous (fun x => c0Time hα (a x) (q x) (f x)) := by have h := continuous_apply_of_strong (fun x => frozenCLM hα (a x).toContinuousMap (ParabolicRadius.radius (q x))) (fun d => continuous_frozenOp_comp hα a (fun x => ParabolicRadius.radius (q x)) ha (ParabolicRadius.continuous_comp q hq) d) (frozenBound (V := V) (β := β) hα) (frozenBound_nonneg hα) (fun x => norm_frozenCLM_le hα (a x).toContinuousMap (ParabolicRadius.radius (q x))) have hh := h.comp (continuous_id.prodMk hf) simpa only [Function.comp_def, id_eq, frozenCLM_apply, c0Time] using hh lemma norm_c0Time_le (hα : 0 < α) (a : V →ᵇ Icc α β) (q : ℝ) (f : C₀(V,ℂ)) : ‖c0Time hα a q f‖ ≤ frozenBound (V := V) (β := β) hα * ‖f‖ := by exact (frozenCLM hα a.toContinuousMap (ParabolicRadius.radius q)).le_opNorm f |>.trans (mul_le_mul_of_nonneg_right (norm_frozenCLM_le hα a.toContinuousMap _) (norm_nonneg _)) end FrozenParametrix namespace ParametrixVolterra section ContinuousSourceMultiply open scoped ZeroAtInfty BoundedContinuousFunction SchwartzMap Laplacian open FrozenParametrix StrongWeightedVolterra KernelPairing variable {V : Type} [NormedAddCommGroup V] variable {T α β : ℝ} variable (A : C(Slab T,V →ᵇ Icc α β)) lemma continuous_sourceMultiply (j : Time T →ᵇ C₀(V,ℂ)) : Continuous (fun t : Time T => C0Multiplier.mul (coefficient (A (now t)).toContinuousMap) (j t)) := C0Multiplier.continuous_mul_joint.comp ((continuous_coefficient_comp (fun t => A (now t)) (A.continuous.comp continuous_now)).prodMk j.continuous) lemma norm_sourceMultiply_le (j : Time T →ᵇ C₀(V,ℂ)) (t : Time T) : ‖C0Multiplier.mul (coefficient (A (now t)).toContinuousMap) (j t)‖ ≤ max |α| |β| * ‖j‖ := (C0Multiplier.norm_mul_le _ _).trans (mul_le_mul (norm_coefficient_le _) (j.norm_coe_le_norm t) (norm_nonneg _) ((abs_nonneg α).trans (le_max_left _ _))) def sourceMultiply (j : Time T →ᵇ C₀(V,ℂ)) : Time T →ᵇ C₀(V,ℂ) := .ofNormedAddCommGroup _ (continuous_sourceMultiply A j) _ (norm_sourceMultiply_le A j) lemma sourceMultiply_apply (j : Time T →ᵇ C₀(V,ℂ)) (t : Time T) : sourceMultiply A j t = C0Multiplier.mul (coefficient (A (now t)).toContinuousMap) (j t) := rfl lemma unweighted_sourceMultiply (j : Time T →ᵇ C₀(V,ℂ)) (t : Time T) : unweighted (sourceMultiply A j) t.val = C0Multiplier.mul (coefficient (A (now t)).toContinuousMap) (unweighted j t.val) := by rw [unweighted_at_time,unweighted_at_time,sourceMultiply_apply] ext x simp only [ZeroAtInftyContinuousMap.smul_apply,C0Multiplier.mul_apply] rw [mul_smul_comm] end ContinuousSourceMultiply open scoped ZeroAtInfty BoundedContinuousFunction SchwartzMap Laplacian open FrozenParametrix StrongWeightedVolterra KernelPairing VolterraIntegral variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {T α β : ℝ} variable (hα : 0 < α) (A : C(Slab T,V →ᵇ Icc α β)) def sourceFrozen (j : Time T →ᵇ C₀(V,ℂ)) (p : ℝ × Time T) : C₀(V,ℂ) := c0Time hα (A (now p.2)) (p.1-p.2.val) (unweighted j p.2.val) lemma continuous_sourceFrozen (j : Time T →ᵇ C₀(V,ℂ)) : Continuous (sourceFrozen hα A j) := continuous_c0Time_comp hα _ _ _ (A.continuous.comp (continuous_now.comp continuous_snd)) (continuous_fst.sub (continuous_subtype_val.comp continuous_snd)) ((continuous_unweighted_on_time j).comp continuous_snd) lemma norm_sourceFrozen_le (j : Time T →ᵇ C₀(V,ℂ)) (p : ℝ × Time T) : ‖sourceFrozen hα A j p‖ ≤ frozenBound (V := V) (β := β) hα * (p.2.val^(-3/4 : ℝ)*‖j‖) := (norm_c0Time_le hα _ _ _).trans (mul_le_mul_of_nonneg_left (norm_unweighted_le j p.2.val p.2.property.1) (frozenBound_nonneg hα)) lemma integrableOn_sourceFrozen_bound (j : Time T →ᵇ C₀(V,ℂ)) : IntegrableOn (fun s : ℝ => frozenBound (V := V) (β := β) hα*(s^(-3/4 : ℝ)*‖j‖)) (Ioc 0 T) := by have h := (intervalIntegral.intervalIntegrable_rpow' (a := 0) (b := T) (by norm_num : (-1:ℝ) < -3/4)).mul_const ‖j‖ |>.const_mul (frozenBound (V := V) (β := β) hα) exact h.1 lemma integrableOn_sourceFrozen_triangle (j : Time T →ᵇ C₀(V,ℂ)) : IntegrableOn (fun p : ℝ × ℝ => if p.2 < p.1 then sourceExtend (sourceFrozen hα A j) p.1 p.2 else 0) (Ioc 0 T ×ˢ Ioc 0 T) := integrableOn_sourceExtend_triangle _ (continuous_sourceFrozen hα A j) _ (integrableOn_sourceFrozen_bound hα j) (fun r s => norm_sourceFrozen_le hα A j (r,s)) lemma intervalIntegrable_sourceFrozen (j : Time T →ᵇ C₀(V,ℂ)) (r t : ℝ) (ht : t ∈ Icc 0 T) : IntervalIntegrable (sourceExtend (sourceFrozen hα A j) r) volume 0 t := by apply (intervalIntegrable_iff_integrableOn_Ioc_of_le ht.1).mpr exact (integrableOn_sourceExtend _ (continuous_sourceFrozen hα A j) _ (integrableOn_sourceFrozen_bound hα j) (fun r s => norm_sourceFrozen_le hα A j (r,s)) r).mono_set (Ioc_subset_Ioc le_rfl ht.2) def duhamel (j : Time T →ᵇ C₀(V,ℂ)) (t : ℝ) : C₀(V,ℂ) := ∫ s in (0 : ℝ)..t, sourceExtend (sourceFrozen hα A j) t s lemma sourceFrozen_eq_frozenRaw (hT : 0 ≤ T) (j : Time T →ᵇ C₀(V,ℂ)) (t : Time T) (s : ℝ) (hs : s ∈ Ioo 0 t.val) : sourceExtend (sourceFrozen hα A j) t.val s = frozenRaw hα A hT t s (unweighted j s) := by have hsT : s ∈ Ioc (0 : ℝ) T := ⟨hs.1,hs.2.le.trans t.property.2⟩ have hsT' : s ∈ Icc (0 : ℝ) T := ⟨hs.1.le,hsT.2⟩ rw [sourceExtend_apply _ _ _ hsT,frozenRaw,ite_eq_left hs,projIcc_of_mem hT hsT'] rfl lemma duhamel_eq_value (hT : 0 ≤ T) (j : Time T →ᵇ C₀(V,ℂ)) (t : Time T) : duhamel hα A j t.val = MildVolterra.value (mildKernel hα A) j t := by rw [duhamel,MildVolterra.value_eq_integral] apply intervalIntegral.integral_congr_Ioo_of_le t.property.1.le intro s hs dsimp only rw [rawKernel_eq_frozenRaw hα A hT] exact sourceFrozen_eq_frozenRaw hα A hT j t s hs lemma continuousOn_duhamel (hT : 0 ≤ T) (j : Time T →ᵇ C₀(V,ℂ)) : ContinuousOn (duhamel hα A j) (Ioc 0 T) := by apply continuousOn_iff_continuous_domRestrict.mpr change Continuous (fun t : Time T => duhamel hα A j t.val) simp_rw [duhamel_eq_value hα A hT j] exact MildVolterra.continuous_value (mildKernel hα A) (frozenBound (V := V) (β := β) hα) (frozenBound_nonneg hα) (norm_mildKernel_le hα A) j lemma continuousAt_duhamel (hT : 0 ≤ T) (j : Time T →ᵇ C₀(V,ℂ)) (t : ℝ) (ht : t ∈ Ioo 0 T) : ContinuousAt (duhamel hα A j) t := (continuousOn_duhamel hα A hT j).continuousAt (Ioc_mem_nhds ht.1 ht.2) lemma integrableOn_pair_sourceFrozen_triangle (j : Time T →ᵇ C₀(V,ℂ)) (L : C₀(V,ℂ) →L[ℂ] ℂ) : IntegrableOn (fun p : ℝ × ℝ => if p.2 < p.1 then L (sourceExtend (sourceFrozen hα A j) p.1 p.2) else 0) (Ioc 0 T ×ˢ Ioc 0 T) := by have hh := L.integrable_comp (integrableOn_sourceFrozen_triangle hα A j) simpa only [IntegrableOn,apply_ite,map_zero] using hh lemma sourceFrozen_weakFTC (j : Time T →ᵇ C₀(V,ℂ)) (φ : SchwartzMap V ℂ) (t : ℝ) (ht : t ∈ Icc 0 T) (s : ℝ) (hs : s ∈ Ioo 0 t) : pairCLM φ φ.integrable (sourceExtend (sourceFrozen hα A j) t s) - pairCLM φ φ.integrable (unweighted j s) = ∫ r in s..t, -pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable (sourceExtend (sourceFrozen hα A (sourceMultiply A j)) r s) := by have hsT : s ∈ Ioc (0 : ℝ) T := ⟨hs.1,hs.2.le.trans ht.2⟩ let s₁ : Time T := ⟨s,hsT⟩ let M := pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable let f₁ := C0Multiplier.mul (coefficient (A (now s₁)).toContinuousMap) (unweighted j s) have he (r : ℝ) : sourceExtend (sourceFrozen hα A (sourceMultiply A j)) r s = c0Time hα (A (now s₁)) (r-s) f₁ := by rw [sourceExtend_apply _ _ _ hsT] change c0Time hα (A (now s₁)) (r-s) (unweighted (sourceMultiply A j) s₁.val) = _ rw [unweighted_sourceMultiply] have hh := pair_c0Time_integral hα (A (now s₁)) (t-s) (sub_nonneg.mpr hs.2.le) (unweighted j s) φ simp_rw [he] change pairCLM φ φ.integrable (sourceExtend (sourceFrozen hα A j) t s) - pairCLM φ φ.integrable (unweighted j s) = ∫ r in s..t, -M (c0Time hα (A (now s₁)) (r-s) f₁) rw [intervalIntegral.integral_neg,intervalIntegral.integral_comp_sub_right (fun q => M (c0Time hα (A (now s₁)) q f₁)) s,sub_self, sourceExtend_apply _ _ _ hsT] change pairCLM φ φ.integrable (c0Time hα (A (now s₁)) (t-s) (unweighted j s)) - pairCLM φ φ.integrable (unweighted j s) = -∫ q in (0 : ℝ)..t-s, M (c0Time hα (A (now s₁)) q f₁) exact sub_eq_iff_eq_add.mpr (by convert hh using 1; abel) lemma pair_duhamel_integral (j : Time T →ᵇ C₀(V,ℂ)) (φ : SchwartzMap V ℂ) (t : ℝ) (ht : t ∈ Icc 0 T) : pairCLM φ φ.integrable (duhamel hα A j t) = (∫ s in (0 : ℝ)..t, pairCLM φ φ.integrable (unweighted j s)) - ∫ r in (0 : ℝ)..t, pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable (duhamel hα A (sourceMultiply A j) r) := by let L := pairCLM φ φ.integrable let M := pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable let K := sourceExtend (sourceFrozen hα A j) let Q := sourceExtend (sourceFrozen hα A (sourceMultiply A j)) have hK := intervalIntegrable_sourceFrozen hα A j t t ht have hu := intervalIntegrable_unweighted j t ht.1 have hA : IntervalIntegrable (fun s => L (K t s)) volume 0 t := ⟨L.integrable_comp hK.1,L.integrable_comp hK.2⟩ have hd : IntervalIntegrable (fun s => L (unweighted j s)) volume 0 t := ⟨L.integrable_comp hu.1,L.integrable_comp hu.2⟩ have hQ := (integrableOn_pair_sourceFrozen_triangle hα A (sourceMultiply A j) M).neg have hQt : IntegrableOn (fun p : ℝ × ℝ => if p.2 < p.1 then -M (Q p.1 p.2) else 0) (Ioc 0 t ×ˢ Ioc 0 t) := by have he : (fun p : ℝ × ℝ => if p.2 < p.1 then -M (Q p.1 p.2) else 0) = (fun p : ℝ × ℝ => -(if p.2 < p.1 then M (Q p.1 p.2) else 0)) := by funext p; split_ifs <;> simp rw [he] exact hQ.mono_set (Set.prod_mono (Ioc_subset_Ioc le_rfl ht.2) (Ioc_subset_Ioc le_rfl ht.2)) have hh := duhamel_integral_identity (fun r s => L (K r s)) (fun r s => -M (Q r s)) (fun s => L (unweighted j s)) ht.1 hA hd hQt (sourceFrozen_weakFTC hα A j φ t ht) have he : (∫ r in (0 : ℝ)..t, ∫ s in (0 : ℝ)..r, -M (Q r s)) = -(∫ r in (0 : ℝ)..t, M (duhamel hα A (sourceMultiply A j) r)) := by rw [← intervalIntegral.integral_neg] apply intervalIntegral.integral_congr_Ioo_of_le ht.1 intro r hr dsimp only rw [intervalIntegral.integral_neg] congr 1 exact M.intervalIntegral_comp_comm (intervalIntegrable_sourceFrozen hα A (sourceMultiply A j) r r ⟨hr.1.le,hr.2.le.trans ht.2⟩) rw [he] at hh rw [L.intervalIntegral_comp_comm hK] at hh exact hh lemma duhamel_zero (j : Time T →ᵇ C₀(V,ℂ)) : duhamel hα A j 0 = 0 := by simp [duhamel] lemma norm_duhamel_le (hT : 0 ≤ T) (j : Time T →ᵇ C₀(V,ℂ)) (t : ℝ) (ht : t ∈ Icc 0 T) : ‖duhamel hα A j t‖ ≤ t^(1/4 : ℝ)* ((∫ u : UnitTime, MildVolterra.massWeight u)*frozenBound (V := V) (β := β) hα*‖j‖) := by rcases eq_or_lt_of_le ht.1 with he | ht0 · subst t; rw [duhamel_zero,norm_zero,Real.zero_rpow (by norm_num : (1/4 : ℝ) ≠ 0),zero_mul] · let t₁ : Time T := ⟨t,ht0,ht.2⟩ change ‖duhamel hα A j t₁.val‖ ≤ _ rw [duhamel_eq_value hα A hT] exact MildVolterra.norm_value_le (mildKernel hα A) (frozenBound (V := V) (β := β) hα) (frozenBound_nonneg hα) (norm_mildKernel_le hα A) j t₁ lemma continuousWithinAt_duhamel_zero (hT : 0 ≤ T) (j : Time T →ᵇ C₀(V,ℂ)) : ContinuousWithinAt (duhamel hα A j) (Icc 0 T) 0 := by change Tendsto (duhamel hα A j) (𝓝[Icc 0 T] 0) (𝓝 (duhamel hα A j 0)) rw [duhamel_zero] apply tendsto_zero_iff_norm_tendsto_zero.mpr apply squeeze_zero' (Eventually.of_forall (fun t => norm_nonneg _)) (show ∀ᶠ t in 𝓝[Icc 0 T] 0, ‖duhamel hα A j t‖ ≤ t^(1/4 : ℝ)* ((∫ u : UnitTime, MildVolterra.massWeight u)*frozenBound (V := V) (β := β) hα*‖j‖) from by filter_upwards [self_mem_nhdsWithin] with t ht exact norm_duhamel_le hα A hT j t ht) have hc : Continuous (fun t : ℝ => t^(1/4 : ℝ)* ((∫ u : UnitTime, MildVolterra.massWeight u)*frozenBound (V := V) (β := β) hα*‖j‖)) := (Real.continuous_rpow_const (by norm_num : (0:ℝ) ≤ 1/4)).mul_const _ simpa only [Real.zero_rpow (by norm_num : (1/4 : ℝ) ≠ 0),zero_mul] using (hc.tendsto 0).mono_left nhdsWithin_le_nhds lemma continuousOn_duhamel_Icc (hT : 0 ≤ T) (j : Time T →ᵇ C₀(V,ℂ)) : ContinuousOn (duhamel hα A j) (Icc 0 T) := by intro t ht rcases eq_or_lt_of_le ht.1 with he | ht0 · subst t; exact continuousWithinAt_duhamel_zero hα A hT j · apply ((continuousOn_duhamel hα A hT j) t ⟨ht0,ht.2⟩).mono_of_mem_nhdsWithin filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Ioi_mem_nhds ht0)] with r hr hr0 exact ⟨hr0,hr.2⟩ lemma intervalIntegrable_duhamel (hT : 0 ≤ T) (j : Time T →ᵇ C₀(V,ℂ)) (t : ℝ) (ht : t ∈ Icc 0 T) : IntervalIntegrable (duhamel hα A j) volume 0 t := by apply ContinuousOn.intervalIntegrable rw [uIcc_of_le ht.1] exact (continuousOn_duhamel_Icc hα A hT j).mono (Icc_subset_Icc le_rfl ht.2) lemma hasDerivAt_pair_duhamel (hT : 0 ≤ T) (j : Time T →ᵇ C₀(V,ℂ)) (φ : SchwartzMap V ℂ) (t : ℝ) (ht : t ∈ Ioo 0 T) : HasDerivAt (fun r => pairCLM φ φ.integrable (duhamel hα A j r)) (pairCLM φ φ.integrable (unweighted j t) - pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable (duhamel hα A (sourceMultiply A j) t)) t := by let L := pairCLM φ φ.integrable let M := pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable have hu := intervalIntegrable_unweighted j t ht.1.le have hD := intervalIntegrable_duhamel hα A hT (sourceMultiply A j) t ⟨ht.1.le,ht.2.le⟩ have huc : ContinuousOn (unweighted j) (Ioc 0 T) := continuousOn_iff_continuous_domRestrict.mpr (continuous_unweighted_on_time j) have hc₁ : ContinuousOn (fun r => L (unweighted j r)) (Ioo 0 T) := L.continuous.comp_continuousOn (huc.mono Ioo_subset_Ioc_self) have hc₂ : ContinuousOn (fun r => M (duhamel hα A (sourceMultiply A j) r)) (Ioo 0 T) := M.continuous.comp_continuousOn ((continuousOn_duhamel hα A hT _).mono Ioo_subset_Ioc_self) have hd₁ := intervalIntegral.integral_hasDerivAt_right (show IntervalIntegrable (fun r => L (unweighted j r)) volume 0 t from ⟨L.integrable_comp hu.1,L.integrable_comp hu.2⟩) (ContinuousOn.stronglyMeasurableAtFilter isOpen_Ioo hc₁ t ht) (hc₁.continuousAt (Ioo_mem_nhds ht.1 ht.2)) have hd₂ := intervalIntegral.integral_hasDerivAt_right (show IntervalIntegrable (fun r => M (duhamel hα A (sourceMultiply A j) r)) volume 0 t from ⟨M.integrable_comp hD.1,M.integrable_comp hD.2⟩) (ContinuousOn.stronglyMeasurableAtFilter isOpen_Ioo hc₂ t ht) (hc₂.continuousAt (Ioo_mem_nhds ht.1 ht.2)) apply (hd₁.sub hd₂).congr_of_eventuallyEq filter_upwards [Ioo_mem_nhds ht.1 ht.2] with r hr exact pair_duhamel_integral hα A j φ r ⟨hr.1.le,hr.2.le⟩ end ParametrixVolterra namespace FrozenParametrix open scoped SchwartzMap Laplacian ZeroAtInfty BoundedContinuousFunction open BiharmonicKernel KernelPairing variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {α β : ℝ} lemma pair_fullResidual (hα : 0 < α) (a₀ a₁ : V →ᵇ Icc α β) (b : V →ᵇ ℂ) (q : ℝ) (hq : 0 < q) (f : C₀(V,ℂ)) (φ ψ : SchwartzMap V ℂ) (hψ : ∀ x, ψ x = ((a₁ x).val : ℂ)*φ x) : pairCLM φ φ.integrable (fullResidualCLM hα a₀.toContinuousMap a₁.toContinuousMap b (ParabolicRadius.radius q) f) = -pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable (c0Time hα a₀ q (C0Multiplier.mul (coefficient a₀.toContinuousMap) f)) + pairCLM (Δ (Δ ψ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ ψ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable (c0Time hα a₀ q f) + pairCLM (fun x => φ x*b x) (integrable_pair φ φ.integrable b) (c0Time hα a₀ q f) := by have h := weak_frozen_residual hα a₀.toContinuousMap a₁.toContinuousMap b q hq f φ ψ hψ rw [(hasDerivAt_pair_timeValue hα a₀.toContinuousMap q hq f φ).deriv] at h simpa only [pairCLM_apply,c0Time_apply hα a₀ q hq] using h.symm end FrozenParametrix namespace ParametrixVolterra open scoped ZeroAtInfty BoundedContinuousFunction SchwartzMap Laplacian open FrozenParametrix StrongWeightedVolterra KernelPairing VolterraIntegral variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {T α β : ℝ} variable (hα : 0 < α) (A : C(Slab T,V →ᵇ Icc α β)) (b : C(Slab T,V →ᵇ ℂ)) def adjointPair (t : Time T) (φ ψ : SchwartzMap V ℂ) : C₀(V,ℂ) →L[ℂ] ℂ := pairCLM (Δ (Δ ψ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ ψ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable + pairCLM (fun x => φ x*b (now t) x) (integrable_pair φ φ.integrable (b (now t))) lemma pair_residualKernel (hT : 0 ≤ T) (j : Time T →ᵇ C₀(V,ℂ)) (t : Time T) (φ ψ : SchwartzMap V ℂ) (hψ : ∀ x, ψ x = ((A (now t) x).val : ℂ)*φ x) (s : ℝ) (hs : s ∈ Ioo 0 t.val) : pairCLM φ φ.integrable (residualKernel hα A b hT t s (unweighted j s)) = -pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable (sourceExtend (sourceFrozen hα A (sourceMultiply A j)) t.val s) + adjointPair b t φ ψ (sourceExtend (sourceFrozen hα A j) t.val s) := by have hsT : s ∈ Ioc (0:ℝ) T := ⟨hs.1,hs.2.le.trans t.property.2⟩ have hsT' : s ∈ Icc (0:ℝ) T := ⟨hs.1.le,hsT.2⟩ let s₁ : Time T := ⟨s,hsT⟩ rw [residualKernel,ite_eq_left hs,projIcc_of_mem hT hsT'] rw [sourceExtend_apply _ _ _ hsT,sourceExtend_apply _ _ _ hsT] change pairCLM φ φ.integrable (actualResidual hα A b t (now s₁) (unweighted j s)) = _ dsimp only [sourceFrozen] rw [show unweighted (sourceMultiply A j) s = C0Multiplier.mul (coefficient (A (now s₁)).toContinuousMap) (unweighted j s) from unweighted_sourceMultiply A j s₁] simpa only [actualResidual,adjointPair,add_apply,now,s₁,add_assoc] using pair_fullResidual hα (A (now s₁)) (A (now t)) (b (now t)) (t.val-s) (sub_pos.mpr hs.2) (unweighted j s) φ ψ hψ lemma pair_integral_residualKernel (hT : 0 ≤ T) (j : Time T →ᵇ C₀(V,ℂ)) (t : Time T) (hi : IntervalIntegrable (fun s => residualKernel hα A b hT t s (unweighted j s)) volume 0 t.val) (φ ψ : SchwartzMap V ℂ) (hψ : ∀ x, ψ x = ((A (now t) x).val : ℂ)*φ x) : pairCLM φ φ.integrable (∫ s in (0:ℝ)..t.val, residualKernel hα A b hT t s (unweighted j s)) = -pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable (duhamel hα A (sourceMultiply A j) t.val) + adjointPair b t φ ψ (duhamel hα A j t.val) := by let L := pairCLM φ φ.integrable let M := pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable let N := adjointPair b t φ ψ have hi₁ := intervalIntegrable_sourceFrozen hα A (sourceMultiply A j) t.val t.val ⟨t.property.1.le,t.property.2⟩ have hi₂ := intervalIntegrable_sourceFrozen hα A j t.val t.val ⟨t.property.1.le,t.property.2⟩ have hm : IntervalIntegrable (fun s => M (sourceExtend (sourceFrozen hα A (sourceMultiply A j)) t.val s)) volume 0 t.val := ⟨M.integrable_comp hi₁.1,M.integrable_comp hi₁.2⟩ have hn : IntervalIntegrable (fun s => N (sourceExtend (sourceFrozen hα A j) t.val s)) volume 0 t.val := ⟨N.integrable_comp hi₂.1,N.integrable_comp hi₂.2⟩ rw [← L.intervalIntegral_comp_comm hi] calc _ = ∫ s in (0:ℝ)..t.val, -M (sourceExtend (sourceFrozen hα A (sourceMultiply A j)) t.val s) + N (sourceExtend (sourceFrozen hα A j) t.val s) := by apply intervalIntegral.integral_congr_Ioo_of_le t.property.1.le intro s hs exact pair_residualKernel hα A b hT j t φ ψ hψ s hs _ = _ := by rw [intervalIntegral.integral_add (f := fun s => -M (sourceExtend (sourceFrozen hα A (sourceMultiply A j)) t.val s)) (g := fun s => N (sourceExtend (sourceFrozen hα A j) t.val s)) hm.neg hn,intervalIntegral.integral_neg, M.intervalIntegral_comp_comm hi₁,N.intervalIntegral_comp_comm hi₂] rfl def mildTime (hT : 0 ≤ T) (d : C₀(V,ℂ)) (j : Time T →ᵇ C₀(V,ℂ)) (t : ℝ) : C₀(V,ℂ) := c0Time hα (A (initial hT)) t d + duhamel hα A j t lemma mildTime_zero (hT : 0 ≤ T) (d : C₀(V,ℂ)) (j : Time T →ᵇ C₀(V,ℂ)) : mildTime hα A hT d j 0 = d := by rw [mildTime,c0Time_zero,duhamel_zero,add_zero] lemma continuousOn_mildTime (hT : 0 ≤ T) (d : C₀(V,ℂ)) (j : Time T →ᵇ C₀(V,ℂ)) : ContinuousOn (mildTime hα A hT d j) (Icc 0 T) := (continuous_c0Time hα (A (initial hT)) d).continuousOn.add (continuousOn_duhamel_Icc hα A hT j) lemma hasDerivAt_pair_mildTime (hT : 0 ≤ T) (d : C₀(V,ℂ)) (j l : Time T →ᵇ C₀(V,ℂ)) (hi : ∀ t : Time T, IntervalIntegrable (fun s => residualKernel hα A b hT t s (unweighted j s)) volume 0 t.val) (hj : ∀ t : Time T, unweighted j t.val + ∫ s in 0..t.val, residualKernel hα A b hT t s (unweighted j s) = l t-actualResidual hα A b t (initial hT) d) (t : Time T) (ht : t.val < T) (φ ψ : SchwartzMap V ℂ) (hψ : ∀ x, ψ x = ((A (now t) x).val : ℂ)*φ x) : HasDerivAt (fun r => pairCLM φ φ.integrable (mildTime hα A hT d j r)) (pairCLM φ φ.integrable (l t)-adjointPair b t φ ψ (mildTime hα A hT d j t.val)) t.val := by let L := pairCLM φ φ.integrable let M := pairCLM (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ) (Δ (Δ φ : SchwartzMap V ℂ) : SchwartzMap V ℂ).integrable let N := adjointPair b t φ ψ have h₁ := hasDerivAt_pair_c0Time hα (A (initial hT)) t.val t.property.1 d φ have h₂ := hasDerivAt_pair_duhamel hα A hT j φ t.val ⟨t.property.1,ht⟩ have he₁ := pair_fullResidual hα (A (initial hT)) (A (now t)) (b (now t)) t.val t.property.1 d φ ψ hψ have he₂ := pair_integral_residualKernel hα A b hT j t (hi t) φ ψ hψ have he := congrArg L (hj t) simp only [map_add,map_sub] at he change L (unweighted j t.val) + L (∫ s in 0..t.val, residualKernel hα A b hT t s (unweighted j s)) = L (l t)-L (actualResidual hα A b t (initial hT) d) at he rw [he₂] at he have he₁' : L (actualResidual hα A b t (initial hT) d) = -M (c0Time hα (A (initial hT)) t.val (C0Multiplier.mul (coefficient (A (initial hT)).toContinuousMap) d)) + N (c0Time hα (A (initial hT)) t.val d) := by simpa only [L,M,N,actualResidual,initial,sub_zero,adjointPair,add_apply,add_assoc] using he₁ rw [he₁'] at he have hd := h₁.add h₂ have heq : (-M (c0Time hα (A (initial hT)) t.val (C0Multiplier.mul (coefficient (A (initial hT)).toContinuousMap) d))) + (L (unweighted j t.val)-M (duhamel hα A (sourceMultiply A j) t.val)) = L (l t)-N (mildTime hα A hT d j t.val) := by rw [mildTime,map_add] linear_combination he rw [heq] at hd apply hd.congr_of_eventuallyEq exact Eventually.of_forall (fun r => L.map_add _ _) end ParametrixVolterra namespace StrongWeightedVolterra variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] variable {T : ℝ} def dampWeight (u : UnitTime) : ℝ := u.val^(-3/4 : ℝ)*(1-u.val)^(-7/8 : ℝ) lemma dampWeight_nonneg (u : UnitTime) : 0 ≤ dampWeight u := mul_nonneg (Real.rpow_nonneg u.property.1.le _) (Real.rpow_nonneg (sub_pos.mpr u.property.2).le _) lemma integrable_dampWeight : Integrable dampWeight := by have hb := (Complex.betaIntegral_convergent (u := (1/4 : ℂ)) (v := (1/8 : ℂ)) (by norm_num) (by norm_num)).norm.1 have hi := hb.mono_set Ioo_subset_Ioc_self have he : EqOn (fun x : ℝ => ‖(x : ℂ)^((1/4 : ℂ)-1) * (1-(x : ℂ))^((1/8 : ℂ)-1)‖) (fun x : ℝ => x^(-3/4 : ℝ)*(1-x)^(-7/8 : ℝ)) (Ioo 0 1) := by intro x hx dsimp only rw [norm_mul] have hq : (1/4 : ℂ)-1 = ((-3/4 : ℝ) : ℂ) := by norm_num have hr : (1/8 : ℂ)-1 = ((-7/8 : ℝ) : ℂ) := by norm_num rw [hq,hr, show (1 : ℂ)-(x : ℂ) = ((1-x : ℝ) : ℂ) by push_cast; rfl, Complex.norm_cpow_real, Complex.norm_cpow_real, Complex.norm_real, Complex.norm_real, Real.norm_eq_abs, Real.norm_eq_abs, abs_of_pos hx.1, abs_of_pos (sub_pos.mpr hx.2)] have hw := hi.congr_fun he measurableSet_Ioo exact (integrableOn_iff_comap_subtypeVal measurableSet_Ioo).mp hw lemma eighth_exp_neg_le_one (x : ℝ) (hx : 0 ≤ x) : x^(1/8 : ℝ)*Real.exp (-x) ≤ 1 := by by_cases h : x ≤ 1 · calc _ ≤ 1*1 := mul_le_mul (Real.rpow_le_one hx h (by norm_num)) (Real.exp_le_one_iff.mpr (by linarith)) (Real.exp_pos _).le (by norm_num) _ = _ := one_mul _ · calc _ ≤ x*Real.exp (-x) := mul_le_mul_of_nonneg_right (Real.rpow_le_self_of_one_le (le_of_not_ge h) (by norm_num)) (Real.exp_pos _).le _ ≤ Real.exp (-1) := Real.mul_exp_neg_le_exp_neg_one x _ ≤ 1 := Real.exp_le_one_iff.mpr (by norm_num) lemma eighth_exp_scale_le (κ t v : ℝ) (hκ : 0 < κ) (ht : 0 < t) (hv : 0 < v) : t^(1/8 : ℝ)*Real.exp (-κ*t*v) ≤ κ^(-1/8 : ℝ)*v^(-1/8 : ℝ) := by have hb := eighth_exp_neg_le_one (κ*t*v) (by positivity) have hcκ : κ^(-1/8 : ℝ)*κ^(1/8 : ℝ)=1 := by rw [← Real.rpow_add hκ]; norm_num have hcv : v^(-1/8 : ℝ)*v^(1/8 : ℝ)=1 := by rw [← Real.rpow_add hv]; norm_num calc _ = (κ^(-1/8 : ℝ)*v^(-1/8 : ℝ))*((κ*t*v)^(1/8 : ℝ)*Real.exp (-(κ*t*v))) := by rw [Real.mul_rpow (mul_pos hκ ht).le hv.le,Real.mul_rpow hκ.le ht.le] calc _ = (κ^(-1/8 : ℝ)*κ^(1/8 : ℝ))*(v^(-1/8 : ℝ)*v^(1/8 : ℝ))* (t^(1/8 : ℝ)*Real.exp (-κ*t*v)) := by rw [hcκ,hcv]; ring _ = _ := by rw [show -(κ*t*v) = -κ*t*v by ring] ring _ ≤ (κ^(-1/8 : ℝ)*v^(-1/8 : ℝ))*1 := mul_le_mul_of_nonneg_left hb (by positivity) _ = _ := mul_one _ lemma damp_scalar_bound (κ : ℝ) (hκ : 0 < κ) (t : Time T) (ht : t.val ≤ 1) (u : UnitTime) : t.val^(1/4 : ℝ)*weight u*Real.exp (-κ*(t.val*(1-u.val))) ≤ κ^(-1/8 : ℝ)*dampWeight u := by have hu : 0 < 1-u.val := sub_pos.mpr u.property.2 have ht8 : t.val^(1/8 : ℝ) ≤ 1 := Real.rpow_le_one t.property.1.le ht (by norm_num) have he := eighth_exp_scale_le κ t.val (1-u.val) hκ t.property.1 hu have hpow : t.val^(1/4 : ℝ) = t.val^(1/8 : ℝ)*t.val^(1/8 : ℝ) := by rw [← Real.rpow_add t.property.1]; norm_num have hmul : weight u*(1-u.val)^(-1/8 : ℝ)=dampWeight u := by rw [weight,dampWeight,mul_assoc,← Real.rpow_add hu]; norm_num calc _ = (t.val^(1/8 : ℝ)*weight u)*(t.val^(1/8 : ℝ)*Real.exp (-κ*t.val*(1-u.val))) := by rw [hpow,show -κ*(t.val*(1-u.val)) = -κ*t.val*(1-u.val) by ring] ring _ ≤ (1*weight u)*(κ^(-1/8 : ℝ)*(1-u.val)^(-1/8 : ℝ)) := mul_le_mul (mul_le_mul_of_nonneg_right ht8 (weight_nonneg u)) he (mul_nonneg (Real.rpow_nonneg t.property.1.le _) (Real.exp_pos _).le) (mul_nonneg (by norm_num) (weight_nonneg u)) _ = _ := by rw [one_mul,← hmul]; ring def dampKernel (N : Kernel T E) (κ : ℝ) : Kernel T E where toFun p := Real.exp (-κ*(p.1.val*(1-p.2.val))) • N p continuous_apply := by change Continuous (fun p : (Time T × UnitTime) × E => Real.exp (-κ*(p.1.1.val*(1-p.1.2.val))) • N p.1 p.2) exact (by fun_prop : Continuous (fun p : (Time T × UnitTime) × E => Real.exp (-κ*(p.1.1.val*(1-p.1.2.val))))).smul N.continuous_apply lemma norm_dampKernel_le (N : Kernel T E) (C : ℝ) (hC : 0 ≤ C) (hN : ∀ p, ‖N p‖ ≤ C) (κ : ℝ) (hκ : 0 ≤ κ) (p : Time T × UnitTime) : ‖dampKernel N κ p‖ ≤ C := by change ‖Real.exp (-κ*(p.1.val*(1-p.2.val))) • N p‖ ≤ _ rw [norm_smul,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] have he : Real.exp (-κ*(p.1.val*(1-p.2.val))) ≤ 1 := Real.exp_le_one_iff.mpr (by rw [neg_mul] exact neg_nonpos.mpr (mul_nonneg hκ (mul_nonneg p.1.property.1.le (sub_pos.mpr p.2.property.2).le))) exact (mul_le_mul_of_nonneg_left (hN p) (Real.exp_pos _).le).trans (mul_le_of_le_one_left hC he) lemma norm_damp_data_le (N : Kernel T E) (C : ℝ) (hC : 0 ≤ C) (hN : ∀ p, ‖N p‖ ≤ C) (κ : ℝ) (hκ : 0 < κ) (ht : T ≤ 1) (j : Time T →ᵇ E) (t : Time T) (u : UnitTime) : ‖t.val^(1/4 : ℝ) • data (dampKernel N κ) j t u‖ ≤ κ^(-1/8 : ℝ)*dampWeight u*(C*‖j‖) := by change ‖t.val^(1/4 : ℝ) • (weight u • ((Real.exp (-κ*(t.val*(1-u.val))) • N (t,u)) (j (lag t u))))‖ ≤ _ rw [smul_apply,smul_smul,smul_smul,norm_smul,Real.norm_eq_abs,abs_of_nonneg (mul_nonneg (mul_nonneg (Real.rpow_nonneg t.property.1.le _) (weight_nonneg u)) (Real.exp_pos _).le)] calc _ ≤ (t.val^(1/4 : ℝ)*weight u*Real.exp (-κ*(t.val*(1-u.val))))*(C*‖j‖) := mul_le_mul_of_nonneg_left ((N (t,u)).le_opNorm _ |>.trans (mul_le_mul (hN _) (j.norm_coe_le_norm _) (norm_nonneg _) hC)) (mul_nonneg (mul_nonneg (Real.rpow_nonneg t.property.1.le _) (weight_nonneg u)) (Real.exp_pos _).le) _ ≤ _ := mul_le_mul_of_nonneg_right (damp_scalar_bound κ hκ t (t.property.2.trans ht) u) (mul_nonneg hC (norm_nonneg _)) lemma norm_damp_value_le (N : Kernel T E) (C : ℝ) (hC : 0 ≤ C) (hN : ∀ p, ‖N p‖ ≤ C) (κ : ℝ) (hκ : 0 < κ) (ht : T ≤ 1) (j : Time T →ᵇ E) (t : Time T) : ‖value (dampKernel N κ) j t‖ ≤ (κ^(-1/8 : ℝ)*(∫ u : UnitTime, dampWeight u)*C)*‖j‖ := by rw [value,← integral_smul] calc _ ≤ ∫ u : UnitTime, κ^(-1/8 : ℝ)*dampWeight u*(C*‖j‖) := norm_integral_le_of_norm_le ((integrable_dampWeight.const_mul _).mul_const _) (Eventually.of_forall (norm_damp_data_le N C hC hN κ hκ ht j t)) _ = _ := by rw [integral_mul_const,integral_const_mul]; ring lemma norm_timeExp_le (c : ℝ) (j : Time T →ᵇ E) (t : Time T) : ‖Real.exp (c*t.val) • j t‖ ≤ Real.exp (|c| * T)*‖j‖ := by rw [norm_smul,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] exact mul_le_mul (Real.exp_le_exp.mpr ((mul_le_mul_of_nonneg_right (le_abs_self c) t.property.1.le).trans (mul_le_mul_of_nonneg_left t.property.2 (abs_nonneg c)))) (j.norm_coe_le_norm t) (norm_nonneg _) (Real.exp_pos _).le def timeExp (c : ℝ) (j : Time T →ᵇ E) : Time T →ᵇ E := .ofNormedAddCommGroup (fun t => Real.exp (c*t.val) • j t) ((Real.continuous_exp.comp (continuous_const.mul continuous_subtype_val)).smul j.continuous) _ (norm_timeExp_le c j) lemma timeExp_apply (c : ℝ) (j : Time T →ᵇ E) (t : Time T) : timeExp c j t = Real.exp (c*t.val) • j t := rfl lemma timeExp_norm_le (c : ℝ) (j : Time T →ᵇ E) : ‖timeExp c j‖ ≤ Real.exp (|c| * T)*‖j‖ := (BoundedContinuousFunction.norm_le (mul_nonneg (Real.exp_pos _).le (norm_nonneg _))).mpr (norm_timeExp_le c j) lemma timeExp_add (c : ℝ) (j k : Time T →ᵇ E) : timeExp c (j+k) = timeExp c j+timeExp c k := by ext t simp only [timeExp_apply,BoundedContinuousFunction.add_apply,smul_add] lemma timeExp_comp (c d : ℝ) (j : Time T →ᵇ E) : timeExp c (timeExp d j) = timeExp (c+d) j := by ext t simp only [timeExp_apply,smul_smul,← Real.exp_add,add_mul] lemma timeExp_zero (j : Time T →ᵇ E) : timeExp 0 j = j := by ext t simp only [timeExp_apply,zero_mul,Real.exp_zero,one_smul] lemma damp_data_conjugation (N : Kernel T E) (κ : ℝ) (j : Time T →ᵇ E) (t : Time T) (u : UnitTime) : data (dampKernel N κ) j t u = Real.exp (-κ*t.val) • data N (timeExp κ j) t u := by change weight u • ((Real.exp (-κ*(t.val*(1-u.val))) • N (t,u)) (j (lag t u))) = _ rw [smul_apply,data,timeExp_apply,ContinuousLinearMap.map_smul_of_tower,smul_smul,smul_smul,smul_smul] congr 1 change weight u*Real.exp (-κ*(t.val*(1-u.val))) = Real.exp (-κ*t.val)*weight u*Real.exp (κ*(t.val*u.val)) calc _ = weight u*Real.exp (-κ*t.val+κ*(t.val*u.val)) := by congr 2; ring _ = _ := by rw [Real.exp_add]; ring lemma damp_value_conjugation (N : Kernel T E) (κ : ℝ) (j : Time T →ᵇ E) (t : Time T) : value (dampKernel N κ) j t = Real.exp (-κ*t.val) • value N (timeExp κ j) t := by unfold value simp_rw [damp_data_conjugation] rw [integral_smul,smul_comm] lemma damp_operator_conjugation (N : Kernel T E) (C : ℝ) (hC : 0 ≤ C) (hN : ∀ p, ‖N p‖ ≤ C) (κ : ℝ) (hκ : 0 ≤ κ) (hT : 0 ≤ T) (j : Time T →ᵇ E) : operator (dampKernel N κ) C hC (norm_dampKernel_le N C hC hN κ hκ) hT j = timeExp (-κ) (operator N C hC hN hT (timeExp κ j)) := by ext t exact damp_value_conjugation N κ j t lemma norm_damp_operator_le (N : Kernel T E) (C : ℝ) (hC : 0 ≤ C) (hN : ∀ p, ‖N p‖ ≤ C) (κ : ℝ) (hκ : 0 < κ) (hT : 0 ≤ T) (ht : T ≤ 1) (j : Time T →ᵇ E) : ‖operator (dampKernel N κ) C hC (norm_dampKernel_le N C hC hN κ hκ.le) hT j‖ ≤ (κ^(-1/8 : ℝ)*(∫ u : UnitTime, dampWeight u)*C)*‖j‖ := by apply (BoundedContinuousFunction.norm_le (mul_nonneg (mul_nonneg (mul_nonneg (Real.rpow_nonneg hκ.le _) (integral_nonneg dampWeight_nonneg)) hC) (norm_nonneg _))).mpr exact norm_damp_value_le N C hC hN κ hκ ht j lemma norm_damp_operatorCLM_le (N : Kernel T E) (C : ℝ) (hC : 0 ≤ C) (hN : ∀ p, ‖N p‖ ≤ C) (κ : ℝ) (hκ : 0 < κ) (hT : 0 ≤ T) (ht : T ≤ 1) : ‖operatorCLM (dampKernel N κ) C hC (norm_dampKernel_le N C hC hN κ hκ.le) hT‖ ≤ κ^(-1/8 : ℝ)*(∫ u : UnitTime, dampWeight u)*C := by apply ContinuousLinearMap.opNorm_le_bound _ (mul_nonneg (mul_nonneg (Real.rpow_nonneg hκ.le _) (integral_nonneg dampWeight_nonneg)) hC) exact norm_damp_operator_le N C hC hN κ hκ hT ht lemma exists_damping_small (C : ℝ) : ∃ κ : ℝ, 0 < κ ∧ κ^(-1/8 : ℝ)*(∫ u : UnitTime, dampWeight u)*C < 1 := by have ht : Tendsto (fun κ : ℝ => κ^(-1/8 : ℝ)*(∫ u : UnitTime, dampWeight u)*C) atTop (𝓝 0) := by simpa only [zero_mul,neg_div] using ((tendsto_rpow_neg_atTop (by norm_num : (0:ℝ) < 1/8)).mul_const (∫ u : UnitTime, dampWeight u)).mul_const C exact ((eventually_gt_atTop (0:ℝ)).and (ht.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1)))).exists theorem exists_unique_solution_full [CompleteSpace E] (N : Kernel T E) (C : ℝ) (hC : 0 ≤ C) (hN : ∀ p, ‖N p‖ ≤ C) (hT : 0 ≤ T) (ht : T ≤ 1) (g : Time T →ᵇ E) : ∃! j : Time T →ᵇ E, j+operator N C hC hN hT j=g := by obtain ⟨κ,hκ,hsmall⟩ := exists_damping_small C let M := dampKernel N κ have hM := norm_dampKernel_le N C hC hN κ hκ.le let A := operatorCLM M C hC hM hT have hA : ‖A‖ < 1 := (norm_damp_operatorCLM_le N C hC hN κ hκ hT ht).trans_lt hsmall let g₀ := timeExp (-κ) g let f : (Time T →ᵇ E) → (Time T →ᵇ E) := fun j => g₀-A j have hf : ContractingWith ‖A‖₊ f := by refine ⟨hA,LipschitzWith.of_dist_le_mul (fun j k => ?_)⟩ rw [dist_eq_norm,dist_eq_norm] change ‖(g₀-A j)-(g₀-A k)‖ ≤ ‖A‖*‖j-k‖ rw [sub_sub_sub_cancel_left,← A.map_sub,norm_sub_rev j k] exact A.le_opNorm (k-j) have hsolve : hf.fixedPoint+operator M C hC hM hT hf.fixedPoint = g₀ := eq_sub_iff_add_eq.mp hf.fixedPoint_isFixedPt.symm let j := timeExp κ hf.fixedPoint have hj : j+operator N C hC hN hT j=g := by have he := congrArg (timeExp κ) hsolve rw [damp_operator_conjugation N C hC hN κ hκ.le hT,timeExp_add,timeExp_comp, add_neg_cancel,timeExp_zero] at he simpa only [g₀,timeExp_comp,add_neg_cancel,timeExp_zero] using he refine ⟨j,hj,fun k hk => ?_⟩ have hk₀ : timeExp (-κ) k+operator M C hC hM hT (timeExp (-κ) k)=g₀ := by rw [damp_operator_conjugation N C hC hN κ hκ.le hT,timeExp_comp,add_neg_cancel,timeExp_zero] rw [← timeExp_add,hk] have hfix : f (timeExp (-κ) k)=timeExp (-κ) k := (eq_sub_iff_add_eq.mpr hk₀).symm have he := congrArg (timeExp κ) (hf.fixedPoint_unique hfix) simpa only [timeExp_comp,add_neg_cancel,timeExp_zero] using he lemma norm_solution_full_le (N : Kernel T E) (C : ℝ) (hC : 0 ≤ C) (hN : ∀ p, ‖N p‖ ≤ C) (κ : ℝ) (hκ : 0 < κ) (hT : 0 ≤ T) (ht : T ≤ 1) (hsmall : κ^(-1/8 : ℝ)*(∫ u : UnitTime, dampWeight u)*C < 1) (g j : Time T →ᵇ E) (hj : j+operator N C hC hN hT j=g) : ‖j‖ ≤ (Real.exp (κ*T))^2/(1-κ^(-1/8 : ℝ)*(∫ u : UnitTime, dampWeight u)*C)*‖g‖ := by let Q := κ^(-1/8 : ℝ)*(∫ u : UnitTime, dampWeight u)*C let M := dampKernel N κ have hM := norm_dampKernel_le N C hC hN κ hκ.le let j₀ := timeExp (-κ) j let g₀ := timeExp (-κ) g have he : j₀+operator M C hC hM hT j₀=g₀ := by rw [damp_operator_conjugation N C hC hN κ hκ.le hT] dsimp only [j₀,g₀] rw [timeExp_comp,add_neg_cancel,timeExp_zero,← timeExp_add,hj] have hb : ‖operator M C hC hM hT j₀‖ ≤ Q*‖j₀‖ := norm_damp_operator_le N C hC hN κ hκ hT ht j₀ have hn := norm_sub_le g₀ (operator M C hC hM hT j₀) rw [← eq_sub_iff_add_eq.mpr he] at hn have hpos : 0 < 1-Q := sub_pos.mpr hsmall have hj₀ : ‖j₀‖ ≤ ‖g₀‖/(1-Q) := (le_div_iff₀ hpos).mpr (by nlinarith) have hj' : timeExp κ j₀=j := by change timeExp κ (timeExp (-κ) j)=j rw [timeExp_comp,add_neg_cancel,timeExp_zero] have hjb := timeExp_norm_le κ j₀ rw [hj',abs_of_pos hκ] at hjb have hgb : ‖g₀‖ ≤ Real.exp (κ*T)*‖g‖ := by simpa only [g₀,abs_neg,abs_of_pos hκ] using timeExp_norm_le (-κ) g calc ‖j‖ ≤ Real.exp (κ*T)*(‖g₀‖/(1-Q)) := hjb.trans (mul_le_mul_of_nonneg_left hj₀ (Real.exp_pos _).le) _ ≤ Real.exp (κ*T)*((Real.exp (κ*T)*‖g‖)/(1-Q)) := mul_le_mul_of_nonneg_left (div_le_div_of_nonneg_right hgb hpos.le) (Real.exp_pos _).le _ = _ := by dsimp only [Q]; ring theorem exists_full_inverse_bound [CompleteSpace E] (N : Kernel T E) (C : ℝ) (hC : 0 ≤ C) (hN : ∀ p, ‖N p‖ ≤ C) (hT : 0 ≤ T) (ht : T ≤ 1) : ∃ D : ℝ, 0 < D ∧ ∀ g : Time T →ᵇ E, ∃ j : Time T →ᵇ E, j+operator N C hC hN hT j=g ∧ ‖j‖ ≤ D*‖g‖ := by obtain ⟨κ,hκ,hsmall⟩ := exists_damping_small C refine ⟨(Real.exp (κ*T))^2/(1-κ^(-1/8 : ℝ)*(∫ u : UnitTime, dampWeight u)*C), div_pos (sq_pos_of_pos (Real.exp_pos _)) (sub_pos.mpr hsmall),fun g => ?_⟩ obtain ⟨j,hj,_⟩ := exists_unique_solution_full N C hC hN hT ht g exact ⟨j,hj,norm_solution_full_le N C hC hN κ hκ hT ht hsmall g j hj⟩ theorem exists_unweighted_solution_full [CompleteSpace E] (N : Kernel T E) (C : ℝ) (hC : 0 ≤ C) (hN : ∀ p, ‖N p‖ ≤ C) (hT : 0 ≤ T) (ht : T ≤ 1) : ∃ D : ℝ, 0 < D ∧ ∀ g : Time T →ᵇ E, ∃ j : Time T →ᵇ E, (∀ t : Time T, IntervalIntegrable (fun r => kernel N t r (unweighted j r)) volume 0 t.val) ∧ (∀ t : Time T, unweighted j t.val + ∫ r in 0..t.val, kernel N t r (unweighted j r) = unweighted g t.val) ∧ ‖j‖ ≤ D*‖g‖ ∧ (∀ t : Time T, ‖unweighted j t.val‖ ≤ t.val^(-3/4 : ℝ)*(D*‖g‖)) := by obtain ⟨D,hD,hd⟩ := exists_full_inverse_bound N C hC hN hT ht refine ⟨D,hD,fun g => ?_⟩ obtain ⟨j,hj,hjb⟩ := hd g refine ⟨j,intervalIntegrable_kernel_unweighted N C hC hN j, unweighted_solves N C hC hN hT g j hj,hjb,fun t => ?_⟩ rw [unweighted_at_time,norm_smul,Real.norm_eq_abs, abs_of_nonneg (Real.rpow_nonneg t.property.1.le _)] exact mul_le_mul_of_nonneg_left ((j.norm_coe_le_norm t).trans hjb) (Real.rpow_nonneg t.property.1.le _) end StrongWeightedVolterra namespace ParametrixVolterra open scoped ZeroAtInfty BoundedContinuousFunction SchwartzMap Laplacian open FrozenParametrix C0Convolution StrongWeightedVolterra KernelPairing variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {T α β : ℝ} theorem exists_parametrix_correction_full (hα : 0 < α) (A : C(Slab T, V →ᵇ Icc α β)) (b : C(Slab T, V →ᵇ ℂ)) (hT₀ : 0 ≤ T) (hT : T ≤ 1) (L B : ℝ) (hL : 0 ≤ L) (hB : 0 ≤ B) (hb : ∀ s, ‖b s‖ ≤ B) (hLip : ∀ s r : Slab T, ∀ x z : V, |(A s x).val-(A r z).val| ≤ L*(‖x-z‖+|s.val-r.val|)) : ∃ C D : ℝ, 0 ≤ C ∧ 0 < D ∧ ∀ (d : C₀(V,ℂ)) (l : Time T →ᵇ C₀(V,ℂ)), ∃ j : Time T →ᵇ C₀(V,ℂ), (∀ t : Time T, IntervalIntegrable (fun r => residualKernel hα A b hT₀ t r (unweighted j r)) volume 0 t.val) ∧ (∀ t : Time T, unweighted j t.val + ∫ r in 0..t.val, residualKernel hα A b hT₀ t r (unweighted j r) = l t-actualResidual hα A b t (initial hT₀) d) ∧ ‖j‖ ≤ D*(‖l‖+C*‖d‖) := by obtain ⟨C,hC,hR⟩ := exists_actualResidual_bound hα A b hT L B hL hB hb hLip obtain ⟨D,hD,hN⟩ := exists_normalized_bound hα A b hT L B hL hB hb hLip have hR₀ : ∀ t : Time T, ‖actualResidual hα A b t (initial hT₀)‖ ≤ C*t.val^(-3/4 : ℝ) := by intro t simpa only [initial,sub_zero] using hR t (initial hT₀) t.property.1 let N : Kernel T C₀(V,ℂ) := Kernel.ofStrong (normalized hα A b) (continuous_normalized_apply hα A b) D hD hN obtain ⟨E,hE,he⟩ := exists_unweighted_solution_full N D hD hN hT₀ hT refine ⟨C,E,hC,hE,fun d l => ?_⟩ let g := weightedSource hα A b hT₀ hT C hC hR₀ d l obtain ⟨j,hi,hj,hbound,_⟩ := he g have hk : ∀ t r, kernel N t r = residualKernel hα A b hT₀ t r := kernel_eq_residualKernel hα A b hT₀ D hD hN simp_rw [hk] at hi hj have hg : ∀ t : Time T, unweighted g t.val = l t-actualResidual hα A b t (initial hT₀) d := unweighted_weightedSource hα A b hT₀ hT C hC hR₀ d l simp_rw [hg] at hj refine ⟨j,hi,hj,hbound.trans ?_⟩ have hgnorm : ‖g‖ ≤ ‖l‖+C*‖d‖ := by apply (BoundedContinuousFunction.norm_le (by positivity)).mpr exact norm_sourceRHS_le hα A b hT₀ hT C hC hR₀ d l exact mul_le_mul_of_nonneg_left hgnorm hE.le def coefficientTest (a : V →ᵇ Icc α β) (ha : ContDiff ℝ ∞ (fun x => (a x).val)) (φ : SchwartzMap V ℂ) (hφ : HasCompactSupport (φ : V → ℂ)) : SchwartzMap V ℂ := HasCompactSupport.toSchwartzMap (f := fun x => ((a x).val : ℂ)*φ x) hφ.mul_left ((Complex.ofRealCLM.contDiff.comp ha).mul (φ.smooth ⊤)) omit [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in lemma coefficientTest_apply (a : V →ᵇ Icc α β) (ha : ContDiff ℝ ∞ (fun x => (a x).val)) (φ : SchwartzMap V ℂ) (hφ : HasCompactSupport (φ : V → ℂ)) (x : V) : coefficientTest a ha φ hφ x = ((a x).val : ℂ)*φ x := rfl theorem exists_weak_evolution (hα : 0 < α) (A : C(Slab T,V →ᵇ Icc α β)) (b : C(Slab T,V →ᵇ ℂ)) (hT₀ : 0 ≤ T) (hT : T ≤ 1) (L B : ℝ) (hL : 0 ≤ L) (hB : 0 ≤ B) (hb : ∀ s, ‖b s‖ ≤ B) (hLip : ∀ s r : Slab T, ∀ x z : V, |(A s x).val-(A r z).val| ≤ L*(‖x-z‖+|s.val-r.val|)) (hA : ∀ s, ContDiff ℝ ∞ (fun x => (A s x).val)) : ∃ K : ℝ, 0 ≤ K ∧ ∀ (d : C₀(V,ℂ)) (l : Time T →ᵇ C₀(V,ℂ)), ∃ u : ℝ → C₀(V,ℂ), u 0=d ∧ ContinuousOn u (Icc 0 T) ∧ (∀ t ∈ Icc 0 T, ‖u t‖ ≤ K*(‖l‖+‖d‖)) ∧ (∀ (t : Time T), t.val < T → ∀ (φ : SchwartzMap V ℂ) (hφ : HasCompactSupport (φ : V → ℂ)), HasDerivAt (fun r => pairCLM φ φ.integrable (u r)) (pairCLM φ φ.integrable (l t)- adjointPair b t φ (coefficientTest (A (now t)) (hA (now t)) φ hφ) (u t.val)) t.val) := by obtain ⟨C,D,hC,hD,hs⟩ := exists_parametrix_correction_full hα A b hT₀ hT L B hL hB hb hLip let M := frozenBound (V := V) (β := β) hα let Q := ∫ v : UnitTime, MildVolterra.massWeight v have hM : 0 ≤ M := frozenBound_nonneg hα have hQ : 0 ≤ Q := integral_nonneg MildVolterra.massWeight_nonneg refine ⟨M+Q*M*D*(1+C),by positivity,fun d l => ?_⟩ obtain ⟨j,hi,hj,hjb⟩ := hs d l refine ⟨mildTime hα A hT₀ d j,mildTime_zero hα A hT₀ d j, continuousOn_mildTime hα A hT₀ d j,fun t ht => ?_,fun t ht φ hφ => ?_⟩ · have ht1 : t^(1/4 : ℝ) ≤ 1 := Real.rpow_le_one ht.1 (ht.2.trans hT) (by positivity) have h0 := norm_c0Time_le hα (A (initial hT₀)) t d have h1 := norm_duhamel_le hα A hT₀ j t ht have hjb' : ‖j‖ ≤ D*((1+C)*(‖l‖+‖d‖)) := hjb.trans (mul_le_mul_of_nonneg_left (by nlinarith [norm_nonneg l,norm_nonneg d]) hD.le) calc ‖mildTime hα A hT₀ d j t‖ ≤ M*‖d‖+t^(1/4 : ℝ)*(Q*M*‖j‖) := (norm_add_le _ _).trans (add_le_add h0 h1) _ ≤ M*‖d‖+Q*M*‖j‖ := add_le_add le_rfl (mul_le_of_le_one_left (mul_nonneg (mul_nonneg hQ hM) (norm_nonneg j)) ht1) _ ≤ M*(‖l‖+‖d‖)+Q*M*(D*((1+C)*(‖l‖+‖d‖))) := add_le_add (mul_le_mul_of_nonneg_left (le_add_of_nonneg_left (norm_nonneg l)) hM) (mul_le_mul_of_nonneg_left hjb' (mul_nonneg hQ hM)) _ = _ := by ring · exact hasDerivAt_pair_mildTime hα A b hT₀ d j l hi hj t ht φ (coefficientTest (A (now t)) (hA (now t)) φ hφ) (coefficientTest_apply _ _ _ _) end ParametrixVolterra open Set Filter MeasureTheory open scoped SchwartzMap Topology namespace SchwartzTimeScale variable {V E : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup E] [NormedSpace ℝ E] def euler (K : SchwartzMap V E) (N : ℕ) : SchwartzMap V E := (-(N : ℝ)/4) • K - (1/4 : ℝ) • (SchwartzMap.bilinLeftCLM (ContinuousLinearMap.apply ℝ E).flip (by fun_prop : Function.HasTemperateGrowth (fun y : V => y)) (SchwartzMap.fderivCLM ℝ V E K)) lemma euler_apply (K : SchwartzMap V E) (N : ℕ) (y : V) : euler K N y = (-(N : ℝ)/4) • K y-(1/4 : ℝ) • (fderiv ℝ K y y) := rfl def atTime (K : SchwartzMap V E) (N : ℕ) (q : ℝ) (y : V) : E := q^(-(N : ℝ)/4) • K (q^(-(1 : ℝ)/4) • y) lemma hasDerivAt_atTime (K : SchwartzMap V E) (N : ℕ) (q : ℝ) (hq : 0 < q) (y : V) : HasDerivAt (fun s => atTime K N s y) (atTime (euler K N) (N+4) q y) q := by have h1 := Real.hasDerivAt_rpow_const (p := -(N : ℝ)/4) (Or.inl hq.ne') have h2 := Real.hasDerivAt_rpow_const (p := -(1 : ℝ)/4) (Or.inl hq.ne') have hK := (K.hasFDerivAt (q^(-(1 : ℝ)/4) • y)).comp_hasDerivAt q (h2.smul_const y) have hd := h1.smul hK convert hd using 1 · rfl · dsimp only [atTime,Function.comp_apply,Pi.smul_apply] rw [euler_apply,smul_sub] simp only [ContinuousLinearMap.map_smul,smul_smul] have he1 : q^(-((N+4 : ℕ) : ℝ)/4) = q^(-(N : ℝ)/4-1) := by congr 1 push_cast ring have he2 : q^(-(N : ℝ)/4) * (-(1 : ℝ)/4*q^(-(1 : ℝ)/4-1)) = -(q^(-(N : ℝ)/4-1)*(1/4))*q^(-(1 : ℝ)/4) := by calc _ = (-(1 : ℝ)/4)*q^(-(N : ℝ)/4+(-(1 : ℝ)/4-1)) := by rw [Real.rpow_add hq]; ring _ = (-(1 : ℝ)/4)*q^((-(N : ℝ)/4-1)+(-(1 : ℝ)/4)) := by congr 2; ring _ = _ := by rw [Real.rpow_add hq]; ring rw [he1,he2] module lemma atTime_eq_scaled (K : SchwartzMap V E) (N : ℕ) (q : ℝ) (hq : 0 < q) (y : V) : atTime K N q y = ((q^(1/4 : ℝ))⁻¹)^N • K ((q^(1/4 : ℝ))⁻¹ • y) := by unfold atTime rw [show -(1 : ℝ)/4 = -(1/4 : ℝ) by ring,Real.rpow_neg hq.le] congr 1 rw [← Real.rpow_natCast,← Real.rpow_neg hq.le,← Real.rpow_mul hq.le] congr 1 ring end SchwartzTimeScale namespace SourceFrozenCancellation open SchwartzTimeScale open scoped BoundedContinuousFunction variable {V E : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] omit [CompleteSpace E] in lemma parameter_envelope (K : SchwartzMap V E) (N : ℕ) (α β : ℝ) (hα : 0 < α) : ∃ B : V → ℝ, Integrable B ∧ Integrable (fun z => (1+‖z‖)*B z) ∧ (∀ z, 0 ≤ B z) ∧ ∀ a ∈ Icc α β, ∀ z, ‖atTime K N a z‖ ≤ B z := by obtain ⟨B,hB,hw,hp,hb⟩ := SchwartzConvolution.scaled_weighted_majorant K (α^(1/4 : ℝ)) (β^(1/4 : ℝ)) (by positivity) N 1 refine ⟨B,hB,by simpa only [pow_one] using hw,hp,?_⟩ intro a ha z rw [atTime_eq_scaled K N a (hα.trans_le ha.1)] exact hb _ ⟨Real.rpow_le_rpow hα.le ha.1 (by norm_num), Real.rpow_le_rpow (hα.trans_le ha.1).le ha.2 (by norm_num)⟩ z omit [CompleteSpace E] in lemma parameter_lipschitz_envelope (K : SchwartzMap V E) (N : ℕ) (α β : ℝ) (hα : 0 < α) : ∃ C : V → ℝ, Integrable C ∧ Integrable (fun z => (1+‖z‖)*C z) ∧ (∀ z, 0 ≤ C z) ∧ ∀ a ∈ Icc α β, ∀ b ∈ Icc α β, ∀ z, ‖atTime K N a z-atTime K N b z‖ ≤ C z*‖a-b‖ := by obtain ⟨C,hC,hw,hp,hb⟩ := parameter_envelope (euler K N) (N+4) α β hα refine ⟨C,hC,hw,hp,fun a ha b hb' z => ?_⟩ exact Convex.norm_image_sub_le_of_norm_hasDerivWithin_le (fun q hq => (hasDerivAt_atTime K N q (hα.trans_le hq.1) z).hasDerivWithinAt) (fun q hq => hb q hq z) (convex_Icc α β) hb' ha omit [CompleteSpace E] in lemma integral_atTime_zero (K : SchwartzMap V E) (N : ℕ) (a : ℝ) (ha : 0 < a) (hK : ∫ z, K z = 0) : ∫ z, atTime K N a z = 0 := by simp_rw [atTime_eq_scaled K N a ha] rw [integral_smul,Measure.integral_comp_inv_smul_of_nonneg volume _ (by positivity)] simp only [hK,smul_zero] omit [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] [CompleteSpace E] in lemma continuous_variable (K : SchwartzMap V E) (N : ℕ) (a : V → ℝ) (ha : Continuous a) (hp : ∀ z, 0 < a z) : Continuous (fun z => atTime K N (a z) z) := by unfold atTime exact (ha.rpow_const (fun z => Or.inl (hp z).ne')).smul (K.continuous.comp ((ha.rpow_const (fun z => Or.inl (hp z).ne')).smul continuous_id)) omit [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] in lemma integrable_envelope_power (B : V → ℝ) (hB : Integrable B) (hW : Integrable (fun z => (1+‖z‖)*B z)) (hp : ∀ z, 0 ≤ B z) (γ : ℝ) (hγ : 0 ≤ γ) (hγ1 : γ ≤ 1) : Integrable (fun z => ‖z‖^γ*B z) := by apply hW.mono' · exact (continuous_norm.rpow_const (fun _ => Or.inr hγ)).aestronglyMeasurable.mul hB.aestronglyMeasurable · filter_upwards with z rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg (Real.rpow_nonneg (norm_nonneg _) _) (hp z))] apply mul_le_mul_of_nonneg_right _ (hp z) by_cases hz : ‖z‖ ≤ 1 · exact (Real.rpow_le_one (norm_nonneg _) hz hγ).trans (by linarith [norm_nonneg z]) · have hh := Real.rpow_le_rpow_of_exponent_le (le_of_not_ge hz) hγ1 rw [Real.rpow_one] at hh exact hh.trans (by linarith) omit [CompleteSpace E] in lemma norm_normalized_source_integral (K : SchwartzMap V E) (N : ℕ) (α β : ℝ) (hα : 0 < α) (hK : ∫ z, K z = 0) (B C : V → ℝ) (hB : Integrable B) (hBw : Integrable (fun z => (1+‖z‖)*B z)) (hBp : ∀ z, 0 ≤ B z) (hC : Integrable C) (hCw : Integrable (fun z => (1+‖z‖)*C z)) (hCp : ∀ z, 0 ≤ C z) (hb : ∀ a ∈ Icc α β, ∀ z, ‖atTime K N a z‖ ≤ B z) (hc : ∀ a ∈ Icc α β, ∀ b ∈ Icc α β, ∀ z, ‖atTime K N a z-atTime K N b z‖ ≤ C z*‖a-b‖) (A : V → ℝ) (hA : Continuous A) (hAr : ∀ y, A y ∈ Icc α β) (L : ℝ) (_hL : 0 ≤ L) (hAL : ∀ x y, ‖A x-A y‖ ≤ L*‖x-y‖) (f : V →ᵇ ℂ) (γ H : ℝ) (hγ : 0 ≤ γ) (hγ1 : γ ≤ 1) (hH : 0 ≤ H) (hhold : ∀ x y, ‖f x-f y‖ ≤ H*‖x-y‖^γ) (r : ℝ) (hr : 0 < r) (x : V) : ‖∫ z, f (x-r • z) • atTime K N (A (x-r • z)) z‖ ≤ H*r^γ*(∫ z, ‖z‖^γ*B z)+‖f‖*L*r*(∫ z, ‖z‖*C z) := by let a : V → ℝ := fun z => A (x-r • z) have ha : Continuous a := hA.comp (continuous_const.sub (continuous_const.smul continuous_id)) have hp (z : V) : 0 < a z := hα.trans_le (hAr _).1 have hi : Integrable (fun z => f (x-r • z) • atTime K N (a z) z) := by apply (hB.const_mul ‖f‖).mono' · exact ((f.continuous.comp (continuous_const.sub (continuous_const.smul continuous_id))).smul (continuous_variable K N a ha hp)).aestronglyMeasurable · filter_upwards with z rw [norm_smul] exact mul_le_mul (f.norm_coe_le_norm _) (hb _ (hAr _) z) (norm_nonneg _) (norm_nonneg _) have hi₀ : Integrable (fun z => f x • atTime K N (A x) z) := by apply (hB.const_mul ‖f x‖).mono' · exact ((continuous_variable K N (fun _ => A x) continuous_const (fun _ => hα.trans_le (hAr x).1)).const_smul (f x)).aestronglyMeasurable · filter_upwards with z rw [norm_smul] exact mul_le_mul_of_nonneg_left (hb _ (hAr x) z) (norm_nonneg _) have hzero : ∫ z, f x • atTime K N (A x) z = 0 := by rw [integral_smul,integral_atTime_zero K N (A x) (hα.trans_le (hAr x).1) hK,smul_zero] have hiB := integrable_envelope_power B hB hBw hBp γ hγ hγ1 have hiC : Integrable (fun z => ‖z‖*C z) := by simpa using integrable_envelope_power C hC hCw hCp 1 (by norm_num) le_rfl have hd (z : V) : ‖(x-r • z)-x‖ = r*‖z‖ := by rw [sub_sub_cancel_left,norm_neg,norm_smul,Real.norm_eq_abs,abs_of_pos hr] calc _ = ‖∫ z, (f (x-r • z) • atTime K N (a z) z-f x • atTime K N (A x) z)‖ := by rw [integral_sub hi hi₀,hzero,sub_zero] _ ≤ ∫ z, ‖f (x-r • z) • atTime K N (a z) z-f x • atTime K N (A x) z‖ := norm_integral_le_integral_norm _ _ ≤ ∫ z, (H*r^γ)*(‖z‖^γ*B z)+(‖f‖*L*r)*(‖z‖*C z) := by apply integral_mono (hi.sub hi₀).norm ((hiB.const_mul _).add (hiC.const_mul _)) intro z have hdec : f (x-r • z) • atTime K N (a z) z-f x • atTime K N (A x) z = (f (x-r • z)-f x) • atTime K N (a z) z+ f x • (atTime K N (a z) z-atTime K N (A x) z) := by module simp only [Pi.sub_apply, Pi.add_apply] rw [hdec] apply (norm_add_le _ _).trans rw [norm_smul,norm_smul] have hF : ‖f (x-r • z)-f x‖ ≤ H*r^γ*‖z‖^γ := by simpa only [hd,Real.mul_rpow hr.le (norm_nonneg _),mul_assoc] using hhold (x-r • z) x have hcoef : ‖atTime K N (a z) z-atTime K N (A x) z‖ ≤ C z*(L*r*‖z‖) := by exact (hc _ (hAr _) _ (hAr x) z).trans (mul_le_mul_of_nonneg_left (by simpa only [hd,mul_assoc] using hAL (x-r • z) x) (hCp z)) calc _ ≤ (H*r^γ*‖z‖^γ)*B z+‖f‖*(C z*(L*r*‖z‖)) := add_le_add (mul_le_mul hF (hb _ (hAr _) z) (norm_nonneg _) (by positivity)) (mul_le_mul (f.norm_coe_le_norm x) hcoef (norm_nonneg _) (norm_nonneg _)) _ = _ := by ring _ = _ := by rw [integral_add (hiB.const_mul _) (hiC.const_mul _),integral_const_mul,integral_const_mul] end SourceFrozenCancellation open Set Filter MeasureTheory open scoped SchwartzMap ContDiff FourierTransform BoundedContinuousFunction Convolution Topology namespace SchwartzDerivatives section Iterated variable {V E : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup E] [NormedSpace ℝ E] def iterated (g : SchwartzMap V E) : (n : ℕ) → SchwartzMap V (ContinuousMultilinearMap ℝ (fun _ : Fin n => V) E) | 0 => g.postcompCLM (continuousMultilinearCurryFin0 ℝ V E).symm.toContinuousLinearEquiv.toContinuousLinearMap | n+1 => (SchwartzMap.fderivCLM ℝ V _ (iterated g n)).postcompCLM (continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (n+1) => V) E).symm.toContinuousLinearEquiv.toContinuousLinearMap lemma iterated_coe (g : SchwartzMap V E) (n : ℕ) : (iterated g n : V → ContinuousMultilinearMap ℝ (fun _ : Fin n => V) E) = iteratedFDeriv ℝ n g := by induction n with | zero => rfl | succ n ih => rw [iteratedFDeriv_succ_eq_comp_left] change ((continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (n+1) => V) E).symm ∘ fderiv ℝ (iterated g n)) = _ rw [ih] lemma iterated_apply (g : SchwartzMap V E) (n : ℕ) (x : V) : iterated g n x = iteratedFDeriv ℝ n g x := congrFun (iterated_coe g n) x end Iterated variable {V E : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] omit [CompleteSpace E] in lemma integral_fderiv (g : SchwartzMap V E) : ∫ x : V, fderiv ℝ g x = 0 := by have h := congrArg (fun f : SchwartzMap V (V →L[ℝ] E) => f 0) (SchwartzMap.fourier_fderivCLM_eq (𝕜 := ℝ) g) simpa [SchwartzMap.fourier_coe, Real.fourier_eq, SchwartzMap.smulRightCLM_apply_apply, SchwartzMap.fderivCLM_apply] using h lemma integral_iterated (g : SchwartzMap V E) (n : ℕ) : ∫ x : V, iteratedFDeriv ℝ (n+1) g x = 0 := by let L := (continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (n+1) => V) E).symm.toContinuousLinearEquiv.toContinuousLinearMap let : ContinuousENorm (V →L[ℝ] ContinuousMultilinearMap ℝ (fun _ : Fin n => V) E) := @SeminormedAddGroup.toContinuousENorm (V →L[ℝ] ContinuousMultilinearMap ℝ (fun _ : Fin n => V) E) inferInstance have hi : Integrable (fun x : V => fderiv ℝ (iterated g n) x) := (SchwartzMap.fderivCLM ℝ V _ (iterated g n)).integrable rw [iteratedFDeriv_succ_eq_comp_left, ← iterated_coe g n] change (∫ x, L (fderiv ℝ (iterated g n) x)) = 0 have hh := ContinuousLinearMap.integral_comp_comm (𝕜 := ℝ) (E := V →L[ℝ] ContinuousMultilinearMap ℝ (fun _ : Fin n => V) E) (Fₗ := ContinuousMultilinearMap ℝ (fun _ : Fin (n+1) => V) E) L hi exact hh.trans (by rw [integral_fderiv, map_zero]) end SchwartzDerivatives namespace ScaledSchwartzConvolution variable {V E : Type} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] [MeasureSpace V] [BorelSpace V] [Measure.IsAddHaarMeasure (volume : Measure V)] [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] variable {a b : ℝ} (ha : 0 < a) (c : V → ℝ) (hc : Continuous c) (hcb : ∀ y, c y ∈ Icc a b) include ha hc hcb lemma iteratedFDeriv_value_smul (f : V →ᵇ ℂ) (g : SchwartzMap V E) (K n : ℕ) : iteratedFDeriv ℝ n (value c (ContinuousLinearMap.lsmul ℝ ℂ) f g K) = value c (ContinuousLinearMap.lsmul ℝ ℂ) f (SchwartzDerivatives.iterated g n) (K+n) := by induction n with | zero => funext x let L := (continuousMultilinearCurryFin0 ℝ V E).symm.toContinuousLinearEquiv.toContinuousLinearMap have hi : Integrable (fun y => f y • ((c y)⁻¹^K • g ((c y)⁻¹ • (x-y)))) := integrable_integrand ha c hc hcb (ContinuousLinearMap.lsmul ℝ ℂ) f g K x change L (∫ y, f y • (c y)⁻¹^K • g ((c y)⁻¹ • (x-y))) = ∫ y, f y • ((c y)⁻¹^K • L (g ((c y)⁻¹ • (x-y)))) rw [← ContinuousLinearMap.integral_comp_comm (𝕜 := ℝ) L hi] apply integral_congr_ae filter_upwards with y ext m rfl | succ n ih => rw [iteratedFDeriv_succ_eq_comp_left, ih, fderiv_value ha c hc hcb] funext x let g' := SchwartzMap.fderivCLM ℝ V _ (SchwartzDerivatives.iterated g n) let L := (continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (n+1) => V) E).symm.toContinuousLinearEquiv.toContinuousLinearMap have hi := integrable_integrand ha c hc hcb (E := V →L[ℝ] ContinuousMultilinearMap ℝ (fun _ : Fin n => V) E) (F := V →L[ℝ] ContinuousMultilinearMap ℝ (fun _ : Fin n => V) E) ((ContinuousLinearMap.lsmul ℝ ℂ).precompR V) f g' (K+n+1) x change L (∫ y, ((ContinuousLinearMap.lsmul ℝ ℂ).precompR V) (f y) ((c y)⁻¹^(K+n+1) • g' ((c y)⁻¹ • (x-y)))) = ∫ y, f y • ((c y)⁻¹^(K+(n+1)) • L (g' ((c y)⁻¹ • (x-y)))) simp only [← Nat.add_assoc] rw [← ContinuousLinearMap.integral_comp_comm (𝕜 := ℝ) L hi] apply integral_congr_ae filter_upwards with y ext m rfl lemma iteratedFDeriv_value_smul_apply (f : V →ᵇ ℂ) (g : SchwartzMap V E) (K n : ℕ) (x : V) : iteratedFDeriv ℝ n (value c (ContinuousLinearMap.lsmul ℝ ℂ) f g K) x = ∫ y, f y • ((c y)⁻¹^(K+n) • iteratedFDeriv ℝ n g ((c y)⁻¹ • (x-y))) := by rw [iteratedFDeriv_value_smul ha c hc hcb] simp only [value,ContinuousLinearMap.lsmul_apply,SchwartzDerivatives.iterated_apply] end ScaledSchwartzConvolution open Set Filter MeasureTheory open scoped Topology SchwartzMap namespace SourceFrozenScaling variable {V E : Type} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] [MeasureSpace V] [BorelSpace V] [Measure.IsAddHaarMeasure (volume : Measure V)] [NormedAddCommGroup E] [NormedSpace ℂ E] lemma normalized_integral (f : V → ℂ) (g : V → E) (c : V → ℝ) (r : ℝ) (hr : 0 < r) (K : ℕ) (x : V) : (∫ y, f y • ((r*c y)⁻¹^K • g ((r*c y)⁻¹ • (x-y)))) = (r^Module.finrank ℝ V*(r⁻¹)^K) • ∫ z, f (x-r • z) • ((c (x-r • z))⁻¹^K • g ((c (x-r • z))⁻¹ • z)) := by rw [← integral_sub_left_eq_self (fun y : V => f y • ((r*c y)⁻¹^K • g ((r*c y)⁻¹ • (x-y)))) volume x] simp only [sub_sub_self] let F : V → E := fun z => f (x-r • z) • ((c (x-r • z))⁻¹^K • g ((c (x-r • z))⁻¹ • z)) have he (y : V) : f (x-y) • ((r*c (x-y))⁻¹^K • g ((r*c (x-y))⁻¹ • y)) = (r⁻¹)^K • F (r⁻¹ • y) := by dsimp [F] rw [smul_inv_smul₀ hr.ne',mul_inv,mul_pow] rw [mul_comm r⁻¹ ((c (x-y))⁻¹),mul_smul] rw [mul_smul,smul_comm (f (x-y)) ((r⁻¹)^K)] simp_rw [he] rw [integral_smul,Measure.integral_comp_inv_smul_of_nonneg volume F hr.le,smul_smul] congr 1 ring lemma dimension_cancel (r : ℝ) (hr : r ≠ 0) (d n : ℕ) : r^d*(r⁻¹)^(d+n) = (r⁻¹)^n := by rw [pow_add,← mul_assoc,← mul_pow,mul_inv_cancel₀ hr,one_pow,one_mul] lemma normalized_integral_order (f : V → ℂ) (g : V → E) (c : V → ℝ) (r : ℝ) (hr : 0 < r) (n : ℕ) (x : V) : (∫ y, f y • ((r*c y)⁻¹^(Module.finrank ℝ V+n) • g ((r*c y)⁻¹ • (x-y)))) = (r⁻¹)^n • ∫ z, f (x-r • z) • ((c (x-r • z))⁻¹^(Module.finrank ℝ V+n) • g ((c (x-r • z))⁻¹ • z)) := by rw [normalized_integral f g c r hr,dimension_cancel r hr.ne'] end SourceFrozenScaling namespace SourceFrozenRows open ScaledSchwartzConvolution SchwartzTimeScale open scoped BoundedContinuousFunction ContDiff variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] def value (K : SchwartzMap V ℂ) (A : V → ℝ) (r : ℝ) (f : V →ᵇ ℂ) : V → ℂ := ScaledSchwartzConvolution.value (fun y => r*(A y)^(1/4 : ℝ)) (ContinuousLinearMap.lsmul ℝ ℂ) f K (Module.finrank ℝ V) def row (K : SchwartzMap V ℂ) (A : V → ℝ) (r : ℝ) (f : V →ᵇ ℂ) (j : ℕ) : V → V [×j]→L[ℝ] ℂ := ScaledSchwartzConvolution.value (fun y => r*(A y)^(1/4 : ℝ)) (ContinuousLinearMap.lsmul ℝ ℂ) f (SchwartzDerivatives.iterated K j) (Module.finrank ℝ V+j) omit [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in lemma radius_bounds (A : V → ℝ) (α β : ℝ) (hα : 0 < α) (hA : ∀ y, A y ∈ Icc α β) (r : ℝ) (hr : 0 < r) (y : V) : r*(A y)^(1/4 : ℝ) ∈ Icc (r*α^(1/4 : ℝ)) (r*β^(1/4 : ℝ)) := by constructor · exact mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hα.le (hA y).1 (by norm_num)) hr.le · exact mul_le_mul_of_nonneg_left (Real.rpow_le_rpow (hα.trans_le (hA y).1).le (hA y).2 (by norm_num)) hr.le lemma contDiff_value (K : SchwartzMap V ℂ) (A : V → ℝ) (hA : Continuous A) (α β : ℝ) (hα : 0 < α) (hAr : ∀ y, A y ∈ Icc α β) (r : ℝ) (hr : 0 < r) (f : V →ᵇ ℂ) : ContDiff ℝ ∞ (value K A r f) := by apply ScaledSchwartzConvolution.contDiff_value (a := r*α^(1/4 : ℝ)) (b := r*β^(1/4 : ℝ)) (by positivity) _ (continuous_const.mul (hA.rpow_const (fun _ => Or.inr (by norm_num)))) (radius_bounds A α β hα hAr r hr) lemma row_eq (K : SchwartzMap V ℂ) (A : V → ℝ) (hA : Continuous A) (α β : ℝ) (hα : 0 < α) (hAr : ∀ y, A y ∈ Icc α β) (r : ℝ) (hr : 0 < r) (f : V →ᵇ ℂ) (j : ℕ) : row K A r f j = iteratedFDeriv ℝ j (value K A r f) := by symm apply ScaledSchwartzConvolution.iteratedFDeriv_value_smul (a := r*α^(1/4 : ℝ)) (b := r*β^(1/4 : ℝ)) (by positivity) _ (continuous_const.mul (hA.rpow_const (fun _ => Or.inr (by norm_num)))) (radius_bounds A α β hα hAr r hr) lemma row_normalized (K : SchwartzMap V ℂ) (A : V → ℝ) (hAp : ∀ y, 0 < A y) (r : ℝ) (hr : 0 < r) (f : V →ᵇ ℂ) (j : ℕ) (x : V) : row K A r f j x = (r⁻¹)^j • ∫ z, f (x-r • z) • atTime (SchwartzDerivatives.iterated K j) (Module.finrank ℝ V+j) (A (x-r • z)) z := by unfold row ScaledSchwartzConvolution.value simp only [ContinuousLinearMap.lsmul_apply] rw [SourceFrozenScaling.normalized_integral_order _ _ _ r hr] congr 1 apply integral_congr_ae filter_upwards with z rw [atTime_eq_scaled _ _ _ (hAp _)] lemma exists_row_holder_bound (K : SchwartzMap V ℂ) (j : ℕ) (hj : 0 < j) (α β : ℝ) (hα : 0 < α) (γ : ℝ) (hγ : 0 ≤ γ) (hγ1 : γ ≤ 1) : ∃ C D : ℝ, 0 ≤ C ∧ 0 ≤ D ∧ ∀ (A : V → ℝ) (_hA : Continuous A) (_hAr : ∀ y, A y ∈ Icc α β) (L : ℝ) (_hL : 0 ≤ L) (_hAL : ∀ x y, ‖A x-A y‖ ≤ L*‖x-y‖) (f : V →ᵇ ℂ) (H : ℝ) (_hH : 0 ≤ H) (_hhold : ∀ x y, ‖f x-f y‖ ≤ H*‖x-y‖^γ) (r : ℝ) (_hr : 0 < r) (x : V), ‖row K A r f j x‖ ≤ (r⁻¹)^j*(H*r^γ*C+‖f‖*L*r*D) := by let K' := SchwartzDerivatives.iterated K j obtain ⟨B,hB,hBw,hBp,hb⟩ := SourceFrozenCancellation.parameter_envelope K' (Module.finrank ℝ V+j) α β hα obtain ⟨D,hD,hDw,hDp,hd⟩ := SourceFrozenCancellation.parameter_lipschitz_envelope K' (Module.finrank ℝ V+j) α β hα refine ⟨∫ z, ‖z‖^γ*B z, ∫ z, ‖z‖*D z, integral_nonneg (fun z => mul_nonneg (Real.rpow_nonneg (norm_nonneg _) _) (hBp z)), integral_nonneg (fun z => mul_nonneg (norm_nonneg _) (hDp z)), ?_⟩ intro A hA hAr L hL hAL f H hH hhold r hr x rw [row_normalized K A (fun y => hα.trans_le (hAr y).1) r hr, norm_smul,Real.norm_eq_abs,abs_of_nonneg (by positivity)] apply mul_le_mul_of_nonneg_left _ (by positivity) have hz : ∫ z, K' z = 0 := by obtain ⟨n,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hj) simp only [K',SchwartzDerivatives.iterated_apply] exact SchwartzDerivatives.integral_iterated K n exact SourceFrozenCancellation.norm_normalized_source_integral K' _ α β hα hz B D hB hBw hBp hD hDw hDp hb hd A hA hAr L hL hAL f γ H hγ hγ1 hH hhold r hr x end SourceFrozenRows namespace SourceFrozenOrder open ScaledSchwartzConvolution SchwartzTimeScale open scoped BoundedContinuousFunction ContDiff variable {V E : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] [NormedAddCommGroup E] [NormedSpace ℂ E] def value (K : SchwartzMap V E) (A : V → ℝ) (r : ℝ) (f : V →ᵇ ℂ) (j : ℕ) : V → E := ScaledSchwartzConvolution.value (fun y => r*(A y)^(1/4 : ℝ)) (ContinuousLinearMap.lsmul ℝ ℂ) f K (Module.finrank ℝ V+j) lemma normalized (K : SchwartzMap V E) (A : V → ℝ) (hAp : ∀ y, 0 < A y) (r : ℝ) (hr : 0 < r) (f : V →ᵇ ℂ) (j : ℕ) (x : V) : value K A r f j x = (r⁻¹)^j • ∫ z, f (x-r • z) • atTime K (Module.finrank ℝ V+j) (A (x-r • z)) z := by unfold value ScaledSchwartzConvolution.value simp only [ContinuousLinearMap.lsmul_apply] rw [SourceFrozenScaling.normalized_integral_order _ _ _ r hr] congr 1 apply integral_congr_ae filter_upwards with z rw [atTime_eq_scaled _ _ _ (hAp _)] lemma exists_norm_bound (K : SchwartzMap V E) (j : ℕ) (α β : ℝ) (hα : 0 < α) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (A : V → ℝ) (_hA : Continuous A) (_hAr : ∀ y, A y ∈ Icc α β) (r : ℝ) (_hr : 0 < r) (f : V →ᵇ ℂ) (x : V), ‖value K A r f j x‖ ≤ (r⁻¹)^j * (‖f‖*C) := by obtain ⟨B,hB,_hw,hBp,hb⟩ := SourceFrozenCancellation.parameter_envelope K (Module.finrank ℝ V+j) α β hα refine ⟨∫ z, B z,integral_nonneg hBp,?_⟩ intro A hA hAr r hr f x rw [normalized K A (fun y => hα.trans_le (hAr y).1) r hr,norm_smul,Real.norm_of_nonneg (by positivity : 0 ≤ (r⁻¹)^j)] apply mul_le_mul_of_nonneg_left _ (by positivity) have hi : Integrable (fun z => f (x-r • z) • atTime K (Module.finrank ℝ V+j) (A (x-r • z)) z) := by apply (hB.const_mul ‖f‖).mono' · exact ((f.continuous.comp (continuous_const.sub (continuous_const.smul continuous_id))).smul (SourceFrozenCancellation.continuous_variable K _ (fun z => A (x-r • z)) (hA.comp (continuous_const.sub (continuous_const.smul continuous_id))) (fun z => hα.trans_le (hAr _).1))).aestronglyMeasurable · filter_upwards with z rw [norm_smul] exact mul_le_mul (f.norm_coe_le_norm _) (hb _ (hAr _) z) (norm_nonneg _) (norm_nonneg _) change ‖∫ z, f (x-r • z) • atTime K _ (A (x-r • z)) z‖ ≤ _ calc _ ≤ ∫ z, ‖f (x-r • z) • atTime K _ (A (x-r • z)) z‖ := norm_integral_le_integral_norm _ _ ≤ ∫ z, ‖f‖*B z := by apply integral_mono hi.norm (hB.const_mul ‖f‖) intro z dsimp only rw [norm_smul] exact mul_le_mul (f.norm_coe_le_norm _) (hb _ (hAr _) z) (norm_nonneg _) (norm_nonneg _) _ = _ := integral_const_mul .. lemma hasFDerivAt_value (K : SchwartzMap V E) (A : V → ℝ) (hA : Continuous A) (α β : ℝ) (hα : 0 < α) (hAr : ∀ y, A y ∈ Icc α β) (r : ℝ) (hr : 0 < r) (f : V →ᵇ ℂ) (j : ℕ) (x : V) : HasFDerivAt (value K A r f j) (value (SchwartzMap.fderivCLM ℝ V E K) A r f (j+1) x) x := by have h := ScaledSchwartzConvolution.hasFDerivAt_value (a := r*α^(1/4 : ℝ)) (b := r*β^(1/4 : ℝ)) (by positivity) (fun y => r*(A y)^(1/4 : ℝ)) (continuous_const.mul (hA.rpow_const (fun _ => Or.inr (by norm_num)))) (SourceFrozenRows.radius_bounds A α β hα hAr r hr) (ContinuousLinearMap.lsmul ℝ ℂ) f K (Module.finrank ℝ V+j) x have he : (ContinuousLinearMap.lsmul ℝ ℂ : ℂ →L[ℝ] E →L[ℝ] E).precompR V = (ContinuousLinearMap.lsmul ℝ ℂ : ℂ →L[ℝ] (V →L[ℝ] E) →L[ℝ] (V →L[ℝ] E)) := by ext c g v rfl simpa only [value, Nat.add_assoc, he] using h lemma exists_bounds (K : SchwartzMap V E) (j : ℕ) (α β : ℝ) (hα : 0 < α) : ∃ C D : ℝ, 0 ≤ C ∧ 0 ≤ D ∧ ∀ (A : V → ℝ) (_hA : Continuous A) (_hAr : ∀ z, A z ∈ Icc α β) (r : ℝ) (_hr : 0 < r) (f : V →ᵇ ℂ), (∀ x, ‖value K A r f j x‖ ≤ (r⁻¹)^j*(‖f‖*C)) ∧ (∀ x y, ‖value K A r f j x-value K A r f j y‖ ≤ ((r⁻¹)^(j+1)*(‖f‖*D))*‖x-y‖) := by obtain ⟨C,hC,hb⟩ := exists_norm_bound K j α β hα obtain ⟨D,hD,hd⟩ := exists_norm_bound (SchwartzMap.fderivCLM ℝ V E K) (j+1) α β hα refine ⟨C,D,hC,hD,?_⟩ intro A hA hAr r hr f refine ⟨hb A hA hAr r hr f,fun x y => ?_⟩ exact (convex_univ : Convex ℝ (univ : Set V)).norm_image_sub_le_of_norm_hasFDerivWithin_le (fun z _ => (hasFDerivAt_value K A hA α β hα hAr r hr f j z).hasFDerivWithinAt) (fun z _ => hd A hA hAr r hr f z) (mem_univ y) (mem_univ x) end SourceFrozenOrder namespace FrozenParametrix open scoped SchwartzMap Laplacian ZeroAtInfty BoundedContinuousFunction open BiharmonicKernel variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] def bilapOne : SchwartzMap V ℂ := Δ (Δ (heatKernel (V := V) 1 zero_lt_one) : SchwartzMap V ℂ) def heatOne : SchwartzMap V ℂ := heatKernel 1 zero_lt_one lemma quarter_root (t : ℝ) (ht : 0 < t) : Real.sqrt (Real.sqrt t) = t^(1/4 : ℝ) := by rw [Real.sqrt_eq_rpow,Real.sqrt_eq_rpow,← Real.rpow_mul ht.le] norm_num lemma bilapKernel_ordered (t : ℝ) (ht : 0 < t) (r : ℝ) (hr : 0 < r) (z : V) : bilapKernel (r^4*t) z = ((r*t^(1/4 : ℝ))⁻¹)^(Module.finrank ℝ V+4) • bilapOne ((r*t^(1/4 : ℝ))⁻¹ • z) := by rw [bilapKernel_scaling t ht r hr,bilapKernel_eq_scaled t ht,quarter_root t ht] dsimp only [bilapOne] rw [mul_inv,mul_pow,smul_smul,smul_smul] congr 2 rw [mul_comm] lemma kernel_ordered (t : ℝ) (ht : 0 < t) (r : ℝ) (hr : 0 < r) (z : V) : kernel (r^4*t) z = ((r*t^(1/4 : ℝ))⁻¹)^(Module.finrank ℝ V) • heatOne ((r*t^(1/4 : ℝ))⁻¹ • z) := by have hs := heatKernel_scaling t ht r hr z rw [heatKernel_eq_kernel,heatKernel_eq_kernel] at hs rw [hs,kernel_eq_scaled t ht,quarter_root t ht] dsimp only [heatOne] rw [mul_inv,mul_pow,smul_smul,smul_smul] simp only [inv_pow] congr 2 rw [mul_comm] lemma integrable_ordered (K : SchwartzMap V ℂ) (j : ℕ) {α β : ℝ} (hα : 0 < α) (a : C(V,Icc α β)) (r : ℝ) (hr : 0 < r) (f : V →ᵇ ℂ) (x : V) : Integrable (fun y => f y • (((r*(a y).val^(1/4 : ℝ))⁻¹)^(Module.finrank ℝ V+j) • K ((r*(a y).val^(1/4 : ℝ))⁻¹ • (x-y)))) := by exact ScaledSchwartzConvolution.integrable_integrand (a := r*α^(1/4 : ℝ)) (b := r*β^(1/4 : ℝ)) (by positivity) (fun y => r*(a y).val^(1/4 : ℝ)) (continuous_const.mul ((continuous_subtype_val.comp a.continuous).rpow_const (fun _ => Or.inr (by norm_num)))) (SourceFrozenRows.radius_bounds (fun y => (a y).val) α β hα (fun y => (a y).property) r hr) (ContinuousLinearMap.lsmul ℝ ℂ) f K (Module.finrank ℝ V+j) x lemma principal_eq_ordered {α β : ℝ} (hα : 0 < α) (a₀ a₁ : C(V,Icc α β)) (r : ℝ) (hr : 0 < r) (f : C₀(V,ℂ)) (x : V) : principalResidual hα a₀ a₁ r f x = coefficient a₁ x * SourceFrozenOrder.value bilapOne (fun y => (a₀ y).val) r f.toBCF 4 x - SourceFrozenOrder.value bilapOne (fun y => (a₀ y).val) r (coefficient a₀*f.toBCF) 4 x := by rw [principalResidual_apply_unscaled hα a₀ a₁ r hr] simp_rw [bilapKernel_ordered _ (hα.trans_le (a₀ _).property.1) r hr] have hi := integrable_ordered bilapOne 4 hα a₀ r hr f.toBCF x have hi' := integrable_ordered bilapOne 4 hα a₀ r hr (coefficient a₀*f.toBCF) x unfold SourceFrozenOrder.value ScaledSchwartzConvolution.value simp only [ContinuousLinearMap.lsmul_apply,smul_eq_mul] at hi hi' ⊢ rw [← integral_const_mul,← integral_sub (hi.const_mul (coefficient a₁ x)) hi'] apply integral_congr_ae filter_upwards with y simp only [coefficient_apply,BoundedContinuousFunction.coe_mul,Pi.mul_apply, Complex.ofReal_sub] change (_-_)*_*f y = _*(f y*_)-(_*f y)*_ ring lemma frozen_eq_ordered {α β : ℝ} (hα : 0 < α) (a₀ : C(V,Icc α β)) (r : ℝ) (hr : 0 < r) (f : C₀(V,ℂ)) (x : V) : frozenOp hα a₀ r f x = SourceFrozenOrder.value heatOne (fun y => (a₀ y).val) r f.toBCF 0 x := by rw [frozenOp,frozen_apply_unscaled _ _ _ _ _ _ _ hr] unfold SourceFrozenOrder.value ScaledSchwartzConvolution.value simp only [Nat.add_zero,ContinuousLinearMap.lsmul_apply,smul_eq_mul] apply integral_congr_ae filter_upwards with y rw [kernel_ordered _ (hα.trans_le (a₀ _).property.1) r hr] exact mul_comm _ _ end FrozenParametrix open Set MeasureTheory namespace ProductLipschitz lemma bound {X : Type*} [NormedAddCommGroup X] (a u : X → ℂ) (B L M N : ℝ) (hB : 0 ≤ B) (hL : 0 ≤ L) (ha : ∀ x, ‖a x‖ ≤ B) (hu : ∀ x, ‖u x‖ ≤ M) (hal : ∀ x y, ‖a x-a y‖ ≤ L*‖x-y‖) (hul : ∀ x y, ‖u x-u y‖ ≤ N*‖x-y‖) (x y : X) : ‖a x*u x-a y*u y‖ ≤ (B*N+L*M)*‖x-y‖ := by have he : a x*u x-a y*u y = a x*(u x-u y)+(a x-a y)*u y := by ring rw [he] calc _ ≤ ‖a x*(u x-u y)‖+‖(a x-a y)*u y‖ := norm_add_le .. _ = ‖a x‖*‖u x-u y‖+‖a x-a y‖*‖u y‖ := by rw [norm_mul,norm_mul] _ ≤ B*(N*‖x-y‖)+(L*‖x-y‖)*M := add_le_add (mul_le_mul (ha x) (hul x y) (norm_nonneg _) hB) (mul_le_mul (hal x y) (hu y) (norm_nonneg _) (mul_nonneg hL (norm_nonneg _))) _ = _ := by ring lemma inv_pow_le {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) {j k : ℕ} (hjk : j ≤ k) : (r⁻¹)^j ≤ (r⁻¹)^k := by have hh : 1 ≤ r⁻¹ := (one_le_inv₀ hr).mpr hr1 exact pow_le_pow_right₀ hh hjk end ProductLipschitz namespace FrozenParametrix open scoped SchwartzMap Laplacian ZeroAtInfty BoundedContinuousFunction open BiharmonicKernel variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in lemma norm_coefficient_le_pos {α β : ℝ} (hα : 0 < α) (a : C(V,Icc α β)) : ‖coefficient a‖ ≤ max β 0 := by apply (BoundedContinuousFunction.norm_le (le_max_right _ _)).mpr intro x rw [coefficient_apply,Complex.norm_real,Real.norm_of_nonneg (hα.le.trans (a x).property.1)] exact (a x).property.2.trans (le_max_left _ _) lemma exists_principal_lipschitz_bound (α β : ℝ) (hα : 0 < α) (L : ℝ) (hL : 0 ≤ L) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (a₀ a₁ : C(V,Icc α β)) (_hLip : ∀ x y, |(a₁ x).val-(a₁ y).val| ≤ L*‖x-y‖) (r : ℝ) (_hr : 0 < r) (_hr1 : r ≤ 1) (f : C₀(V,ℂ)) (x y : V), ‖principalResidual hα a₀ a₁ r f x-principalResidual hα a₀ a₁ r f y‖ ≤ ((r⁻¹)^5*‖f‖*C)*‖x-y‖ := by obtain ⟨C,D,hC,hD,hb⟩ := SourceFrozenOrder.exists_bounds (bilapOne : SchwartzMap V ℂ) 4 α β hα let B := max β 0 have hB : 0 ≤ B := le_max_right _ _ refine ⟨L*C+2*B*D,by positivity,?_⟩ intro a₀ a₁ hLip r hr hr1 f x y let A := fun z => (a₀ z).val have hA : Continuous A := continuous_subtype_val.comp a₀.continuous have hAr : ∀ z, A z ∈ Icc α β := fun z => (a₀ z).property obtain ⟨hu,hul⟩ := hb A hA hAr r hr f.toBCF obtain ⟨_hv,hvl⟩ := hb A hA hAr r hr (coefficient a₀*f.toBCF) have ha : ∀ z, ‖coefficient a₁ z‖ ≤ B := fun z => ((coefficient a₁).norm_coe_le_norm z).trans (norm_coefficient_le_pos hα a₁) have hal : ∀ z w, ‖coefficient a₁ z-coefficient a₁ w‖ ≤ L*‖z-w‖ := by intro z w simpa only [coefficient_apply,← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs] using hLip z w have hmul : ‖coefficient a₀*f.toBCF‖ ≤ B*‖f‖ := by simpa only [ZeroAtInftyContinuousMap.norm_toBCF_eq_norm] using (norm_mul_le (coefficient a₀) f.toBCF).trans (mul_le_mul_of_nonneg_right (norm_coefficient_le_pos hα a₀) (norm_nonneg f.toBCF)) have hp := ProductLipschitz.bound (coefficient a₁) (SourceFrozenOrder.value bilapOne A r f.toBCF 4) B L ((r⁻¹)^4*(‖f.toBCF‖*C)) ((r⁻¹)^5*(‖f.toBCF‖*D)) hB hL ha hu hal hul x y simp_rw [principal_eq_ordered hα a₀ a₁ r hr] calc _ = ‖(coefficient a₁ x*SourceFrozenOrder.value bilapOne A r f.toBCF 4 x- coefficient a₁ y*SourceFrozenOrder.value bilapOne A r f.toBCF 4 y)- (SourceFrozenOrder.value bilapOne A r (coefficient a₀*f.toBCF) 4 x- SourceFrozenOrder.value bilapOne A r (coefficient a₀*f.toBCF) 4 y)‖ := by congr 1; ring _ ≤ ‖coefficient a₁ x*SourceFrozenOrder.value bilapOne A r f.toBCF 4 x- coefficient a₁ y*SourceFrozenOrder.value bilapOne A r f.toBCF 4 y‖+ ‖SourceFrozenOrder.value bilapOne A r (coefficient a₀*f.toBCF) 4 x- SourceFrozenOrder.value bilapOne A r (coefficient a₀*f.toBCF) 4 y‖ := norm_sub_le .. _ ≤ (B*((r⁻¹)^5*(‖f.toBCF‖*D))+L*((r⁻¹)^4*(‖f.toBCF‖*C)))*‖x-y‖+ ((r⁻¹)^5*(‖coefficient a₀*f.toBCF‖*D))*‖x-y‖ := add_le_add hp (hvl x y) _ ≤ (B*((r⁻¹)^5*(‖f‖*D))+L*((r⁻¹)^5*(‖f‖*C)))*‖x-y‖+ ((r⁻¹)^5*((B*‖f‖)*D))*‖x-y‖ := by simp only [ZeroAtInftyContinuousMap.norm_toBCF_eq_norm] gcongr <;> first | exact hmul | exact (one_le_inv₀ hr).mpr hr1 | norm_num _ = _ := by ring lemma exists_frozen_lipschitz_bound (α β : ℝ) (hα : 0 < α) (B L : ℝ) (hB : 0 ≤ B) (hL : 0 ≤ L) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (a : C(V,Icc α β)) (b : V →ᵇ ℂ) (_hb : ‖b‖ ≤ B) (_hLip : ∀ x y, ‖b x-b y‖ ≤ L*‖x-y‖) (r : ℝ) (_hr : 0 < r) (_hr1 : r ≤ 1) (f : C₀(V,ℂ)) (x y : V), ‖b x*frozenOp hα a r f x-b y*frozenOp hα a r f y‖ ≤ ((r⁻¹)^5*‖f‖*C)*‖x-y‖ := by obtain ⟨C,D,hC,hD,hbound⟩ := SourceFrozenOrder.exists_bounds (heatOne : SchwartzMap V ℂ) 0 α β hα refine ⟨B*D+L*C,by positivity,?_⟩ intro a b hb hLip r hr hr1 f x y obtain ⟨hu,hul⟩ := hbound (fun z => (a z).val) (continuous_subtype_val.comp a.continuous) (fun z => (a z).property) r hr f.toBCF have hh := ProductLipschitz.bound b (SourceFrozenOrder.value heatOne (fun z => (a z).val) r f.toBCF 0) B L ((r⁻¹)^0*(‖f.toBCF‖*C)) ((r⁻¹)^1*(‖f.toBCF‖*D)) hB hL (fun z => (b.norm_coe_le_norm z).trans hb) hu hLip hul x y simp_rw [frozen_eq_ordered hα a r hr] apply hh.trans calc _ ≤ (B*((r⁻¹)^5*(‖f‖*D))+L*((r⁻¹)^5*(‖f‖*C)))*‖x-y‖ := by simp only [ZeroAtInftyContinuousMap.norm_toBCF_eq_norm] gcongr <;> first | exact (one_le_inv₀ hr).mpr hr1 | norm_num _ = _ := by ring lemma exists_fullResidual_lipschitz_bound (α β : ℝ) (hα : 0 < α) (B L : ℝ) (hB : 0 ≤ B) (hL : 0 ≤ L) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (a₀ a₁ : C(V,Icc α β)) (b : V →ᵇ ℂ) (_hb : ‖b‖ ≤ B) (_hLipa : ∀ x y, |(a₁ x).val-(a₁ y).val| ≤ L*‖x-y‖) (_hLipb : ∀ x y, ‖b x-b y‖ ≤ L*‖x-y‖) (r : ℝ) (_hr : 0 < r) (_hr1 : r ≤ 1) (f : C₀(V,ℂ)) (x y : V), ‖fullResidualCLM hα a₀ a₁ b r f x-fullResidualCLM hα a₀ a₁ b r f y‖ ≤ ((r⁻¹)^5*‖f‖*C)*‖x-y‖ := by obtain ⟨C,hC,hp⟩ := exists_principal_lipschitz_bound (V := V) α β hα L hL obtain ⟨D,hD,hf⟩ := exists_frozen_lipschitz_bound (V := V) α β hα B L hB hL refine ⟨C+D,add_nonneg hC hD,?_⟩ intro a₀ a₁ b hb ha hl r hr hr1 f x y have he : fullResidualCLM hα a₀ a₁ b r f x-fullResidualCLM hα a₀ a₁ b r f y = (principalResidual hα a₀ a₁ r f x-principalResidual hα a₀ a₁ r f y)+ (b x*frozenOp hα a₀ r f x-b y*frozenOp hα a₀ r f y) := by simp only [fullResidualCLM_apply, ZeroAtInftyContinuousMap.add_apply, C0Multiplier.mul_apply] ring rw [he] calc _ ≤ ‖principalResidual hα a₀ a₁ r f x-principalResidual hα a₀ a₁ r f y‖+ ‖b x*frozenOp hα a₀ r f x-b y*frozenOp hα a₀ r f y‖ := norm_add_le .. _ ≤ ((r⁻¹)^5*‖f‖*C)*‖x-y‖+((r⁻¹)^5*‖f‖*D)*‖x-y‖ := add_le_add (hp a₀ a₁ ha r hr hr1 f x y) (hf a₀ b hb hl r hr hr1 f x y) _ = _ := by ring end FrozenParametrix open scoped Topology namespace HolderInterpolation lemma scalar {z A B δ γ : ℝ} (hz : 0 ≤ z) (hA : 0 ≤ A) (hB : 0 ≤ B) (hδ : 0 ≤ δ) (hγ : 0 < γ) (hγ1 : γ < 1) (ha : z ≤ A) (hb : z ≤ B*δ) : z ≤ A^(1-γ)*B^γ*δ^γ := by by_cases hz0 : z = 0 · rw [hz0]; positivity have hz' : 0 < z := lt_of_le_of_ne hz (Ne.symm hz0) calc _ = z^(1-γ)*z^γ := by rw [← Real.rpow_add hz']; simp _ ≤ A^(1-γ)*(B*δ)^γ := mul_le_mul (Real.rpow_le_rpow hz ha (by linarith)) (Real.rpow_le_rpow hz hb hγ.le) (Real.rpow_nonneg hz _) (Real.rpow_nonneg hA _) _ = _ := by rw [Real.mul_rpow hB hδ]; ring lemma scaled {z C δ q γ p₀ p₁ : ℝ} (hz : 0 ≤ z) (hC : 0 ≤ C) (hδ : 0 ≤ δ) (hq : 0 < q) (hγ : 0 < γ) (hγ1 : γ < 1) (ha : z ≤ C*q^p₀) (hb : z ≤ (C*q^p₁)*δ) : z ≤ C*q^(p₀*(1-γ)+p₁*γ)*δ^γ := by by_cases hC0 : C = 0 · simpa only [hC0,zero_mul] using ha have hC' : 0 < C := lt_of_le_of_ne hC (Ne.symm hC0) have he := scalar hz (mul_nonneg hC (Real.rpow_nonneg hq.le _)) (mul_nonneg hC (Real.rpow_nonneg hq.le _)) hδ hγ hγ1 ha hb rw [Real.mul_rpow hC (Real.rpow_nonneg hq.le _), Real.mul_rpow hC (Real.rpow_nonneg hq.le _), ← Real.rpow_mul hq.le,← Real.rpow_mul hq.le] at he have heq : C^(1-γ)*q^(p₀*(1-γ))*(C^γ*q^(p₁*γ))*δ^γ = C*q^(p₀*(1-γ)+p₁*γ)*δ^γ := by calc _ = (C^(1-γ)*C^γ)*(q^(p₀*(1-γ))*q^(p₁*γ))*δ^γ := by ring _ = _ := by rw [← Real.rpow_add hC',← Real.rpow_add hq]; simp exact heq ▸ he lemma residual {z C δ q γ : ℝ} (hz : 0 ≤ z) (hC : 0 ≤ C) (hδ : 0 ≤ δ) (hq : 0 < q) (hγ : 0 < γ) (hγ1 : γ < 1) (ha : z ≤ C*q^(-(3/4 : ℝ))) (hb : z ≤ (C*q^(-(5/4 : ℝ)))*δ) : z ≤ C*q^(-(3+2*γ)/4)*δ^γ := by simpa only [show (-(3/4 : ℝ))*(1-γ)+ (-(5/4 : ℝ))*γ = -(3+2*γ)/4 by ring] using (scaled hz hC hδ hq hγ hγ1 ha hb) end HolderInterpolation namespace ParabolicRadius lemma inv_pow_five {q : ℝ} (hq : 0 ≤ q) : (radius q)⁻¹^5 = q^(-5/4 : ℝ) := by unfold radius rw [← Real.rpow_natCast,← Real.rpow_neg hq,← Real.rpow_mul hq] congr 1 norm_num end ParabolicRadius namespace ParametrixVolterra open scoped ZeroAtInfty BoundedContinuousFunction open FrozenParametrix C0Convolution StrongWeightedVolterra variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {T α β : ℝ} (hα : 0 < α) (A : C(Slab T, V →ᵇ Icc α β)) (b : C(Slab T, V →ᵇ ℂ)) lemma exists_actualResidual_holder_bound (hT : T ≤ 1) (L B : ℝ) (hL : 0 ≤ L) (hB : 0 ≤ B) (hb : ∀ s, ‖b s‖ ≤ B) (hLip : ∀ s r : Slab T, ∀ x z : V, |(A s x).val-(A r z).val| ≤ L*(‖x-z‖+|s.val-r.val|)) (hLipb : ∀ s : Slab T, ∀ x z : V, ‖b s x-b s z‖ ≤ L*‖x-z‖) (γ : ℝ) (hγ : 0 < γ) (hγ1 : γ < 1) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (t : Time T) (s : Slab T) (_hst : s.val by change |(A (now t) z).val-(A (now t) w).val| ≤ L*‖z-w‖ simpa only [sub_self,abs_zero,add_zero] using hLip (now t) (now t) z w) (hLipb _) (ParabolicRadius.radius q) (ParabolicRadius.pos hq) (ParabolicRadius.le_one hq.le hq1) f x y rw [ParabolicRadius.inv_pow_five hq.le] at h apply h.trans calc _ = D*‖f‖*q^(-5/4 : ℝ)*‖x-y‖ := by ring _ ≤ _ := by gcongr; nlinarith simpa only [neg_div] using HolderInterpolation.residual (norm_nonneg _) (show 0 ≤ (2*C+D)*‖f‖ by positivity) (norm_nonneg (x-y)) hq hγ hγ1 (by simpa only [neg_div] using hs) (by simpa only [neg_div] using hl) end ParametrixVolterra open Set MeasureTheory namespace BetaWeights def density (σ ρ t s : ℝ) : ℝ := s^(-σ)*(t-s)^(-ρ) lemma density_nonneg (σ ρ t s : ℝ) (hs : 0 ≤ s) (hst : s ≤ t) : 0 ≤ density σ ρ t s := by exact mul_nonneg (Real.rpow_nonneg hs _) (Real.rpow_nonneg (sub_nonneg.mpr hst) _) lemma intervalIntegrable_one (σ ρ : ℝ) (hσ : σ < 1) (hρ : ρ < 1) : IntervalIntegrable (density σ ρ 1) volume 0 1 := by have hb := (Complex.betaIntegral_convergent (u := ((1-σ : ℝ) : ℂ)) (v := ((1-ρ : ℝ) : ℂ)) (by simpa using sub_pos.mpr hσ) (by simpa using sub_pos.mpr hρ)).norm apply hb.congr_ae rw [uIoc_of_le (by norm_num : (0:ℝ) ≤ 1),← restrict_Ioo_eq_restrict_Ioc] filter_upwards [ae_restrict_mem measurableSet_Ioo] with x hx dsimp only [density] rw [norm_mul] have hq : ((1-σ : ℝ) : ℂ)-1 = ((-σ : ℝ) : ℂ) := by push_cast; ring have hr : ((1-ρ : ℝ) : ℂ)-1 = ((-ρ : ℝ) : ℂ) := by push_cast; ring rw [hq,hr,show (1 : ℂ)-(x : ℂ) = ((1-x : ℝ) : ℂ) by push_cast; rfl, Complex.norm_cpow_real,Complex.norm_cpow_real,Complex.norm_real,Complex.norm_real, Real.norm_eq_abs,Real.norm_eq_abs,abs_of_pos hx.1,abs_of_pos (sub_pos.mpr hx.2)] lemma density_scaled (σ ρ t s : ℝ) (ht : 0 < t) (hs : s ∈ Ioo 0 1) : density σ ρ t (t*s) = (t^(-σ)*t^(-ρ))*density σ ρ 1 s := by unfold density rw [show t-t*s = t*(1-s) by ring,Real.mul_rpow ht.le hs.1.le, Real.mul_rpow ht.le (sub_nonneg.mpr hs.2.le)] ring lemma intervalIntegrable (σ ρ : ℝ) (hσ : σ < 1) (hρ : ρ < 1) (t : ℝ) (ht : 0 < t) : IntervalIntegrable (density σ ρ t) volume 0 t := by rw [← IntervalIntegrable.comp_mul_left_iff ht.ne'] simp only [zero_div,div_self ht.ne'] apply ((intervalIntegrable_one σ ρ hσ hρ).const_mul (t^(-σ)*t^(-ρ))).congr_ae rw [uIoc_of_le (by norm_num : (0:ℝ) ≤ 1)] have hmem : ∀ᵐ s ∂volume.restrict (Ioc (0:ℝ) 1), s ∈ Ioo 0 1 := by rw [← restrict_Ioo_eq_restrict_Ioc] exact ae_restrict_mem measurableSet_Ioo filter_upwards [hmem] with s hs exact (density_scaled σ ρ t s ht hs).symm lemma integral_scale (σ ρ t : ℝ) (ht : 0 < t) : (∫ s in 0..t, density σ ρ t s) = t^(1-σ-ρ)*(∫ s in 0..1, density σ ρ 1 s) := by have hc := intervalIntegral.smul_integral_comp_mul_left (a := 0) (b := 1) (density σ ρ t) t simp only [mul_zero,mul_one,smul_eq_mul] at hc rw [← hc] have he : (∫ s in 0..1, density σ ρ t (t*s)) = (t^(-σ)*t^(-ρ))*(∫ s in 0..1, density σ ρ 1 s) := by rw [← intervalIntegral.integral_const_mul] apply intervalIntegral.integral_congr_ae rw [uIoc_of_le (by norm_num : (0:ℝ) ≤ 1),← ae_restrict_iff' measurableSet_Ioc,← restrict_Ioo_eq_restrict_Ioc] filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs exact density_scaled σ ρ t s ht hs rw [he,← mul_assoc] congr 1 calc _ = t^(1 : ℝ)*(t^(-σ)*t^(-ρ)) := by rw [Real.rpow_one] _ = _ := by rw [← Real.rpow_add ht,← Real.rpow_add ht]; congr 1; ring end BetaWeights namespace HolderIntegral open scoped ZeroAtInfty BoundedContinuousFunction variable {V : Type*} [TopologicalSpace V] def eval (x : V) : C₀(V,ℂ) →L[ℝ] ℂ := LinearMap.mkContinuous { toFun := fun f => f x map_add' := fun _ _ => rfl map_smul' := fun _ _ => rfl } 1 (fun f => by change ‖f.toBCF x‖ ≤ 1*‖f‖ simpa only [one_mul,ZeroAtInftyContinuousMap.norm_toBCF_eq_norm] using f.toBCF.norm_coe_le_norm x) lemma intervalIntegral_apply (f : ℝ → C₀(V,ℂ)) {a b : ℝ} (hf : IntervalIntegrable f volume a b) (x : V) : (∫ s in a..b, f s) x = ∫ s in a..b, f s x := ((eval x).intervalIntegral_comp_comm hf).symm lemma beta_holder {x y : V} (f : ℝ → C₀(V,ℂ)) (t : ℝ) (ht : 0 < t) (hf : IntervalIntegrable f volume 0 t) (σ ρ : ℝ) (hσ : σ < 1) (hρ : ρ < 1) (C δ : ℝ) (h : ∀ s ∈ Ioo 0 t, ‖f s x-f s y‖ ≤ C*BetaWeights.density σ ρ t s*δ) : ‖(∫ s in 0..t, f s) x-(∫ s in 0..t, f s) y‖ ≤ C*t^(1-σ-ρ)*(∫ s in 0..1, BetaWeights.density σ ρ 1 s)*δ := by rw [intervalIntegral_apply f hf x,intervalIntegral_apply f hf y, ← intervalIntegral.integral_sub (show IntervalIntegrable (fun s => f s x) volume 0 t from ⟨(eval x).integrable_comp hf.1,(eval x).integrable_comp hf.2⟩) (show IntervalIntegrable (fun s => f s y) volume 0 t from ⟨(eval y).integrable_comp hf.1,(eval y).integrable_comp hf.2⟩)] calc _ ≤ ∫ s in 0..t, C*BetaWeights.density σ ρ t s*δ := by apply intervalIntegral.norm_integral_le_of_norm_le ht.le · rw [← ae_restrict_iff' measurableSet_Ioc,← restrict_Ioo_eq_restrict_Ioc] filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs exact h s hs · exact ((BetaWeights.intervalIntegrable σ ρ hσ hρ t ht).const_mul C).mul_const δ _ = _ := by rw [intervalIntegral.integral_mul_const,intervalIntegral.integral_const_mul, BetaWeights.integral_scale σ ρ t ht] ring end HolderIntegral namespace ParametrixVolterra open scoped ZeroAtInfty BoundedContinuousFunction open FrozenParametrix C0Convolution StrongWeightedVolterra variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable {T α β : ℝ} (hα : 0 < α) (A : C(Slab T, V →ᵇ Icc α β)) (b : C(Slab T, V →ᵇ ℂ)) lemma holder_residual_integral (hT₀ : 0 ≤ T) (C : ℝ) (hC : 0 ≤ C) (hR : ∀ (t : Time T) (s : Slab T), s.val residualKernel hα A b hT₀ t r (unweighted j r)) volume 0 t.val) (x y : V) : ‖(∫ r in 0..t.val, residualKernel hα A b hT₀ t r (unweighted j r)) x- (∫ r in 0..t.val, residualKernel hα A b hT₀ t r (unweighted j r)) y‖ ≤ (C*‖j‖)*t.val^(-5/8 : ℝ)*(∫ s in 0..1, BetaWeights.density (3/4) (7/8) 1 s)*‖x-y‖^(1/4 : ℝ) := by have hh := HolderIntegral.beta_holder (x := x) (y := y) (fun r => residualKernel hα A b hT₀ t r (unweighted j r)) t.val t.property.1 hi (3/4) (7/8) (by norm_num) (by norm_num) (C*‖j‖) (‖x-y‖^(1/4 : ℝ)) apply (show 1-(3/4 : ℝ)-7/8 = -5/8 by norm_num) ▸ hh intro r hr let s : Slab T := ⟨r,hr.1.le,hr.2.le.trans t.property.2⟩ have he : projIcc 0 T hT₀ r = s := projIcc_of_mem hT₀ s.property have hu : ‖unweighted j r‖ ≤ r^(-3/4 : ℝ)*‖j‖ := by let u : Time T := ⟨r,hr.1,hr.2.le.trans t.property.2⟩ change ‖unweighted j u.val‖ ≤ _ rw [unweighted_at_time,norm_smul,Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg hr.1.le _)] exact mul_le_mul_of_nonneg_left (j.norm_coe_le_norm u) (Real.rpow_nonneg hr.1.le _) simp only [residualKernel,ite_eq_left hr,he] calc _ ≤ C*‖unweighted j r‖*(t.val-r)^(-7/8 : ℝ)*‖x-y‖^(1/4 : ℝ) := hR t s hr.2 _ x y _ ≤ C*(r^(-3/4 : ℝ)*‖j‖)*(t.val-r)^(-7/8 : ℝ)*‖x-y‖^(1/4 : ℝ) := by gcongr exact Real.rpow_nonneg (sub_nonneg.mpr hr.2.le) _ _ = _ := by unfold BetaWeights.density; ring_nf lemma correction_holder (hT₀ : 0 ≤ T) (hT : T ≤ 1) (C : ℝ) (hC : 0 ≤ C) (hR : ∀ (t : Time T) (s : Slab T), s.val residualKernel hα A b hT₀ t r (unweighted j r)) volume 0 t.val) (hj : ∀ t : Time T, unweighted j t.val + ∫ r in 0..t.val, residualKernel hα A b hT₀ t r (unweighted j r) = l t-actualResidual hα A b t (initial hT₀) d) : ∃ D : ℝ, 0 ≤ D ∧ ∀ t : Time T, ∀ x y, ‖unweighted j t.val x-unweighted j t.val y‖ ≤ D*t.val^(-7/8 : ℝ)*‖x-y‖^(1/4 : ℝ) := by let Q := ∫ s in 0..1, BetaWeights.density (3/4) (7/8) 1 s have hQ : 0 ≤ Q := intervalIntegral.integral_nonneg (by norm_num : (0:ℝ) ≤ 1) (fun s hs => BetaWeights.density_nonneg _ _ _ _ hs.1 hs.2) refine ⟨H+C*‖d‖+C*‖j‖*Q,by positivity,fun t x y => ?_⟩ let v := ∫ r in 0..t.val, residualKernel hα A b hT₀ t r (unweighted j r) have he : unweighted j t.val = l t-actualResidual hα A b t (initial hT₀) d-v := eq_sub_of_add_eq (hj t) have hd := hR t (initial hT₀) t.property.1 d x y simp only [initial,sub_zero] at hd have hv := holder_residual_integral hα A b hT₀ C hC hR j t (hi t) x y have ht1 : t.val ≤ 1 := t.property.2.trans hT have hp : 1 ≤ t.val^(-7/8 : ℝ) := Real.one_le_rpow_of_pos_of_le_one_of_nonpos t.property.1 ht1 (by norm_num) have hq : t.val^(-5/8 : ℝ) ≤ t.val^(-7/8 : ℝ) := Real.rpow_le_rpow_of_exponent_ge t.property.1 ht1 (by norm_num) rw [he] change ‖(l t x-actualResidual hα A b t (initial hT₀) d x-v x)- (l t y-actualResidual hα A b t (initial hT₀) d y-v y)‖ ≤ _ calc _ = ‖(l t x-l t y)-(actualResidual hα A b t (initial hT₀) d x- actualResidual hα A b t (initial hT₀) d y)-(v x-v y)‖ := by congr 1; ring _ ≤ ‖l t x-l t y‖+‖actualResidual hα A b t (initial hT₀) d x- actualResidual hα A b t (initial hT₀) d y‖+‖v x-v y‖ := (norm_sub_le _ _).trans (add_le_add (norm_sub_le _ _) le_rfl) _ ≤ H*‖x-y‖^(1/4 : ℝ)+C*‖d‖*t.val^(-7/8 : ℝ)*‖x-y‖^(1/4 : ℝ)+ (C*‖j‖)*t.val^(-5/8 : ℝ)*Q*‖x-y‖^(1/4 : ℝ) := add_le_add (add_le_add (hl t x y) hd) hv _ ≤ H*t.val^(-7/8 : ℝ)*‖x-y‖^(1/4 : ℝ)+C*‖d‖*t.val^(-7/8 : ℝ)*‖x-y‖^(1/4 : ℝ)+ (C*‖j‖)*t.val^(-7/8 : ℝ)*Q*‖x-y‖^(1/4 : ℝ) := by gcongr; exact le_mul_of_one_le_right hH hp _ = _ := by ring end ParametrixVolterra namespace SourceFrozenRows section ExistsRowZeroBound open ScaledSchwartzConvolution SchwartzTimeScale open scoped BoundedContinuousFunction ContDiff variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] lemma exists_row_zero_bound (K : SchwartzMap V ℂ) (α β : ℝ) (hα : 0 < α) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (A : V → ℝ) (_hA : Continuous A) (_hAr : ∀ y, A y ∈ Icc α β) (r : ℝ) (_hr : 0 < r) (f : V →ᵇ ℂ) (x : V), ‖row K A r f 0 x‖ ≤ ‖f‖*C := by let K' := SchwartzDerivatives.iterated K 0 obtain ⟨B,hB,_hw,hBp,hb⟩ := SourceFrozenCancellation.parameter_envelope K' (Module.finrank ℝ V+0) α β hα refine ⟨∫ z, B z,integral_nonneg hBp,?_⟩ intro A hA hAr r hr f x rw [row_normalized K A (fun y => hα.trans_le (hAr y).1) r hr,pow_zero,one_smul] have hi : Integrable (fun z => f (x-r • z) • atTime K' (Module.finrank ℝ V+0) (A (x-r • z)) z) := by apply (hB.const_mul ‖f‖).mono' · exact ((f.continuous.comp (continuous_const.sub (continuous_const.smul continuous_id))).smul (SourceFrozenCancellation.continuous_variable K' _ (fun z => A (x-r • z)) (hA.comp (continuous_const.sub (continuous_const.smul continuous_id))) (fun z => hα.trans_le (hAr _).1))).aestronglyMeasurable · filter_upwards with z rw [norm_smul] exact mul_le_mul (f.norm_coe_le_norm _) (hb _ (hAr _) z) (norm_nonneg _) (norm_nonneg _) change ‖∫ z, f (x-r • z) • atTime K' _ (A (x-r • z)) z‖ ≤ _ calc _ ≤ ∫ z, ‖f (x-r • z) • atTime K' _ (A (x-r • z)) z‖ := norm_integral_le_integral_norm _ _ ≤ ∫ z, ‖f‖*B z := by apply integral_mono hi.norm (hB.const_mul ‖f‖) intro z dsimp only rw [norm_smul] exact mul_le_mul (f.norm_coe_le_norm _) (hb _ (hAr _) z) (norm_nonneg _) (norm_nonneg _) _ = _ := integral_const_mul .. lemma continuous_row (K : SchwartzMap V ℂ) (A : V → ℝ) (hA : Continuous A) (α β : ℝ) (hα : 0 < α) (hAr : ∀ y, A y ∈ Icc α β) (r : ℝ) (hr : 0 < r) (f : V →ᵇ ℂ) (j : ℕ) : Continuous (row K A r f j) := by rw [row_eq K A hA α β hα hAr r hr] have hd : ContDiff ℝ (j+1) (value K A r f) := by simpa only [Nat.cast_add,Nat.cast_one] using (contDiff_infty.mp (contDiff_value K A hA α β hα hAr r hr f) (j+1)) exact hd.continuous_iteratedFDeriv (by exact_mod_cast Nat.le_succ j) lemma hasFDerivAt_row (K : SchwartzMap V ℂ) (A : V → ℝ) (hA : Continuous A) (α β : ℝ) (hα : 0 < α) (hAr : ∀ y, A y ∈ Icc α β) (r : ℝ) (hr : 0 < r) (f : V →ᵇ ℂ) (j : ℕ) (x : V) : HasFDerivAt (row K A r f j) ((continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (j+1) => V) ℂ) (row K A r f (j+1) x)) x := by rw [row_eq K A hA α β hα hAr r hr f j, row_eq K A hA α β hα hAr r hr f (j+1)] have hd : ContDiff ℝ (j+1) (value K A r f) := by simpa only [Nat.cast_add,Nat.cast_one] using (contDiff_infty.mp (contDiff_value K A hA α β hα hAr r hr f) (j+1)) have hd' := (hd.differentiable_iteratedFDeriv (m := j) (by exact_mod_cast Nat.lt_succ_self j) x).hasFDerivAt simpa only [fderiv_iteratedFDeriv, Function.comp_apply] using hd' lemma stronglyMeasurable_time_row (K : SchwartzMap V ℂ) (A : ℝ → V → ℝ) (hA : Continuous (Function.uncurry A)) (f : ℝ → (V →ᵇ ℂ)) (hf : Continuous f) (j : ℕ) (t : ℝ) (x : V) : StronglyMeasurable (fun s : ℝ => row K (A s) ((t-s)^(1/4 : ℝ)) (f s) j x) := by let c : ℝ × V → ℝ := fun p => (((t-p.1)^(1/4 : ℝ))*(A p.1 p.2)^(1/4 : ℝ))⁻¹ have hc : Measurable c := by dsimp [c] exact (((measurable_const.sub measurable_fst).pow_const _).mul (hA.measurable.pow_const _)).inv have he : Continuous (fun p : ℝ × V => f p.1 p.2) := by fun_prop have hg := (SchwartzDerivatives.iterated K j).continuous have hm : StronglyMeasurable (fun p : ℝ × V => f p.1 p.2 • (c p ^ (Module.finrank ℝ V+j) • SchwartzDerivatives.iterated K j (c p • (x-p.2)))) := he.stronglyMeasurable.smul ((hc.pow_const _).stronglyMeasurable.smul (hg.comp_stronglyMeasurable (hc.stronglyMeasurable.smul (continuous_const.sub continuous_snd).stronglyMeasurable))) exact hm.integral_prod_right' omit [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in lemma radius_weight (q : ℝ) (hq : 0 < q) (j : ℕ) (p : ℝ) : ((q^(1/4 : ℝ))⁻¹)^j*(q^(1/4 : ℝ))^p = q^((p-j)/4) := by rw [inv_pow,← Real.rpow_natCast,← Real.rpow_mul hq.le,← Real.rpow_neg hq.le, ← Real.rpow_mul hq.le,← Real.rpow_add hq] congr 1 ring end ExistsRowZeroBound section IntegrableOnLagRpow open SchwartzTimeScale open scoped BoundedContinuousFunction ContDiff variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] lemma integrableOn_lag_rpow (t p : ℝ) (ht : 0 ≤ t) (hp : -1 < p) : IntegrableOn (fun s : ℝ => (t-s)^p) (Ioo 0 t) := by have hi := (intervalIntegral.intervalIntegrable_rpow' (a := 0) (b := t) hp).comp_sub_left t simp only [sub_zero,sub_self] at hi exact (intervalIntegrable_iff_integrableOn_Ioo_of_le ht).mp hi.symm lemma exists_integrable_row_bound (K : SchwartzMap V ℂ) (A : ℝ → V → ℝ) (hA : Continuous (Function.uncurry A)) (α β : ℝ) (hα : 0 < α) (hAr : ∀ s y, A s y ∈ Icc α β) (L : ℝ) (hL : 0 ≤ L) (hAL : ∀ s x y, ‖A s x-A s y‖ ≤ L*‖x-y‖) (f : ℝ → (V →ᵇ ℂ)) (hf : Continuous f) (γ H t : ℝ) (ht : 0 ≤ t) (hγ : 0 < γ) (hγ1 : γ ≤ 1) (hH : 0 ≤ H) (hhold : ∀ s ∈ Ioo 0 t, ∀ x y : V, ‖f s x-f s y‖ ≤ H*‖x-y‖^γ) (j : ℕ) (hj4 : j ≤ 4) : ∃ b : ℝ → ℝ, IntegrableOn b (Ioo 0 t) ∧ ∀ s ∈ Ioo 0 t, ∀ x : V, ‖row K (A s) ((t-s)^(1/4 : ℝ)) (f s) j x‖ ≤ b s := by obtain ⟨M,hM⟩ := (isCompact_Icc (a := (0 : ℝ)) (b := t)).exists_bound_of_continuousOn hf.continuousOn have hAs (s : ℝ) : Continuous (A s) := hA.comp (continuous_const.prodMk continuous_id) by_cases hj : j = 0 · subst j obtain ⟨C,hC,hb⟩ := exists_row_zero_bound K α β hα refine ⟨fun _ => M*C,integrableOn_const (hs := measure_Ioo_lt_top.ne),?_⟩ intro s hs x exact (hb (A s) (hAs s) (hAr s) _ (Real.rpow_pos_of_pos (sub_pos.mpr hs.2) _) (f s) x).trans (mul_le_mul_of_nonneg_right (hM s ⟨hs.1.le,hs.2.le⟩) hC) · obtain ⟨C,D,hC,hD,hb⟩ := exists_row_holder_bound K j (Nat.pos_of_ne_zero hj) α β hα γ hγ.le hγ1 have hp : -1 < (γ-j)/4 := by have : (j : ℝ) ≤ 4 := by exact_mod_cast hj4 linarith have hp' : -1 < ((1 : ℝ)-j)/4 := by have : (j : ℝ) ≤ 4 := by exact_mod_cast hj4 linarith refine ⟨fun s => H*C*(t-s)^((γ-j)/4)+M*L*D*(t-s)^(((1 : ℝ)-j)/4), ((integrableOn_lag_rpow t _ ht hp).const_mul _).add ((integrableOn_lag_rpow t _ ht hp').const_mul _),?_⟩ intro s hs x have hq : 0 < t-s := sub_pos.mpr hs.2 have hr : 0 < (t-s)^(1/4 : ℝ) := Real.rpow_pos_of_pos hq _ have he := hb (A s) (hAs s) (hAr s) L hL (hAL s) (f s) H hH (hhold s hs) _ hr x have heq : (((t-s)^(1/4 : ℝ))⁻¹)^j * (H*((t-s)^(1/4 : ℝ))^γ*C+‖f s‖*L*((t-s)^(1/4 : ℝ))*D) = H*C*(t-s)^((γ-j)/4)+‖f s‖*L*D*(t-s)^(((1 : ℝ)-j)/4) := by rw [mul_add] have h1 := radius_weight (t-s) hq j γ have h2 := radius_weight (t-s) hq j 1 rw [Real.rpow_one] at h2 calc _ = H*C*((((t-s)^(1/4 : ℝ))⁻¹)^j*((t-s)^(1/4 : ℝ))^γ) + ‖f s‖*L*D*((((t-s)^(1/4 : ℝ))⁻¹)^j*(t-s)^(1/4 : ℝ)) := by ring _ = _ := by rw [h1,h2] rw [heq] at he apply he.trans exact add_le_add le_rfl (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right (hM s ⟨hs.1.le,hs.2.le⟩) hL) hD) (Real.rpow_nonneg hq.le _)) def integralRow (K : SchwartzMap V ℂ) (A : ℝ → V → ℝ) (f : ℝ → (V →ᵇ ℂ)) (t : ℝ) (j : ℕ) (x : V) : V [×j]→L[ℝ] ℂ := ∫ s in Ioo 0 t, row K (A s) ((t-s)^(1/4 : ℝ)) (f s) j x variable (K : SchwartzMap V ℂ) (A : ℝ → V → ℝ) (hA : Continuous (Function.uncurry A)) (α β : ℝ) (hα : 0 < α) (hAr : ∀ s y, A s y ∈ Icc α β) (L : ℝ) (hL : 0 ≤ L) (hAL : ∀ s x y, ‖A s x-A s y‖ ≤ L*‖x-y‖) (f : ℝ → (V →ᵇ ℂ)) (hf : Continuous f) (γ H t : ℝ) (ht : 0 ≤ t) (hγ : 0 < γ) (hγ1 : γ ≤ 1) (hH : 0 ≤ H) (hhold : ∀ s ∈ Ioo 0 t, ∀ x y : V, ‖f s x-f s y‖ ≤ H*‖x-y‖^γ) include hA hα hAr hf hAL hL hH ht hγ hγ1 hhold lemma integrableOn_row (j : ℕ) (hj4 : j ≤ 4) (x : V) : IntegrableOn (fun s => row K (A s) ((t-s)^(1/4 : ℝ)) (f s) j x) (Ioo 0 t) := by obtain ⟨b,hi,hb⟩ := exists_integrable_row_bound K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold j hj4 exact hi.mono' (stronglyMeasurable_time_row K A hA f hf j t x).aestronglyMeasurable (by filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs; exact hb s hs x) lemma continuous_integralRow (j : ℕ) (hj4 : j ≤ 4) : Continuous (integralRow K A f t j) := by obtain ⟨b,hbi,hb⟩ := exists_integrable_row_bound K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold j hj4 apply continuous_of_dominated (bound := b) · intro x exact (stronglyMeasurable_time_row K A hA f hf j t x).aestronglyMeasurable · intro x filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs exact hb s hs x · exact hbi · filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs exact continuous_row K (A s) (hA.comp (continuous_const.prodMk continuous_id)) α β hα (hAr s) _ (Real.rpow_pos_of_pos (sub_pos.mpr hs.2) _) (f s) j lemma hasFDerivAt_integralRow (j : ℕ) (hj4 : j+1 ≤ 4) (x : V) : HasFDerivAt (integralRow K A f t j) ((continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (j+1) => V) ℂ) (integralRow K A f t (j+1) x)) x := by let C := (continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (j+1) => V) ℂ).toContinuousLinearEquiv.toContinuousLinearMap let : ContinuousENorm (V →L[ℝ] V [×j]→L[ℝ] ℂ) := @SeminormedAddGroup.toContinuousENorm (V →L[ℝ] V [×j]→L[ℝ] ℂ) inferInstance have hi := integrableOn_row K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold (j+1) hj4 x have hcomm := ContinuousLinearMap.integral_comp_comm (𝕜 := ℝ) (E := V [×(j+1)]→L[ℝ] ℂ) (Fₗ := V →L[ℝ] V [×j]→L[ℝ] ℂ) C hi change HasFDerivAt (fun x => ∫ s in Ioo 0 t, row K (A s) ((t-s)^(1/4 : ℝ)) (f s) j x) (C (∫ s in Ioo 0 t, row K (A s) ((t-s)^(1/4 : ℝ)) (f s) (j+1) x)) x rw [← hcomm] obtain ⟨b,hbi,hb⟩ := exists_integrable_row_bound K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold (j+1) hj4 apply hasFDerivAt_integral_of_dominated_of_fderiv_le (s := univ) (F' := fun x s => C (row K (A s) ((t-s)^(1/4 : ℝ)) (f s) (j+1) x)) (bound := b) (univ_mem : univ ∈ 𝓝 x) · exact Eventually.of_forall fun z => (stronglyMeasurable_time_row K A hA f hf j t z).aestronglyMeasurable · exact integrableOn_row K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold j (by omega) x · exact (C.continuous.comp_stronglyMeasurable (stronglyMeasurable_time_row K A hA f hf (j+1) t x)).aestronglyMeasurable · filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs z _ change ‖(continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (j+1) => V) ℂ) (row K (A s) ((t-s)^(1/4 : ℝ)) (f s) (j+1) z)‖ ≤ b s rw [LinearIsometryEquiv.norm_map] exact hb s hs z · exact hbi · filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs z _ exact hasFDerivAt_row K (A s) (hA.comp (continuous_const.prodMk continuous_id)) α β hα (hAr s) _ (Real.rpow_pos_of_pos (sub_pos.mpr hs.2) _) (f s) j z lemma contDiff_integralRow (j n : ℕ) (hjn : j+n ≤ 4) : ContDiff ℝ n (integralRow K A f t j) := by induction n generalizing j with | zero => rw [Nat.cast_zero, contDiff_zero] exact continuous_integralRow K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold j (by omega) | succ n ih => rw [Nat.cast_add, Nat.cast_one, contDiff_succ_iff_hasFDerivAt] let C : (V [×(j+1)]→L[ℝ] ℂ) →L[ℝ] (V →L[ℝ] V [×j]→L[ℝ] ℂ) := (continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (j+1) => V) ℂ).toContinuousLinearEquiv.toContinuousLinearMap refine ⟨fun x => C (integralRow K A f t (j+1) x), ?_, ?_⟩ · exact contDiff_const.clm_apply (ih (j+1) (by omega)) · intro x exact hasFDerivAt_integralRow K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold j (by omega) x end IntegrableOnLagRpow section Duhamel open scoped BoundedContinuousFunction ContDiff variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] def duhamel (K : SchwartzMap V ℂ) (A : ℝ → V → ℝ) (f : ℝ → (V →ᵇ ℂ)) (t : ℝ) (x : V) : ℂ := ∫ s in Ioo 0 t, value K (A s) ((t-s)^(1/4 : ℝ)) (f s) x variable (K : SchwartzMap V ℂ) (A : ℝ → V → ℝ) (hA : Continuous (Function.uncurry A)) (α β : ℝ) (hα : 0 < α) (hAr : ∀ s y, A s y ∈ Icc α β) (L : ℝ) (hL : 0 ≤ L) (hAL : ∀ s x y, ‖A s x-A s y‖ ≤ L*‖x-y‖) (f : ℝ → (V →ᵇ ℂ)) (hf : Continuous f) (γ H t : ℝ) (ht : 0 ≤ t) (hγ : 0 < γ) (hγ1 : γ ≤ 1) (hH : 0 ≤ H) (hhold : ∀ s ∈ Ioo 0 t, ∀ x y : V, ‖f s x-f s y‖ ≤ H*‖x-y‖^γ) include hA hα hAr hf hAL hL hH ht hγ hγ1 hhold lemma duhamel_eq_integralRow_zero (x : V) : duhamel K A f t x = (continuousMultilinearCurryFin0 ℝ V ℂ) (integralRow K A f t 0 x) := by let C := (continuousMultilinearCurryFin0 ℝ V ℂ).toContinuousLinearEquiv.toContinuousLinearMap have hc := ContinuousLinearMap.integral_comp_comm (𝕜 := ℝ) (E := V [×0]→L[ℝ] ℂ) (Fₗ := ℂ) C (integrableOn_row K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold 0 (by omega) x) change (∫ s in Ioo 0 t, value K (A s) ((t-s)^(1/4 : ℝ)) (f s) x) = C (∫ s in Ioo 0 t, row K (A s) ((t-s)^(1/4 : ℝ)) (f s) 0 x) rw [← hc] apply integral_congr_ae filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs rw [row_eq K (A s) (hA.comp (continuous_const.prodMk continuous_id)) α β hα (hAr s) _ (Real.rpow_pos_of_pos (sub_pos.mpr hs.2) _),iteratedFDeriv_zero_eq_comp] exact (LinearIsometryEquiv.apply_symm_apply _ _).symm lemma contDiff_duhamel : ContDiff ℝ 4 (duhamel K A f t) := by have he : duhamel K A f t = fun x => (continuousMultilinearCurryFin0 ℝ V ℂ) (integralRow K A f t 0 x) := funext (duhamel_eq_integralRow_zero K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold) rw [he] let C : (V [×0]→L[ℝ] ℂ) →L[ℝ] ℂ := (continuousMultilinearCurryFin0 ℝ V ℂ).toContinuousLinearEquiv.toContinuousLinearMap change ContDiff ℝ 4 (fun x => C (integralRow K A f t 0 x)) exact contDiff_const.clm_apply (contDiff_integralRow K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold 0 4 (by omega)) lemma iteratedFDeriv_duhamel (j : ℕ) (hj : j ≤ 4) : iteratedFDeriv ℝ j (duhamel K A f t) = integralRow K A f t j := by induction j with | zero => funext x rw [iteratedFDeriv_zero_eq_comp] change (continuousMultilinearCurryFin0 ℝ V ℂ).symm (duhamel K A f t x) = _ rw [duhamel_eq_integralRow_zero K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold, LinearIsometryEquiv.symm_apply_apply] | succ j ih => rw [iteratedFDeriv_succ_eq_comp_left,ih (by omega)] funext x dsimp only [Function.comp_apply] rw [(hasFDerivAt_integralRow K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold j hj x).fderiv] exact LinearIsometryEquiv.symm_apply_apply _ _ lemma exists_norm_iteratedFDeriv_duhamel_bound (j : ℕ) (hj : j ≤ 4) : ∃ C : ℝ, ∀ x, ‖iteratedFDeriv ℝ j (duhamel K A f t) x‖ ≤ C := by obtain ⟨b,hi,hb⟩ := exists_integrable_row_bound K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold j hj refine ⟨∫ s in Ioo 0 t, b s,fun x => ?_⟩ rw [iteratedFDeriv_duhamel K A hA α β hα hAr L hL hAL f hf γ H t ht hγ hγ1 hH hhold j hj] apply norm_integral_le_of_norm_le hi filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs exact hb s hs x end Duhamel open ScaledSchwartzConvolution SchwartzTimeScale open scoped BoundedContinuousFunction ContDiff variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] lemma stronglyMeasurable_time_row_of_stronglyMeasurable (K : SchwartzMap V ℂ) (A : ℝ → V → ℝ) (hA : Continuous (Function.uncurry A)) (f : ℝ → (V →ᵇ ℂ)) (hf : StronglyMeasurable f) (j : ℕ) (t : ℝ) (x : V) : StronglyMeasurable (fun s : ℝ => row K (A s) ((t-s)^(1/4 : ℝ)) (f s) j x) := by let c : ℝ × V → ℝ := fun p => (((t-p.1)^(1/4 : ℝ))*(A p.1 p.2)^(1/4 : ℝ))⁻¹ have hc : Measurable c := by dsimp [c] exact (((measurable_const.sub measurable_fst).pow_const _).mul (hA.measurable.pow_const _)).inv have he' : Continuous (fun p : (V →ᵇ ℂ) × V => p.1 p.2) := by fun_prop have he : StronglyMeasurable (fun p : ℝ × V => f p.1 p.2) := he'.comp_stronglyMeasurable ((hf.comp_measurable measurable_fst).prodMk continuous_snd.stronglyMeasurable) have hg := (SchwartzDerivatives.iterated K j).continuous have hm : StronglyMeasurable (fun p : ℝ × V => f p.1 p.2 • (c p ^ (Module.finrank ℝ V+j) • SchwartzDerivatives.iterated K j (c p • (x-p.2)))) := he.smul ((hc.pow_const _).stronglyMeasurable.smul (hg.comp_stronglyMeasurable (hc.stronglyMeasurable.smul (continuous_const.sub continuous_snd).stronglyMeasurable))) exact hm.integral_prod_right' lemma exists_integrable_row_bound_weighted (K : SchwartzMap V ℂ) (A : ℝ → V → ℝ) (hA : Continuous (Function.uncurry A)) (α β : ℝ) (hα : 0 < α) (hAr : ∀ s y, A s y ∈ Icc α β) (L : ℝ) (hL : 0 ≤ L) (hAL : ∀ s x y, ‖A s x-A s y‖ ≤ L*‖x-y‖) (f : ℝ → (V →ᵇ ℂ)) (σ τ M H γ t : ℝ) (hσ : σ < 1) (hτ : τ < 1) (_hM : 0 ≤ M) (hH : 0 ≤ H) (ht : 0 < t) (hγ : 0 < γ) (hγ1 : γ ≤ 1) (hfn : ∀ s ∈ Ioo 0 t, ‖f s‖ ≤ M*s^(-σ)) (hhold : ∀ s ∈ Ioo 0 t, ∀ x y : V, ‖f s x-f s y‖ ≤ (H*s^(-τ))*‖x-y‖^γ) (j : ℕ) (hj4 : j ≤ 4) : ∃ b : ℝ → ℝ, IntegrableOn b (Ioo 0 t) ∧ ∀ s ∈ Ioo 0 t, ∀ x : V, ‖row K (A s) ((t-s)^(1/4 : ℝ)) (f s) j x‖ ≤ b s := by have hAs (s : ℝ) : Continuous (A s) := hA.comp (continuous_const.prodMk continuous_id) by_cases hj : j = 0 · subst j obtain ⟨C,hC,hb⟩ := exists_row_zero_bound K α β hα have hi := (intervalIntegrable_iff_integrableOn_Ioo_of_le ht.le).mp (BetaWeights.intervalIntegrable σ 0 hσ (by norm_num) t ht) refine ⟨fun s => (M*C)*BetaWeights.density σ 0 t s,hi.const_mul _,?_⟩ intro s hs x have hh := (hb (A s) (hAs s) (hAr s) ((t-s)^(1/4 : ℝ)) (Real.rpow_pos_of_pos (sub_pos.mpr hs.2) _) (f s) x).trans (mul_le_mul_of_nonneg_right (hfn s hs) hC) simpa only [BetaWeights.density,neg_zero,Real.rpow_zero,mul_one,mul_assoc,mul_comm,mul_left_comm] using hh · obtain ⟨C,D,hC,hD,hb⟩ := exists_row_holder_bound K j (Nat.pos_of_ne_zero hj) α β hα γ hγ.le hγ1 have hp : ((j : ℝ)-γ)/4 < 1 := by have : (j : ℝ) ≤ 4 := by exact_mod_cast hj4 linarith have hp' : ((j : ℝ)-1)/4 < 1 := by have : (j : ℝ) ≤ 4 := by exact_mod_cast hj4 linarith have hi := (intervalIntegrable_iff_integrableOn_Ioo_of_le ht.le).mp (BetaWeights.intervalIntegrable τ ((j-γ)/4) hτ hp t ht) have hi' := (intervalIntegrable_iff_integrableOn_Ioo_of_le ht.le).mp (BetaWeights.intervalIntegrable σ ((j-1)/4) hσ hp' t ht) refine ⟨fun s => (H*C)*BetaWeights.density τ ((j-γ)/4) t s+ (M*L*D)*BetaWeights.density σ ((j-1)/4) t s, (hi.const_mul _).add (hi'.const_mul _),?_⟩ intro s hs x have hq : 0 < t-s := sub_pos.mpr hs.2 have hr : 0 < (t-s)^(1/4 : ℝ) := Real.rpow_pos_of_pos hq _ have hsτ : 0 ≤ H*s^(-τ) := mul_nonneg hH (Real.rpow_nonneg hs.1.le _) have he := hb (A s) (hAs s) (hAr s) L hL (hAL s) (f s) (H*s^(-τ)) hsτ (hhold s hs) _ hr x have heq : (((t-s)^(1/4 : ℝ))⁻¹)^j * ((H*s^(-τ))*((t-s)^(1/4 : ℝ))^γ*C+‖f s‖*L*((t-s)^(1/4 : ℝ))*D) = (H*s^(-τ))*C*(t-s)^((γ-j)/4)+‖f s‖*L*D*(t-s)^(((1 : ℝ)-j)/4) := by rw [mul_add] have h1 := radius_weight (t-s) hq j γ have h2 := radius_weight (t-s) hq j 1 rw [Real.rpow_one] at h2 calc _ = (H*s^(-τ))*C*((((t-s)^(1/4 : ℝ))⁻¹)^j*((t-s)^(1/4 : ℝ))^γ) + ‖f s‖*L*D*((((t-s)^(1/4 : ℝ))⁻¹)^j*(t-s)^(1/4 : ℝ)) := by ring _ = _ := by rw [h1,h2] rw [heq] at he apply he.trans have hn := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right (hfn s hs) hL) hD) (Real.rpow_nonneg hq.le (((1 : ℝ)-j)/4)) calc _ ≤ (H*s^(-τ))*C*(t-s)^((γ-j)/4)+(M*s^(-σ))*L*D*(t-s)^(((1 : ℝ)-j)/4) := add_le_add le_rfl hn _ = _ := by unfold BetaWeights.density rw [show -((↑j-γ)/4) = (γ-j)/4 by ring,show -(((j:ℝ)-1)/4) = ((1:ℝ)-j)/4 by ring] ring end SourceFrozenRows namespace WeightedSourceFrozenRows open SourceFrozenRows open scoped BoundedContinuousFunction ContDiff variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] variable (K : SchwartzMap V ℂ) (A : ℝ → V → ℝ) (hA : Continuous (Function.uncurry A)) (α β : ℝ) (hα : 0 < α) (hAr : ∀ s y, A s y ∈ Icc α β) (L : ℝ) (hL : 0 ≤ L) (hAL : ∀ s x y, ‖A s x-A s y‖ ≤ L*‖x-y‖) (f : ℝ → (V →ᵇ ℂ)) (hf : StronglyMeasurable f) (σ τ M H γ t : ℝ) (hσ : σ < 1) (hτ : τ < 1) (hM : 0 ≤ M) (hH : 0 ≤ H) (ht : 0 < t) (hγ : 0 < γ) (hγ1 : γ ≤ 1) (hfn : ∀ s ∈ Ioo 0 t, ‖f s‖ ≤ M*s^(-σ)) (hhold : ∀ s ∈ Ioo 0 t, ∀ x y : V, ‖f s x-f s y‖ ≤ (H*s^(-τ))*‖x-y‖^γ) include hA hα hAr hf hAL hL hH ht hγ hγ1 hhold hσ hτ hM hfn lemma integrableOn_row (j : ℕ) (hj4 : j ≤ 4) (x : V) : IntegrableOn (fun s => row K (A s) ((t-s)^(1/4 : ℝ)) (f s) j x) (Ioo 0 t) := by obtain ⟨b,hi,hb⟩ := SourceFrozenRows.exists_integrable_row_bound_weighted K A hA α β hα hAr L hL hAL f σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold j hj4 exact hi.mono' (stronglyMeasurable_time_row_of_stronglyMeasurable K A hA f hf j t x).aestronglyMeasurable (by filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs; exact hb s hs x) lemma continuous_integralRow (j : ℕ) (hj4 : j ≤ 4) : Continuous (integralRow K A f t j) := by obtain ⟨b,hbi,hb⟩ := SourceFrozenRows.exists_integrable_row_bound_weighted K A hA α β hα hAr L hL hAL f σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold j hj4 apply continuous_of_dominated (bound := b) · intro x exact (stronglyMeasurable_time_row_of_stronglyMeasurable K A hA f hf j t x).aestronglyMeasurable · intro x filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs exact hb s hs x · exact hbi · filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs exact continuous_row K (A s) (hA.comp (continuous_const.prodMk continuous_id)) α β hα (hAr s) _ (Real.rpow_pos_of_pos (sub_pos.mpr hs.2) _) (f s) j lemma hasFDerivAt_integralRow (j : ℕ) (hj4 : j+1 ≤ 4) (x : V) : HasFDerivAt (integralRow K A f t j) ((continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (j+1) => V) ℂ) (integralRow K A f t (j+1) x)) x := by let C := (continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (j+1) => V) ℂ).toContinuousLinearEquiv.toContinuousLinearMap let : ContinuousENorm (V →L[ℝ] V [×j]→L[ℝ] ℂ) := @SeminormedAddGroup.toContinuousENorm (V →L[ℝ] V [×j]→L[ℝ] ℂ) inferInstance have hi := integrableOn_row K A hA α β hα hAr L hL hAL f hf σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold (j+1) hj4 x have hcomm := ContinuousLinearMap.integral_comp_comm (𝕜 := ℝ) (E := V [×(j+1)]→L[ℝ] ℂ) (Fₗ := V →L[ℝ] V [×j]→L[ℝ] ℂ) C hi change HasFDerivAt (fun x => ∫ s in Ioo 0 t, row K (A s) ((t-s)^(1/4 : ℝ)) (f s) j x) (C (∫ s in Ioo 0 t, row K (A s) ((t-s)^(1/4 : ℝ)) (f s) (j+1) x)) x rw [← hcomm] obtain ⟨b,hbi,hb⟩ := SourceFrozenRows.exists_integrable_row_bound_weighted K A hA α β hα hAr L hL hAL f σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold (j+1) hj4 apply hasFDerivAt_integral_of_dominated_of_fderiv_le (s := univ) (F' := fun x s => C (row K (A s) ((t-s)^(1/4 : ℝ)) (f s) (j+1) x)) (bound := b) (univ_mem : univ ∈ 𝓝 x) · exact Eventually.of_forall fun z => (stronglyMeasurable_time_row_of_stronglyMeasurable K A hA f hf j t z).aestronglyMeasurable · exact integrableOn_row K A hA α β hα hAr L hL hAL f hf σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold j (by omega) x · exact (C.continuous.comp_stronglyMeasurable (stronglyMeasurable_time_row_of_stronglyMeasurable K A hA f hf (j+1) t x)).aestronglyMeasurable · filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs z _ change ‖(continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (j+1) => V) ℂ) (row K (A s) ((t-s)^(1/4 : ℝ)) (f s) (j+1) z)‖ ≤ b s rw [LinearIsometryEquiv.norm_map] exact hb s hs z · exact hbi · filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs z _ exact hasFDerivAt_row K (A s) (hA.comp (continuous_const.prodMk continuous_id)) α β hα (hAr s) _ (Real.rpow_pos_of_pos (sub_pos.mpr hs.2) _) (f s) j z lemma contDiff_integralRow (j n : ℕ) (hjn : j+n ≤ 4) : ContDiff ℝ n (integralRow K A f t j) := by induction n generalizing j with | zero => rw [Nat.cast_zero, contDiff_zero] exact continuous_integralRow K A hA α β hα hAr L hL hAL f hf σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold j (by omega) | succ n ih => rw [Nat.cast_add, Nat.cast_one, contDiff_succ_iff_hasFDerivAt] let C : (V [×(j+1)]→L[ℝ] ℂ) →L[ℝ] (V →L[ℝ] V [×j]→L[ℝ] ℂ) := (continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (j+1) => V) ℂ).toContinuousLinearEquiv.toContinuousLinearMap refine ⟨fun x => C (integralRow K A f t (j+1) x), ?_, ?_⟩ · exact contDiff_const.clm_apply (ih (j+1) (by omega)) · intro x exact hasFDerivAt_integralRow K A hA α β hα hAr L hL hAL f hf σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold j (by omega) x lemma duhamel_eq_integralRow_zero (x : V) : duhamel K A f t x = (continuousMultilinearCurryFin0 ℝ V ℂ) (integralRow K A f t 0 x) := by let C := (continuousMultilinearCurryFin0 ℝ V ℂ).toContinuousLinearEquiv.toContinuousLinearMap have hc := ContinuousLinearMap.integral_comp_comm (𝕜 := ℝ) (E := V [×0]→L[ℝ] ℂ) (Fₗ := ℂ) C (integrableOn_row K A hA α β hα hAr L hL hAL f hf σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold 0 (by omega) x) change (∫ s in Ioo 0 t, value K (A s) ((t-s)^(1/4 : ℝ)) (f s) x) = C (∫ s in Ioo 0 t, row K (A s) ((t-s)^(1/4 : ℝ)) (f s) 0 x) rw [← hc] apply integral_congr_ae filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs rw [row_eq K (A s) (hA.comp (continuous_const.prodMk continuous_id)) α β hα (hAr s) _ (Real.rpow_pos_of_pos (sub_pos.mpr hs.2) _),iteratedFDeriv_zero_eq_comp] exact (LinearIsometryEquiv.apply_symm_apply _ _).symm lemma contDiff_duhamel : ContDiff ℝ 4 (duhamel K A f t) := by have he : duhamel K A f t = fun x => (continuousMultilinearCurryFin0 ℝ V ℂ) (integralRow K A f t 0 x) := funext (duhamel_eq_integralRow_zero K A hA α β hα hAr L hL hAL f hf σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold) rw [he] let C : (V [×0]→L[ℝ] ℂ) →L[ℝ] ℂ := (continuousMultilinearCurryFin0 ℝ V ℂ).toContinuousLinearEquiv.toContinuousLinearMap change ContDiff ℝ 4 (fun x => C (integralRow K A f t 0 x)) exact contDiff_const.clm_apply (contDiff_integralRow K A hA α β hα hAr L hL hAL f hf σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold 0 4 (by omega)) lemma iteratedFDeriv_duhamel (j : ℕ) (hj : j ≤ 4) : iteratedFDeriv ℝ j (duhamel K A f t) = integralRow K A f t j := by induction j with | zero => funext x rw [iteratedFDeriv_zero_eq_comp] change (continuousMultilinearCurryFin0 ℝ V ℂ).symm (duhamel K A f t x) = _ rw [duhamel_eq_integralRow_zero K A hA α β hα hAr L hL hAL f hf σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold, LinearIsometryEquiv.symm_apply_apply] | succ j ih => rw [iteratedFDeriv_succ_eq_comp_left,ih (by omega)] funext x dsimp only [Function.comp_apply] rw [(hasFDerivAt_integralRow K A hA α β hα hAr L hL hAL f hf σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold j hj x).fderiv] exact LinearIsometryEquiv.symm_apply_apply _ _ lemma exists_norm_iteratedFDeriv_duhamel_bound (j : ℕ) (hj : j ≤ 4) : ∃ C : ℝ, ∀ x, ‖iteratedFDeriv ℝ j (duhamel K A f t) x‖ ≤ C := by obtain ⟨b,hi,hb⟩ := SourceFrozenRows.exists_integrable_row_bound_weighted K A hA α β hα hAr L hL hAL f σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold j hj refine ⟨∫ s in Ioo 0 t, b s,fun x => ?_⟩ rw [iteratedFDeriv_duhamel K A hA α β hα hAr L hL hAL f hf σ τ M H γ t hσ hτ hM hH ht hγ hγ1 hfn hhold j hj] apply norm_integral_le_of_norm_le hi filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs exact hb s hs x end WeightedSourceFrozenRows end end OAI