| import OAI.Analysis.AncientFlat.Kernel.ClassicalBase |
| import OAI.Analysis.AncientFlat.Evolution.ChangingPair |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ContDiff BoundedContinuousFunction |
| namespace FlatDuality |
| open BiharmonicKernel |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| def kernelTest (θ γ : ℝ → ℝ) (z : V) (r : ℝ) (x : V) : ℂ := |
| (γ r : ℂ)*kernel (θ r) (x-z) |
| def kernelTestDeriv (θ γ : ℝ → ℝ) (z : V) (r : ℝ) (x : V) : ℂ := |
| (Complex.ofReal (deriv γ r))*kernel (θ r) (x-z)+ |
| (γ r : ℂ)*(-bilapKernel (θ r) (x-z)*(Complex.ofReal (deriv θ r))) |
| lemma hasDerivAt_kernelTest (θ γ : ℝ → ℝ) (z : V) (r : ℝ) |
| (hθ : DifferentiableAt ℝ θ r) (hγ : DifferentiableAt ℝ γ r) (hr : 0<θ r) (x : V) : |
| HasDerivAt (fun s => kernelTest θ γ z s x) (kernelTestDeriv θ γ z r x) r := by |
| simpa only [kernelTest,kernelTestDeriv,Function.comp_def,Complex.real_smul,mul_comm (Complex.ofReal (deriv θ r))] using! |
| hγ.hasDerivAt.ofReal_comp.mul ((hasDerivAt_kernel_bilap (θ r) hr (x-z)).scomp r hθ.hasDerivAt) |
| lemma continuous_kernelTest (θ γ : ℝ → ℝ) (z : V) (r : ℝ) (hr : 0<θ r) : |
| Continuous (kernelTest θ γ z r) := |
| continuous_const.mul (continuous_kernel_comp (fun _ : V => θ r) (fun x => x-z) |
| continuous_const (continuous_id.sub continuous_const) (fun _ => hr)) |
| lemma continuous_kernelTestDeriv (θ γ : ℝ → ℝ) (z : V) (r : ℝ) (hr : 0<θ r) : |
| Continuous (kernelTestDeriv θ γ z r) := |
| (continuous_const.mul (continuous_kernel_comp (fun _ : V => θ r) (fun x => x-z) |
| continuous_const (continuous_id.sub continuous_const) (fun _ => hr))).add |
| (continuous_const.mul (((continuous_bilapKernel_comp (fun _ : V => θ r) (fun x => x-z) |
| continuous_const (continuous_id.sub continuous_const) (fun _ => hr)).neg).mul continuous_const)) |
|
|
| lemma hasDerivAt_kernel_pair (θ γ : ℝ → ℝ) (hθ : ContDiff ℝ 1 θ) (hγ : ContDiff ℝ 1 γ) |
| (z : V) {u : ℝ → V →ᵇ ℂ} {p : V →ᵇ ℂ} {t : ℝ} |
| (hu : HasDerivAt u p t) (ht : 0<θ t) : |
| HasDerivAt (fun r => ∫ x, kernelTest θ γ z r x*u r x) |
| (∫ x, kernelTestDeriv θ γ z t x*u t x+kernelTest θ γ z t x*p x) t := by |
| let a := θ t/2 |
| let b := 2*θ t |
| let G := ‖γ t‖+1 |
| let D := ‖deriv γ t‖+1 |
| let Q := ‖deriv θ t‖+1 |
| have ha : 0<a := by dsimp [a]; positivity |
| have hG : 0<G := by dsimp [G]; positivity |
| have hD : 0<D := by dsimp [D]; positivity |
| have hQ : 0<Q := by dsimp [Q]; positivity |
| have hθn : ∀ᶠ r in 𝓝 t, θ r ∈ Ioo a b := |
| hθ.continuous.continuousAt.eventually (Ioo_mem_nhds (by dsimp [a]; linarith) (by dsimp [b]; linarith)) |
| have hγn : ∀ᶠ r in 𝓝 t, ‖γ r‖<G := |
| hγ.continuous.continuousAt.norm.eventually (gt_mem_nhds (lt_add_one _)) |
| have hDn : ∀ᶠ r in 𝓝 t, ‖deriv γ r‖<D := |
| hγ.continuous_deriv_one.continuousAt.norm.eventually (gt_mem_nhds (lt_add_one _)) |
| have hQn : ∀ᶠ r in 𝓝 t, ‖deriv θ r‖<Q := |
| hθ.continuous_deriv_one.continuousAt.norm.eventually (gt_mem_nhds (lt_add_one _)) |
| obtain ⟨δ,hδ,hball⟩ := Metric.mem_nhds_iff.mp (hθn.and (hγn.and (hDn.and hQn))) |
| have hb0 : t ∈ Metric.ball t δ := Metric.mem_ball_self hδ |
| obtain ⟨A,hAi,hAn,hAb⟩ := FrozenParametrix.kernel_majorant_nonneg (V := V) a b ha |
| obtain ⟨B,hBi,_,hBn,hBb⟩ := bilapKernel_weighted_majorant (V := V) a b ha |
| let A₁ : V → ℝ := fun x => G*A (x-z) |
| let B₁ : V → ℝ := fun x => D*A (x-z)+(G*Q)*B (x-z) |
| have hAbound : ∀ r ∈ Metric.ball t δ, ∀ x, ‖kernelTest θ γ z r x‖≤A₁ x := by |
| intro r hr x |
| obtain ⟨hq,hg,_,_⟩ := hball hr |
| change ‖(γ r : ℂ)*kernel (θ r) (x-z)‖≤G*A (x-z) |
| rw [norm_mul,Complex.norm_real] |
| exact mul_le_mul hg.le (hAb _ ⟨hq.1.le,hq.2.le⟩ _) (norm_nonneg _) hG.le |
| have hBbound : ∀ r ∈ Metric.ball t δ, ∀ x, ‖kernelTestDeriv θ γ z r x‖≤B₁ x := by |
| intro r hr x |
| obtain ⟨hq,hg,hd,hq'⟩ := hball hr |
| unfold kernelTestDeriv |
| apply (norm_add_le _ _).trans |
| rw [norm_mul,norm_mul,norm_mul,norm_neg,Complex.norm_real,Complex.norm_real,Complex.norm_real] |
| calc |
| _ ≤ D*A (x-z)+G*(B (x-z)*Q) := add_le_add |
| (mul_le_mul hd.le (hAb _ ⟨hq.1.le,hq.2.le⟩ _) (norm_nonneg _) hD.le) |
| (mul_le_mul hg.le (mul_le_mul (hBb _ ⟨hq.1.le,hq.2.le⟩ _) hq'.le |
| (norm_nonneg _) (hBn _)) (mul_nonneg (norm_nonneg _) (norm_nonneg _)) hG.le) |
| _ = B₁ x := by dsimp [B₁]; ring |
| refine hasDerivAt_changing_pair hu (kernelTest θ γ z) (kernelTestDeriv θ γ z t) |
| (Metric.ball t δ) (Metric.ball_mem_nhds t hδ) |
| ?_ ?_ ?_ A₁ B₁ ?_ ?_ ?_ ?_ ?_ ?_ |
| · intro r hr |
| exact (continuous_kernelTest θ γ z r (ha.trans (hball hr).1.1)).aestronglyMeasurable |
| · exact (continuous_kernelTestDeriv θ γ z t ht).aestronglyMeasurable |
| · exact hasDerivAt_kernelTest θ γ z t ((hθ.differentiable (by norm_num)) t) ((hγ.differentiable (by norm_num)) t) ht |
| · exact (hAi.comp_sub_right z).const_mul G |
| · exact ((hAi.comp_sub_right z).const_mul D).add ((hBi.comp_sub_right z).const_mul (G*Q)) |
| · exact fun x => mul_nonneg hG.le (hAn _) |
| · exact fun x => add_nonneg (mul_nonneg hD.le (hAn _)) (mul_nonneg (mul_nonneg hG.le hQ.le) (hBn _)) |
| · exact hAbound |
| · intro r hr x |
| exact Convex.norm_image_sub_le_of_norm_hasDerivWithin_le |
| (fun s hs => (hasDerivAt_kernelTest θ γ z s ((hθ.differentiable (by norm_num)) s) |
| ((hγ.differentiable (by norm_num)) s) (ha.trans (hball hs).1.1) x).hasDerivWithinAt) |
| (fun s hs => hBbound s hs x) (convex_ball t δ) hb0 hr |
| end FlatDuality |
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| end |
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| end OAI |
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