| import OAI.Analysis.AncientFlat.Kernel.ClassicalBase |
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|
| namespace OAI |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped ContDiff SchwartzMap |
| namespace FlatDuality |
| open BiharmonicKernel |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| lemma schwartz_jet_majorant (n : ℕ) {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (g : SchwartzMap V E) (a b : ℝ) (ha : 0<a) (K : ℕ) : |
| ∃ B : V → ℝ, Integrable B ∧ Integrable (fun x => (1+‖x‖)*B x) ∧ |
| (∀ x, 0≤B x) ∧ ∀ c ∈ Icc a b, ∀ x, |
| ‖c⁻¹^K • iteratedFDeriv ℝ n g (c⁻¹ • x)‖≤B x := by |
| induction n generalizing E with |
| | zero => |
| obtain ⟨B,hi,hw,hn,hb⟩ := SchwartzConvolution.scaled_weighted_majorant g a b ha K 1 |
| refine ⟨B,hi,by simpa only [pow_one] using hw,hn,fun c hc x => ?_⟩ |
| simpa only [norm_smul,norm_iteratedFDeriv_zero] using hb c hc x |
| | succ n ih => |
| obtain ⟨B,hi,hw,hn,hb⟩ := ih (SchwartzMap.fderivCLM ℝ V E g) |
| refine ⟨B,hi,hw,hn,fun c hc x => ?_⟩ |
| have he : ((SchwartzMap.fderivCLM ℝ V E g : SchwartzMap V (V →L[ℝ] E)) : V → (V →L[ℝ] E)) = fderiv ℝ g := rfl |
| simpa only [norm_smul,he,norm_iteratedFDeriv_fderiv] using hb c hc x |
| lemma kernel_jet_weighted_majorant (n : ℕ) (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, |
| ‖iteratedFDeriv ℝ n (kernel (V := V) t) x‖≤B x := by |
| let r := fun t : ℝ => Real.sqrt (Real.sqrt t) |
| obtain ⟨B,hi,hw,hn,hb⟩ := schwartz_jet_majorant n (heatKernel (V := V) 1 zero_lt_one) |
| (r a) (r b) (Real.sqrt_pos.mpr (Real.sqrt_pos.mpr ha)) (Module.finrank ℝ V+n) |
| refine ⟨B,hi,hw,hn,fun t ht x => ?_⟩ |
| have hp : 0<t := ha.trans_le ht.1 |
| have hr : 0<r t := Real.sqrt_pos.mpr (Real.sqrt_pos.mpr hp) |
| have hrs : r t^4=t := by |
| dsimp [r] |
| nlinarith [Real.sq_sqrt (Real.sqrt_nonneg t),Real.sq_sqrt hp.le] |
| have he : (kernel (V := V) t)=(heatKernel t hp : V → ℂ) := by |
| exact (heatKernel_eq_kernel t hp).symm |
| rw [he] |
| have hs := heatKernel_deriv_scaling (V := V) 1 zero_lt_one (r t) hr n x |
| simp only [mul_one,hrs] at hs |
| rw [hs,←inv_pow,←pow_add] |
| exact hb (r t) ⟨Real.sqrt_le_sqrt (Real.sqrt_le_sqrt ht.1), |
| Real.sqrt_le_sqrt (Real.sqrt_le_sqrt ht.2)⟩ x |
| lemma kernel_four_jet_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, ∀ n : ℕ, n≤4 → ∀ x, |
| ‖iteratedFDeriv ℝ n (kernel (V := V) t) x‖≤B x := by |
| choose B hi hw hn hb using fun n => kernel_jet_weighted_majorant (V := V) n a b ha |
| refine ⟨fun x => ∑ n ∈ Finset.range 5,B n x,integrable_finsetSum _ (fun n _ => hi n),?_, |
| fun x => Finset.sum_nonneg (fun n _ => hn n x),?_⟩ |
| · simp_rw [Finset.mul_sum] |
| exact integrable_finsetSum _ (fun n _ => hw n) |
| · intro t ht n hn' x |
| exact (hb n t ht x).trans |
| (Finset.single_le_sum (fun i _ => hn i x) (show n ∈ Finset.range 5 by simpa only [Finset.mem_range] using Nat.lt_succ_of_le hn')) |
| end FlatDuality |
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| end |
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| end OAI |
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