| import OAI.Analysis.AncientFlat.Kernel.ClassicalBase |
| import OAI.Analysis.AncientFlat.Kernel.SchwartzIBP |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ContDiff SchwartzMap ZeroAtInfty |
| namespace FlatDuality |
| open BiharmonicKernel ParametrixUniqueness |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| def kernelMoment : ℝ := ∫ x : V, ‖x‖*‖heatKernel 1 zero_lt_one x‖ |
| lemma kernelMoment_nonneg : 0≤kernelMoment (V := V) := integral_nonneg (fun point => mul_nonneg (norm_nonneg point) (norm_nonneg _)) |
| lemma kernel_moment_integrable (t : ℝ) (ht : 0<t) : Integrable (fun x : V => ‖x‖*‖kernel t x‖) := by |
| rw [←heatKernel_eq_kernel t ht] |
| simpa only [pow_one] using (heatKernel (V := V) t ht).integrable_pow_mul volume 1 |
| lemma kernel_moment_scaling (r : ℝ) (hr : 0<r) : |
| (∫ x : V, ‖x‖*‖kernel (r^4) x‖)=r*kernelMoment (V := V) := by |
| have he (x : V) : kernel (r^4) x=(r^Module.finrank ℝ V)⁻¹ • heatKernel 1 zero_lt_one (r⁻¹ • x) := by |
| rw [←heatKernel_eq_kernel _ (by positivity)] |
| simpa only [mul_one] using heatKernel_scaling (V := V) 1 zero_lt_one r hr x |
| have hf : (fun x : V => ‖x‖*‖kernel (r^4) x‖)=fun x : V => |
| ((r^Module.finrank ℝ V)⁻¹*r)*(‖r⁻¹ • x‖*‖heatKernel 1 zero_lt_one (r⁻¹ • x)‖) := by |
| funext x |
| rw [he,norm_smul,norm_smul,Real.norm_of_nonneg (by positivity),Real.norm_of_nonneg (by positivity)] |
| field_simp |
| rw [hf,integral_const_mul,Measure.integral_comp_inv_smul_of_nonneg volume |
| (fun x : V => ‖x‖*‖heatKernel 1 zero_lt_one x‖) hr.le] |
| dsimp [kernelMoment] |
| field_simp [hr.ne'] |
| lemma norm_kernel_pair_sub_le (r : ℝ) (hr : 0<r) (g : C₀(V,ℂ)) (K : ℝ) (_hK : 0≤K) |
| (hg : ∀ x y, ‖g x-g y‖≤K*‖x-y‖) (z : V) : |
| ‖(∫ x, kernel (r^4) (x-z)*g x)-g z‖≤K*r*kernelMoment (V := V) := by |
| have ht : 0<r^4 := by positivity |
| have hk : Integrable (fun x : V => kernel (r^4) (x-z)) := by |
| rw [←heatKernel_eq_kernel _ ht] |
| exact (heatKernel _ ht).integrable.comp_sub_right z |
| have hmass : (∫ x : V, kernel (r^4) (x-z))=1 := by |
| rw [integral_sub_right_eq_self,←heatKernel_eq_kernel _ ht,integral_heatKernel] |
| have hi : Integrable (fun x => kernel (r^4) (x-z)*g x) := |
| integrable_mul_bounded hk g.continuous ⟨‖g‖,norm_nonneg _,C0Convolution.apply_norm_le g⟩ |
| have he : (∫ x, kernel (r^4) (x-z)*g x)-g z=∫ x, kernel (r^4) (x-z)*(g x-g z) := by |
| calc |
| _ = (∫ x, kernel (r^4) (x-z)*g x)-(∫ x, kernel (r^4) (x-z)*g z) := by rw [integral_mul_const,hmass,one_mul] |
| _ = _ := by |
| rw [←integral_sub hi (hk.mul_const (g z))] |
| apply integral_congr_ae |
| filter_upwards with x |
| ring |
| rw [he] |
| calc |
| _ ≤ ∫ x, K*(‖x-z‖*‖kernel (r^4) (x-z)‖) := by |
| apply norm_integral_le_of_norm_le ((kernel_moment_integrable _ ht).comp_sub_right z |>.const_mul K) |
| filter_upwards with x |
| rw [norm_mul] |
| calc |
| _ ≤ ‖kernel (r^4) (x-z)‖*(K*‖x-z‖) := mul_le_mul_of_nonneg_left (hg x z) (norm_nonneg _) |
| _=_ := by ring |
| _ = K*r*kernelMoment (V := V) := by |
| rw [integral_const_mul] |
| have he' := integral_sub_right_eq_self (μ := volume) (fun x : V => ‖x‖*‖kernel (r^4) x‖) z |
| rw [he',kernel_moment_scaling r hr] |
| ring |
| end FlatDuality |
|
|
| end |
|
|
| end OAI |
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