openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Graphs /OutgoingWeightedKernelNorm.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.InverseEndpointCoefficient | |
| namespace OAI | |
| /-! Graph energies and pair estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4PrefixShell | |
| open MeasureTheory Set | |
| open scoped ContDiff | |
| variable {n : ℕ} | |
| local notation "V" => EuclideanSpace ℝ (Fin (n+1)) | |
| lemma outgoingWeightedKernelND_norm (r s : ℝ) (b : V × V → SmoothFunctionND V) | |
| (c : V × V → ℂ) (z : V × V) (hs : 0 ≤ s) : | |
| ‖c z*outgoingWeightedKernelND r s b z‖ = | |
| (dist z.1 z.2*s)^(-(n:ℝ)/2)* | |
| ‖c z*(b z).1 ((dist z.1 z.2)⁻¹ • (z.1-z.2))‖ := by | |
| have hW := Real.rpow_nonneg (mul_nonneg (dist_nonneg (x := z.1) (y := z.2)) hs) (-(n:ℝ)/2) | |
| simp only [outgoingWeightedKernelND,norm_mul,Complex.norm_real,Real.norm_eq_abs, | |
| abs_of_nonneg hW] | |
| have he : ‖Complex.exp (Complex.I*(r*dist z.1 z.2))‖=1 := by simp [Complex.norm_exp] | |
| rw [he] | |
| ring | |
| lemma outgoingWeightedKernelND_measurable (s : ℝ) (b : V × V → SmoothFunctionND V) | |
| (c : V × V → ℂ) | |
| (hc : Measurable (fun z => c z*(b z).1 ((dist z.1 z.2)⁻¹ • (z.1-z.2)))) : | |
| Measurable (fun q : ℝ × (V × V) => c q.2*outgoingWeightedKernelND q.1 s b q.2) := by | |
| have hw : Measurable (fun q : ℝ × (V × V) => | |
| (((dist q.2.1 q.2.2*s)^(-(n:ℝ)/2):ℝ):ℂ)) := by fun_prop | |
| have hp : Measurable (fun q : ℝ × (V × V) => | |
| Complex.exp (Complex.I*(q.1*dist q.2.1 q.2.2))) := by fun_prop | |
| have hm := (hw.mul hp).mul (hc.comp measurable_snd) | |
| convert hm using 1 | |
| funext q | |
| simp only [outgoingWeightedKernelND, Pi.mul_apply, Function.comp_apply] | |
| ring | |
| lemma outgoingWeightedKernelND_integrable (s : ℝ) (hs : 0 < s) | |
| (b : V × V → SmoothFunctionND V) (c : V × V → ℂ) | |
| (μ : Measure (V × V)) [IsFiniteMeasure μ] (r : ℝ) {δ C : ℝ} | |
| (hδ : 0 < δ) | |
| (hsep : ∀ᵐ z ∂μ, δ ≤ dist z.1 z.2*s) | |
| (hm : Measurable (fun z => c z*(b z).1 ((dist z.1 z.2)⁻¹ • (z.1-z.2)))) | |
| (hb : ∀ᵐ z ∂μ, ‖c z*(b z).1 ((dist z.1 z.2)⁻¹ • (z.1-z.2))‖ ≤ C) : | |
| Integrable (fun z => c z*outgoingWeightedKernelND r s b z) μ := by | |
| apply Integrable.of_bound ((outgoingWeightedKernelND_measurable s b c hm).comp | |
| (measurable_const.prodMk measurable_id)).aestronglyMeasurable (δ^(-(n:ℝ)/2)*C) | |
| filter_upwards [hsep,hb] with z hz hzb | |
| change ‖c z*outgoingWeightedKernelND r s b z‖ ≤ _ | |
| rw [outgoingWeightedKernelND_norm r s b c z hs.le] | |
| apply mul_le_mul (Real.rpow_le_rpow_of_nonpos hδ hz | |
| (div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (Nat.cast_nonneg n)) (by norm_num))) hzb | |
| (norm_nonneg _) (Real.rpow_nonneg hδ.le _) | |
| lemma weighted_pair_integral_measurableND (s : ℝ) | |
| (b : V × V → SmoothFunctionND V) (c : V × V → ℂ) | |
| (μ : Measure (V × V)) [SFinite μ] (L : ℝ) | |
| (hm : Measurable (fun z => c z*(b z).1 ((dist z.1 z.2)⁻¹ • (z.1-z.2)))) : | |
| Measurable (fun r => L • ∫ z, c z*outgoingWeightedKernelND r s b z ∂μ) := | |
| ((outgoingWeightedKernelND_measurable s b c hm).stronglyMeasurable.integral_prod_right' (ν := μ)).measurable.const_smul L | |
| end Attempt4PrefixShell | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |