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7f37411 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | 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
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