File size: 2,541 Bytes
b17ead8 | 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 | import OAI.MeasureTheory.Falconer.Angular.AngularCellSpatialIntegral
import OAI.MeasureTheory.Falconer.Angular.AngularRestrictedIntegralSum
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
attribute [local instance] Classical.propDecidable
theorem angular_contribution_spatial_sum (Q₁ Q₂ : DyadicSquare)
(ρ ν : Measure Plane) [IsFiniteMeasure ρ] [IsFiniteMeasure ν]
(A B : CellPath) (n : ℕ) (a v : ℝ) {Ψ : ℝ → ℝ}
(hΨ : Measurable Ψ) (hpos : ∀ t, 0 ≤ Ψ t)
(hs : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4:ℝ) 4)
{W : ℝ} (hW : 0 ≤ W) (hw : ∀ t, Ψ t ≤ W)
(Ma₁ Ma₂ : ℝ) {Mp₁ Mp₂ : ℝ} (hMp₁ : 0 < Mp₁) (hMp₂ : 0 < Mp₂)
(β γ : Plane → UnitCircle → ℂ)
(hβ : Measurable (fun z : Plane × UnitCircle => β z.1 z.2))
(hγ : Measurable (fun z : Plane × UnitCircle => γ z.1 z.2))
{D₁ D₂ : ℝ} (hD₁ : 0 ≤ D₁) (hD₂ : 0 ≤ D₂)
(hbβ : ∀ x u, ‖β x u‖ ≤ D₁) (hbγ : ∀ x u, ‖γ x u‖ ≤ D₂)
(G : (CellPath × CellPath) → (CellPath × CellPath) → Prop) :
angularCellContribution Q₁ Q₂ ρ ν A B n a v Ψ Ma₁ Ma₂ Mp₁ Mp₂ β γ G =
(((2:ℝ)^(2*(v-a)))/(Ma₁^2*Ma₂^2)) •
∑ row ∈ sourceProductVertices Q₁ Q₂ ρ ν A B n,
∑ col ∈ sourceProductVertices Q₁ Q₂ ρ ν A B n,
if G row col then angularCellSpatialIntegral Q₁ Q₂ ρ ν β γ row col v Ψ else 0 := by
let := sourceAngularShellMeasure_finite v Ψ hW hw (fun t ht => by
by_contra hn
have := (hs t hn).2
linarith)
unfold angularCellContribution
rw [angularRestrictedIntegral_sum a v Ψ _ G _ _
(angularCellTransform_measurable Q₁ ρ Mp₁ β hβ)
(angularCellTransform_measurable Q₂ ν Mp₂ γ hγ)
(B := D₁*(ρ univ).toReal/Mp₁) (D := D₂*(ν univ).toReal/Mp₂) (by positivity) (by positivity)
(fun P z => sourceAngularTransform_global_bound ρ _ hMp₁ β hD₁ hbβ z.1 z.2)
(fun P z => sourceAngularTransform_global_bound ν _ hMp₂ γ hD₂ hbγ z.1 z.2)]
simp_rw [Finset.smul_sum]
apply Finset.sum_congr rfl
intro row _
apply Finset.sum_congr rfl
intro col _
by_cases hg : G row col
· simp only [hg,↓reduceIte]
rw [angular_pair_spatial_integral Q₁ Q₂ ρ ν hMp₁ hMp₂ β γ hβ hγ hD₁ hD₂ hbβ hbγ
row col a v hΨ hpos hs hW hw,smul_smul,smul_smul]
congr 1
simp only [mul_inv_rev,pow_two,div_eq_mul_inv]
field_simp
· simp [hg]
end PlanarFalconer
end OAI
|