File size: 5,442 Bytes
d4317d7 | 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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.MeasureTheory.Falconer.Graphs.DistanceGraphPointwise
import OAI.MeasureTheory.Falconer.Measures.DistanceSchurIntegral
import OAI.MeasureTheory.Falconer.Graphs.ProductGraphEnergy
namespace OAI
namespace PlanarFalconer
open Set MeasureTheory
open scoped ComplexConjugate
theorem original_distance_graph_integral_bound
(Q₁ Q₂ : DyadicSquare) (ρ ν : Measure Plane) [IsFiniteMeasure ρ] [IsFiniteMeasure ν]
(A B : CellPath) (N a n : ℕ) (sig t : ℝ) (hA : A.length = a) (hB : B.length = a)
{θ : ℝ} (hθ : 0 < θ) (hscale : θ = graphUncappedAngularScale N sig t) (axis : ℝ) (I : Finset GraphNonedgeIndex)
(hI : I ⊆ sourceParentGraphIndices N sig t (Q₁.descendant A) (Q₂.descendant B) ρ ν n A B)
{d : ℝ} (hd : 0 ≤ d)
(hdegree : ∀ j k s reverse row,
((activeDistanceMatrixNeighbors I (a+n) j k θ axis s reverse row).card : ℝ) ≤ d)
{M₁ M₂ : ℝ} (hM₁ : 0 < M₁) (hM₂ : 0 < M₂)
(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))
{B₁ B₂ : ℝ} (hB₁ : 0 ≤ B₁) (hB₂ : 0 ≤ B₂)
(hbβ : ∀ x u, ‖β x u‖ ≤ B₁) (hbγ : ∀ x u, ‖γ x u‖ ≤ B₂)
(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) :
let U := fun P : CellPath => fun z : ℝ × UnitCircle =>
sourceAngularTransform ρ (Q₁.descendant P).carrier M₁ β z.1 z.2
let V := fun Q : CellPath => fun z : ℝ × UnitCircle =>
sourceAngularTransform ν (Q₂.descendant Q).carrier M₂ γ z.1 z.2
‖((2:ℝ)^(2*(v-(a+n)))) •
∑ e ∈ I, graphOriginalTerm ![ρ,ν,ν,ρ]
(fun l => (graphQuadCells ![Q₁,Q₂,Q₂,Q₁] e.1 l).carrier)
![M₁,M₂,M₂,M₁] ![β,γ,γ,β] true e.2.2 axis θ v e.2.1.1 e.2.1.2 Ψ‖ ≤
d * ∑ row ∈ sourceProductVertices Q₁ Q₂ ρ ν A B n,
(sourceAngularH (a+n) v Ψ (U row.1) (V row.2)).toReal := by
classical
let : ∀ l : Fin 4, IsFiniteMeasure (![ρ,ν,ν,ρ] l) := by
intro l; fin_cases l <;> dsimp <;> infer_instance
have hM : ∀ l : Fin 4, 0 < (![M₁,M₂,M₂,M₁] l) := by
intro l; fin_cases l <;> first | exact hM₁ | exact hM₂
have hb : ∀ l : Fin 4, Measurable (fun z : Plane × UnitCircle => (![β,γ,γ,β] l) z.1 z.2) := by
intro l; fin_cases l <;> first | exact hβ | exact hγ
have hbound : ∀ (l : Fin 4) x u, ‖(![β,γ,γ,β] l) x u‖ ≤ (![B₁,B₂,B₂,B₁] l) := by
intro l; fin_cases l <;> first | exact hbβ | exact hbγ
have hB4 : ∀ l : Fin 4, 0 ≤ (![B₁,B₂,B₂,B₁] l) := by
intro l; fin_cases l <;> first | exact hB₁ | exact hB₂
let H := graphSupportedDistanceIndices I θ axis
let F := fun z : ℝ × (UnitCircle × UnitCircle) => ∑ e ∈ H,
sourceGraphFourTransform ![ρ,ν,ν,ρ]
(fun l => (graphQuadCells ![Q₁,Q₂,Q₂,Q₁] e.1 l).carrier)
![M₁,M₂,M₂,M₁] ![β,γ,γ,β]
(graphRotatedDistanceWeight axis (angularBinCount θ) e.2.1.1 e.2.1.2) z.1 z.2.1 z.2.2
let U := fun P : CellPath => fun z : ℝ × UnitCircle =>
sourceAngularTransform ρ (Q₁.descendant P).carrier M₁ β z.1 z.2
let V := fun Q : CellPath => fun z : ℝ × UnitCircle =>
sourceAngularTransform ν (Q₂.descendant Q).carrier M₂ γ z.1 z.2
have hpoint (r : ℝ) (u w : UnitCircle) : ‖F (r,(u,w))‖ ≤
d/2 * ∑ row ∈ sourceProductVertices Q₁ Q₂ ρ ν A B n,
(‖U row.1 (r,u)‖^2*‖V row.2 (r,w)‖^2 + ‖V row.2 (r,u)‖^2*‖U row.1 (r,w)‖^2) :=
distance_graph_pointwise_bound Q₁ Q₂ ρ ν A B N a n sig t hA hB
I hI hθ hscale axis d hd hdegree (fun P u => U P (r,u)) (fun Q w => V Q (r,w)) u w
have hmain := original_distance_schur_integral (sourceProductVertices Q₁ Q₂ ρ ν A B n)
(fun row => U row.1) (fun row => V row.2)
(fun row _ => measurable_sourceAngularTransform ρ (Q₁.descendant row.1).carrier M₁ β hβ)
(fun row _ => measurable_sourceAngularTransform ν (Q₂.descendant row.2).carrier M₂ γ hγ)
(fun row => B₁*(ρ (Q₁.descendant row.1).carrier).toReal/M₁)
(fun row => B₂*(ν (Q₂.descendant row.2).carrier).toReal/M₂)
(fun row _ => by positivity) (fun row _ => by positivity)
(fun row _ z => sourceAngularTransform_norm_le ρ _ M₁ hM₁ β B₁ (fun x _ => hbβ x) z.1 z.2)
(fun row _ z => sourceAngularTransform_norm_le ν _ M₂ hM₂ γ B₂ (fun x _ => hbγ x) z.1 z.2)
F hd hpoint (a+n) v hΨ hpos hs hW hw
have hsum := sourceGraphFourTransform_original_sum H ![ρ,ν,ν,ρ]
(fun e l => (graphQuadCells ![Q₁,Q₂,Q₂,Q₁] e.1 l).carrier)
![M₁,M₂,M₂,M₁] hM ![β,γ,γ,β]
(fun e => graphRotatedDistanceWeight axis (angularBinCount θ) e.2.1.1 e.2.1.2) hb
(fun e => (graphRotatedDistanceWeight_continuous _ _ _ _).measurable)
(fun e => graphRotatedDistanceWeight_norm_le _ _ _ _)
![B₁,B₂,B₂,B₁] hB4 hbound hΨ hs hW
(fun t => by rw [abs_of_nonneg (hpos t)]; exact hw t) v
dsimp only [F] at hmain
rw [hsum] at hmain
dsimp only
rw [graphOriginalDistanceSum_supported]
simpa only [Finset.smul_sum,graphOriginalTerm,graphSourceWeight,
↓reduceIte,H,U,V] using hmain
end PlanarFalconer
end OAI
|