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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 | import OAI.MeasureTheory.Falconer.Shells.ClassPairDistance
import OAI.MeasureTheory.Falconer.Shells.FilteredClassShell
namespace OAI
/-! # The assembled class law inherits the local shell bound -/
namespace Falconer.Regularization
open Set MeasureTheory
open scoped ENNReal NNReal
variable {d : ℕ}
theorem regularizationPairDistance_shell_le {k : ℕ} (hdim : 1 ≤ d)
(Q₁ Q₂ : DyadicCube d) (hQ₁ : 0 < Q₁.side) (hQ₂ : 0 < Q₂.side)
(μ ν : Measure (Space d)) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(N B₁ B₂ : ℕ) (t₁ t₂ : ℝ≥0∞) (ht₁ : 0 < t₁) (ht₂ : 0 < t₂) (σ κ : ℝ)
(p₁ p₂ : List ℕ → CellPath d)
(he₁ : ∀ T ∈ regularizationIndex Q₁ μ N B₁ t₁, regularizationLabel Q₁ μ N B₁ (p₁ T) = T)
(he₂ : ∀ T ∈ regularizationIndex Q₂ ν N B₂ t₂, regularizationLabel Q₂ ν N B₂ (p₂ T) = T)
(c : List ℕ × List ℕ → ℕ) (A D : Fin k → Set (Space d))
(hm : ∀ j, MeasurableSet (A j) ∧ MeasurableSet (D j))
(hd : Pairwise fun j l => Disjoint (A j ×ˢ D j) (A l ×ˢ D l))
{δ : ℝ} (hδ : 0 < δ) (hsep : ∀ᵐ z ∂μ.prod ν, δ ≤ dist z.1 z.2)
{E : ℝ} (hE : 0 ≤ E)
(hlocal : ∀ i ∈ (regularizationIndex Q₁ μ N B₁ t₁).product (regularizationIndex Q₂ ν N B₂ t₂),
∀ j : Fin k,
(∫ r in Icc ((2:ℝ)^((N:ℝ)-1)) ((2:ℝ)^((N:ℝ)+1)),
‖charFun (endpointWeightedDistance ((sourceClass Q₁ μ N B₁ (p₁ i.1)).restrict (A j))
((sourceClass Q₂ ν N B₂ (p₂ i.2)).restrict (D j)) 1 1 1
(sourceOuterMask Q₁ μ N B₁ σ κ (p₁ i.1) (c i))
(sourceOuterMask Q₂ ν N B₂ σ κ (p₂ i.2) (c i))) r‖^2) ≤ E) :
PlanarFalconer.shellEnergy
(regularizationPairDistance Q₁ Q₂ μ ν N B₁ B₂ t₁ t₂ σ κ p₁ p₂ c (⋃ j, A j ×ˢ D j)) N ≤
2*(k:ℝ)^2*E := by
let : ∀ T, IsFiniteMeasure (sourceClass Q₁ μ N B₁ (p₁ T)) := fun _ => sourceClass_finite ..
let : ∀ T, IsFiniteMeasure (sourceClass Q₂ ν N B₂ (p₂ T)) := fun _ => sourceClass_finite ..
let τ := fun (i : List ℕ × List ℕ) (j : Fin k) =>
endpointWeightedDistance ((sourceClass Q₁ μ N B₁ (p₁ i.1)).restrict (A j))
((sourceClass Q₂ ν N B₂ (p₂ i.2)).restrict (D j)) 1 1 1
(sourceOuterMask Q₁ μ N B₁ σ κ (p₁ i.1) (c i))
(sourceOuterMask Q₂ ν N B₂ σ κ (p₂ i.2) (c i))
let : ∀ i j, IsFiniteMeasure (τ i j) := by
intro i j
apply endpointWeightedDistance_finite hdim _ _ zero_lt_one zero_lt_one hδ
_ _ (fun x w => regularOuterMask_bounds _ _ _ _ _ _ _ x w)
(fun x w => regularOuterMask_bounds _ _ _ _ _ _ _ x w)
have hμ : (sourceClass Q₁ μ N B₁ (p₁ i.1)).restrict (A j) ≪ μ :=
(Measure.absolutelyContinuous_of_le Measure.restrict_le_self).trans (sourceClass_absolutelyContinuous ..)
have hν : (sourceClass Q₂ ν N B₂ (p₂ i.2)).restrict (D j) ≪ ν :=
(Measure.absolutelyContinuous_of_le Measure.restrict_le_self).trans (sourceClass_absolutelyContinuous ..)
filter_upwards [(hμ.prod hν).ae_le hsep] with z hz
simpa only [div_one] using hz
rw [regularizationPairDistance_eq_mixture Q₁ Q₂ μ ν N B₁ B₂ t₁ t₂ ht₁ ht₂ σ κ p₁ p₂ he₁ he₂ c A D hm hd]
exact shellEnergy_filter_mixture _ (classPairWeight Q₁ Q₂ μ ν N B₁ B₂) k N τ
(classPairWeight_sum_le_one Q₁ Q₂ hQ₁ hQ₂ μ ν N B₁ B₂ t₁ t₂) hE hlocal
end Falconer.Regularization
end OAI
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