| 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 |
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|
| end OAI |
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