| import OAI.MeasureTheory.Falconer.Estimates.EndpointTwoSidedScale |
| import OAI.MeasureTheory.Falconer.Radial.FiniteRadialRefinement |
| import OAI.MeasureTheory.Falconer.Radial.RadialScaleWindow |
| import OAI.MeasureTheory.Falconer.Radial.ReverseSlabsAllRadii |
|
|
| namespace OAI |
|
|
| /-! # The same original pair set can be improved throughout a dyadic scale window -/ |
|
|
| namespace Falconer.Regularization |
| open Set MeasureTheory NormedSpace |
| open scoped ENNReal BigOperators |
|
|
| theorem endpoint_scale_window_gain (hLW : PublishedFiniteLoomisWhitney) |
| {h : ℕ} {α t q ζ C D s : ℝ} |
| (hq : q = (h:ℝ)+1) (ht : (h:ℝ)+3/2 < t) (ht' : t < (h:ℝ)+2) |
| (hζ : 0 < ζ) (htα : t < α) (hC : 0 ≤ C) |
| (Q : DyadicCube (h+3)) (hQ : Q.side = 1) |
| (hD : 1 ≤ D) (hDdim : Real.sqrt (h+3:ℕ) ≤ D) (hs : 0 < s) : |
| ∃ ε gain : ℝ, 0 < ε ∧ 0 < gain ∧ ∀ K : ℕ, 0 < K → |
| ∃ c : ℝ, 0 < c ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀, |
| ∀ (μ ν : Measure (Space (h+3))) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν], |
| μ Q.carrierᶜ = 0 → ν Q.carrierᶜ = 0 → |
| (∀ y : Space (h+3), ∀ r > 0, μ (Metric.ball y r) ≤ ENNReal.ofReal (C*r^α)) → |
| (∀ y : Space (h+3), ∀ r > 0, ν (Metric.ball y r) ≤ ENNReal.ofReal (C*r^α)) → |
| ∀ A P : Set (Space (h+3)), MeasurableSet A → MeasurableSet P → |
| μ Aᶜ = 0 → ν Pᶜ = 0 → |
| (∀ z ∈ A, ∀ x ∈ P, s ≤ ‖z-x‖) → |
| (∀ y ∈ Q.carrier, ∀ x ∈ A ∪ P, ‖y-x‖ ≤ D) → |
| ∀ G : Set (Space (h+3) × Space (h+3)), MeasurableSet G → |
| (∀ᵐ z ∂μ, ∀ r : ℝ, 1/(2:ℝ)^N ≤ r → r ≤ 1 → |
| ∀ u : Space (h+3), ‖u‖ = 1 → |
| ν {x | (z,x) ∈ G ∧ dist (normalize (x-z)) u < r} ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(ε/(2*(K:ℝ)))*r^q)) → |
| (∀ᵐ z ∂μ, ∀ u : Space (h+3), ‖u‖ = 1 → ∀ r : ℝ, |
| 1/(2:ℝ)^N ≤ r → r ≤ 1 → |
| ν {x | (z,x) ∈ G ∧ |inner ℝ u (normalize (x-z))| ≤ r} ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(ε/(2*(K:ℝ)))*r^ζ)) → |
| (∀ᵐ x ∂ν, ∀ r : ℝ, 1/(2:ℝ)^N ≤ r → r ≤ 1 → |
| ∀ u : Space (h+3), ‖u‖ = 1 → |
| μ {z | (z,x) ∈ G ∧ dist (normalize (z-x)) u < r} ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(ε/(2*(K:ℝ)))*r^q)) → |
| (∀ᵐ x ∂ν, ∀ u : Space (h+3), ‖u‖ = 1 → ∀ r : ℝ, |
| 1/(2:ℝ)^N ≤ r → r ≤ 1 → |
| μ {z | (z,x) ∈ G ∧ |inner ℝ u (normalize (z-x))| ≤ r} ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(ε/(2*(K:ℝ)))*r^ζ)) → |
| ∃ G' : Set (Space (h+3) × Space (h+3)), MeasurableSet G' ∧ G' ⊆ G ∧ |
| (μ.prod ν) (G \ G') ≤ 2*ENNReal.ofReal (((2:ℝ)^N)^(-c)) ∧ |
| ∀ j : ℕ, N/K ≤ j → j ≤ N → |
| (∀ x u : Space (h+3), μ {z | (z,x) ∈ G' ∧ ‖normalize (z-x)-u‖ ≤ 1/(2:ℝ)^j} ≤ |
| ENNReal.ofReal ((1/(2:ℝ)^j)^(q+gain))) ∧ |
| (∀ z u : Space (h+3), ν {x | (z,x) ∈ G' ∧ ‖normalize (x-z)-u‖ ≤ 1/(2:ℝ)^j} ≤ |
| ENNReal.ofReal ((1/(2:ℝ)^j)^(q+gain))) := by |
| obtain ⟨ε,gain,hε,hgain,Ns,hscale⟩ := endpoint_two_sided_one_scale hLW hq ht ht' hζ |
| htα hC Q hQ hD hDdim hs |
| refine ⟨ε,gain,hε,hgain,?_⟩ |
| intro K hK |
| obtain ⟨c,hc,Nc,hcost⟩ := radial_scale_window_cost K hK hgain |
| refine ⟨c,hc,max (K*Ns) (max (2*K) Nc),?_⟩ |
| intro N hN μ ν _ _ hμroot hνroot hμfrost hνfrost A P hA hP hμA hνP hsep hpin G hG hcap hslab hcap' hslab' |
| simp only [max_le_iff] at hN |
| obtain ⟨hNs,hNK,hNc⟩ := hN |
| have hstart : Ns ≤ N/K := (Nat.le_div_iff_mul_le hK).mpr (by simpa only [Nat.mul_comm] using hNs) |
| have hcapSwap : ∀ᵐ z ∂ν, ∀ r : ℝ, 1/(2:ℝ)^N ≤ r → r ≤ 1 → |
| ∀ u : Space (h+3), ‖u‖ = 1 → |
| μ {x | (z,x) ∈ Prod.swap ⁻¹' G ∧ dist (normalize (x-z)) u < r} ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(ε/(2*(K:ℝ)))*r^q) := by |
| simpa only [mem_preimage,Prod.swap_prod_mk] using hcap' |
| have hslabSwap : ∀ᵐ z ∂ν, ∀ u : Space (h+3), ‖u‖ = 1 → ∀ r : ℝ, |
| 1/(2:ℝ)^N ≤ r → r ≤ 1 → |
| μ {x | (z,x) ∈ Prod.swap ⁻¹' G ∧ |inner ℝ u (normalize (x-z))| ≤ r} ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(ε/(2*(K:ℝ)))*r^ζ) := by |
| simpa only [mem_preimage,Prod.swap_prod_mk] using hslab' |
| obtain ⟨H,hH,hsub,hdel,hbounds⟩ := finite_radial_refinement μ ν G hG (Finset.Icc (N/K) N) |
| (fun j => 2*ENNReal.ofReal (((2:ℝ)^j)^(-gain))) |
| (fun j => ENNReal.ofReal ((1/(2:ℝ)^j)^(q+gain))) (fun j => 1/(2:ℝ)^j) (by |
| intro j hj |
| obtain ⟨hjlo,hjhi⟩ := Finset.mem_Icc.mp hj |
| have hloss := radial_scale_window_loss hK hNK hjlo hε.le |
| have hc₁ := reverse_caps_all_radii μ μ ν Measure.AbsolutelyContinuous.rfl |
| (Subset.rfl (s := G)) hjhi hloss hcap |
| have hc₂ := reverse_caps_all_radii ν ν μ Measure.AbsolutelyContinuous.rfl |
| (Subset.rfl (s := Prod.swap ⁻¹' G)) hjhi hloss hcapSwap |
| have hs₁ := reverse_slabs_all_radii μ μ ν Measure.AbsolutelyContinuous.rfl |
| (Subset.rfl (s := G)) hjhi hloss hslab |
| have hs₂ := reverse_slabs_all_radii ν ν μ Measure.AbsolutelyContinuous.rfl |
| (Subset.rfl (s := Prod.swap ⁻¹' G)) hjhi hloss hslabSwap |
| obtain ⟨H,hH,hsub,hdel,hbounds⟩ := hscale j (hstart.trans hjlo) μ ν |
| hμroot hνroot hμfrost hνfrost A P hA hP hμA hνP hsep hpin G hG hc₁ hs₁ |
| (by simpa only [mem_preimage,Prod.swap_prod_mk] using hc₂) |
| (by simpa only [mem_preimage,Prod.swap_prod_mk] using hs₂) |
| refine ⟨H,hH,hsub,hdel,?_⟩ |
| have he : ((2:ℝ)^j)^(-q-gain) = (1/(2:ℝ)^j)^(q+gain) := by |
| rw [one_div,Real.inv_rpow (by positivity),← Real.rpow_neg (by positivity)] |
| congr 1 |
| ring |
| simpa only [he] using hbounds) |
| refine ⟨H,hH,hsub,?_,fun j hjlo hjhi => hbounds j (Finset.mem_Icc.mpr ⟨hjlo,hjhi⟩)⟩ |
| apply hdel.trans |
| rw [← Finset.mul_sum] |
| exact mul_le_mul' le_rfl (hcost N hNc) |
|
|
| end Falconer.Regularization |
|
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| end OAI |
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