| import OAI.MeasureTheory.Falconer.Radial.RadialZoomBounds |
| import OAI.MeasureTheory.Falconer.Geometry.FiniteScaleBall |
| import OAI.MeasureTheory.Falconer.Recursion.ProfileIncrementFrostman |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| namespace Falconer.Regularization |
| open Set MeasureTheory |
| open scoped ENNReal |
| variable {d : ℕ} |
|
|
| theorem cell_normalization_map (ρ : Measure (Space d)) (p : ℕ) (c : Space d) |
| {b : ℝ} (hb : 0 < b) : |
| Measure.map (radialZoom (b*(2:ℝ)^p) (by positivity) c) ρ = |
| Measure.map (radialZoom b hb.ne' 0) (Measure.map (cellScaleMap p c) ρ) := by |
| rw [Measure.map_map (radialZoom b hb.ne' 0).measurable (cellScaleMap_continuous p c).measurable] |
| congr 1 |
| ext x |
| simp only [Function.comp_apply,radialZoom_apply,cellScaleMap,sub_zero,smul_smul] |
|
|
| theorem cell_normalization_support (Q : DyadicCube d) (hQ : Q.side = 1) |
| (ρ : Measure (Space d)) (P : CellPath d) {b : ℝ} (hb : 0 < b) : |
| ∀ᵐ z ∂Measure.map (radialZoom (b*(2:ℝ)^P.length) (by positivity) (Q.descendant P).center) |
| (ProbabilityTheory.cond ρ (Q.descendant P).carrier), ‖z‖ ≤ b*Real.sqrt d := by |
| rw [cell_normalization_map _ _ _ hb] |
| apply radialZoom_ae_bound _ hb 0 |
| simpa only [sub_zero] using rescaled_conditional_support Q hQ ρ P |
|
|
| theorem cell_normalization_quarter_point (Q : DyadicCube d) (hQ : Q.side = 1) |
| (P : CellPath d) {b : ℝ} (hb : 0 < b) (hsmall : b*Real.sqrt d ≤ 1/4) |
| {z : Space d} (hz : z ∈ (Q.descendant P).carrier) : |
| ‖z-(Q.descendant P).center‖ ≤ 1/(4*(b*(2:ℝ)^P.length)) := by |
| have hh := rescaled_cell_norm_bound Q hQ P hz |
| rw [cellScaleMap,norm_smul,Real.norm_eq_abs,abs_of_pos (by positivity : 0 < (2:ℝ)^P.length)] at hh |
| have hm := (mul_le_mul_of_nonneg_left hh hb.le).trans hsmall |
| apply (le_div_iff₀ (by positivity : 0 < 4*(b*(2:ℝ)^P.length))).mpr |
| nlinarith |
|
|
| theorem conditional_frostman_coefficient_ge_one (d N : ℕ) {s E lam : ℝ} |
| (hs : 0 ≤ s) (hE : 0 ≤ E) (hlam : 0 ≤ lam) : |
| 1 ≤ (((2*Nat.ceil (2+4*Real.sqrt d)+1)^d:ℕ):ℝ)*(2:ℝ)^s* |
| ((2:ℝ)^N)^(E+2*((d:ℝ)+1)*lam) := by |
| have hdim : (1:ℝ) ≤ (((2*Nat.ceil (2+4*Real.sqrt d)+1)^d:ℕ):ℝ) := by |
| exact_mod_cast (one_le_pow₀ (by omega : 1 ≤ 2*Nat.ceil (2+4*Real.sqrt d)+1) : |
| 1 ≤ (2*Nat.ceil (2+4*Real.sqrt d)+1)^d) |
| have htwo : 1 ≤ (2:ℝ)^s := Real.one_le_rpow (by norm_num) hs |
| have hR : 1 ≤ ((2:ℝ)^N)^(E+2*((d:ℝ)+1)*lam) := |
| Real.one_le_rpow (one_le_pow₀ (by norm_num)) (by positivity) |
| exact one_le_mul_of_one_le_of_one_le (one_le_mul_of_one_le_of_one_le hdim htwo) hR |
|
|
| theorem sourceClass_normalized_conditional_frostman (Q : DyadicCube d) (hQ : Q.side = 1) |
| (μ : Measure (Space d)) [IsProbabilityMeasure μ] {N B : ℕ} (hN : 0 < N) (hB : 0 < B) |
| {lam s E b : ℝ} (hs : 0 ≤ s) (hE : 0 ≤ E) (hlam : 0 ≤ lam) |
| (hb : 0 < b) (hb1 : b ≤ 1) (hwidth : (B:ℝ) ≤ lam*(N:ℝ)) |
| (hlarge : (2:ℝ)^(N/B+2) ≤ ((2:ℝ)^N)^lam) |
| {p₀ : CellPath d} (hp₀ : p₀ ∈ regularizationLeaves Q μ N) |
| {p t : ℕ} (hpt : p ≤ t) (htN : t ≤ N) {P : CellPath d} (hP : P.length = p) |
| (hPpos : 0 < sourceClass Q μ N B p₀ (Q.descendant P).carrier) |
| (hinc : ∀ n, p ≤ n → n ≤ t → |
| s*(n-p:ℕ)/N-E ≤ sourceProfile Q μ N B p₀ n-sourceProfile Q μ N B p₀ p) : |
| ∀ x : Space d, ∀ r : ℝ, 0 < r → 1/(2:ℝ)^(t-p) ≤ r → |
| Measure.map (radialZoom (b*(2:ℝ)^p) (by positivity) (Q.descendant P).center) |
| (ProbabilityTheory.cond (sourceClass Q μ N B p₀) (Q.descendant P).carrier) |
| (Metric.ball x r) ≤ |
| ENNReal.ofReal (((((2*Nat.ceil (2+4*Real.sqrt d)+1)^d:ℕ):ℝ)*(2:ℝ)^s* |
| ((2:ℝ)^N)^(E+2*((d:ℝ)+1)*lam))*b^(-s)*r^s) := by |
| have hside : 0 < Q.side := by rw [hQ]; norm_num |
| have : IsProbabilityMeasure (ProbabilityTheory.cond (sourceClass Q μ N B p₀) |
| (Q.descendant P).carrier) := sourceClass_conditional_probability Q hside μ N B hp₀ P hPpos |
| intro x r hr hδ |
| rw [cell_normalization_map _ _ _ hb] |
| apply radialZoom_frostman_bound _ hb hb1 |
| (conditional_frostman_coefficient_ge_one d N hs hE hlam) hs (by simp) 0 |
| (sourceClass_conditional_frostman_of_increment Q hQ μ hN hB hs hwidth hlarge |
| hp₀ hpt htN hP hPpos hinc) x r hr hδ |
|
|
| theorem sourceClass_scaled_dyadic_ball (Q : DyadicCube d) (hQ : Q.side = 1) |
| (μ : Measure (Space d)) [IsProbabilityMeasure μ] {N B : ℕ} (hN : 0 < N) (hB : 0 < B) |
| {lam s E b K : ℝ} (hs : 0 ≤ s) (hE : 0 ≤ E) (hlam : 0 ≤ lam) |
| (hb : 0 < b) (hb1 : b ≤ 1) (hwidth : (B:ℝ) ≤ lam*(N:ℝ)) |
| (hlarge : (2:ℝ)^(N/B+2) ≤ ((2:ℝ)^N)^lam) |
| {p₀ : CellPath d} (hp₀ : p₀ ∈ regularizationLeaves Q μ N) |
| {p t M : ℕ} (hpt : p ≤ t) (htN : t ≤ N) (hM : M ≤ t-p) |
| {P : CellPath d} (hP : P.length = p) |
| (hPpos : 0 < sourceClass Q μ N B p₀ (Q.descendant P).carrier) |
| (hinc : ∀ n, p ≤ n → n ≤ t → |
| s*(n-p:ℕ)/N-E ≤ sourceProfile Q μ N B p₀ n-sourceProfile Q μ N B p₀ p) |
| (hcoef : (((((2*Nat.ceil (2+4*Real.sqrt d)+1)^d:ℕ):ℝ)*(2:ℝ)^s* |
| ((2:ℝ)^N)^(E+2*((d:ℝ)+1)*lam))*b^(-s)) ≤ K) : |
| ∀ x : Space d, ∀ j : ℕ, j ≤ M → |
| ProbabilityTheory.cond (sourceClass Q μ N B p₀) (Q.descendant P).carrier |
| (Metric.ball x ((1/(2:ℝ)^j)/(b*(2:ℝ)^p))) ≤ |
| ENNReal.ofReal (K*(1/(2:ℝ)^j)^s) := by |
| intro x j hj |
| have ha : 0 < b*(2:ℝ)^p := by positivity |
| have hr : 1/(2:ℝ)^(t-p) ≤ 1/(2:ℝ)^j := |
| one_div_le_one_div_of_le (by positivity) (pow_le_pow_right₀ (by norm_num) (hj.trans hM)) |
| have hh := sourceClass_normalized_conditional_frostman Q hQ μ hN hB hs hE hlam hb hb1 |
| hwidth hlarge hp₀ hpt htN hP hPpos hinc |
| (radialZoom (b*(2:ℝ)^p) ha.ne' (Q.descendant P).center x) |
| (1/(2:ℝ)^j) (by positivity) hr |
| rw [radialZoom_ball_apply _ ha] at hh |
| have hc : (Q.descendant P).center+(b*(2:ℝ)^p)⁻¹ • |
| radialZoom (b*(2:ℝ)^p) ha.ne' (Q.descendant P).center x = x := by |
| simp only [radialZoom_apply,smul_smul,inv_mul_cancel₀ ha.ne',one_smul] |
| abel |
| rw [hc] at hh |
| apply hh.trans |
| exact ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right hcoef (by positivity)) |
|
|
| end Falconer.Regularization |
|
|
| end OAI |
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|
