| import OAI.MeasureTheory.Falconer.Recursion.CriticalPotential |
| import Mathlib.MeasureTheory.Integral.Lebesgue.Map |
|
|
| namespace OAI |
|
|
| /-! # Critical dyadic inverse potentials of a nonnegative statistic -/ |
|
|
| namespace Falconer |
| open Set Metric MeasureTheory |
| open scoped ENNReal |
| variable {X : Type*} [MeasurableSpace X] |
|
|
| theorem regularized_scalar_potential_critical (μ : Measure X) (a : X → ℝ) |
| (ha : Measurable a) (ha0 : ∀ x, 0 ≤ a x) |
| {K q : ℝ} (hq : 0 ≤ q) (g k : ℕ) |
| (hsmall : ∀ j : ℕ, j ≤ g+k → |
| μ {x | a x < 1/(2:ℝ)^j} ≤ ENNReal.ofReal (K*(1/(2:ℝ)^j)^q)) : |
| (∫⁻ x in {x | a x < 1/(2:ℝ)^g}, |
| ENNReal.ofReal ((1/(a x+1/(2:ℝ)^(g+k)))^q) ∂μ) ≤ |
| ENNReal.ofReal K+(k:ℝ≥0∞)*ENNReal.ofReal (K*(2:ℝ)^q) := by |
| have hpre (j : ℕ) : a ⁻¹' ball (0:ℝ) (1/(2:ℝ)^j) = {x | a x < 1/(2:ℝ)^j} := by |
| ext x |
| simp only [mem_preimage,mem_ball,Real.dist_eq,sub_zero,abs_of_nonneg (ha0 x),mem_ofPred_eq] |
| have hball (j : ℕ) (hj : j ≤ g+k) : |
| (Measure.map a μ) (ball (0:ℝ) (1/(2:ℝ)^j)) ≤ |
| ENNReal.ofReal (K*(1/(2:ℝ)^j)^q) := by |
| rw [Measure.map_apply ha measurableSet_ball,hpre] |
| exact hsmall j hj |
| have hh := regularized_potential_critical (Measure.map a μ) (0:ℝ) hq g k hball |
| have hf : Measurable (fun z : ℝ => |
| ENNReal.ofReal ((1/(dist 0 z+1/(2:ℝ)^(g+k)))^q)) := by fun_prop |
| have he : (∫⁻ z in ball (0:ℝ) (1/(2:ℝ)^g), |
| ENNReal.ofReal ((1/(dist 0 z+1/(2:ℝ)^(g+k)))^q) ∂Measure.map a μ) = |
| ∫⁻ x in {x | a x < 1/(2:ℝ)^g}, |
| ENNReal.ofReal ((1/(a x+1/(2:ℝ)^(g+k)))^q) ∂μ := by |
| rw [← lintegral_indicator measurableSet_ball,lintegral_map |
| (hf.indicator measurableSet_ball) ha, |
| ← lintegral_indicator (measurableSet_lt ha measurable_const)] |
| apply lintegral_congr |
| intro x |
| by_cases hx : a x < 1/(2:ℝ)^g |
| · have hz : a x ∈ ball (0:ℝ) (1/(2:ℝ)^g) := by |
| change x ∈ a ⁻¹' ball (0:ℝ) (1/(2:ℝ)^g) |
| rw [hpre] |
| exact hx |
| rw [indicator_of_mem hz,indicator_of_mem (show x ∈ {x | a x < 1/(2:ℝ)^g} from hx)] |
| simp only [Real.dist_eq,zero_sub,abs_neg,abs_of_nonneg (ha0 x)] |
| · have hz : a x ∉ ball (0:ℝ) (1/(2:ℝ)^g) := by |
| change x ∉ a ⁻¹' ball (0:ℝ) (1/(2:ℝ)^g) |
| rw [hpre] |
| exact hx |
| rw [indicator_of_notMem hz,indicator_of_notMem (show x ∉ {x | a x < 1/(2:ℝ)^g} from hx)] |
| rw [he] at hh |
| exact hh |
|
|
| end Falconer |
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| end OAI |
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