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d4317d7 | 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 | 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
end OAI
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