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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 | import OAI.MeasureTheory.Falconer.Measures.OriginalConditionalFrostman
namespace OAI
/-! # Original conditional laws stay within the actual dyadic cell diameter -/
namespace Falconer.Regularization
open Set MeasureTheory
open scoped ENNReal
variable {d : ℕ}
theorem conditional_cell_pair_geometry (Q : DyadicCube d) (hQ : Q.side = 1)
(ρ : Measure (Space d)) (P : CellPath d) {g : ℕ} (hP : P.length = g)
{D : ℝ} (hD : Real.sqrt d ≤ D)
(G : Set (Space d × Space d))
(hpin : ∀ z x, (z,x) ∈ G → ∀ y ∈ (Q.descendant P).carrier, ‖y-x‖ ≤ D) :
∀ᵐ z ∂ProbabilityTheory.cond ρ (Q.descendant P).carrier,
∀ᵐ y ∂ProbabilityTheory.cond ρ (Q.descendant P).carrier,
‖y-z‖ ≤ D/(2:ℝ)^g ∧ ∀ x, (z,x) ∈ G → ‖y-x‖ ≤ D := by
apply ProbabilityTheory.ae_cond_of_forall_mem (Q.descendant P).measurable
intro z hz
apply ProbabilityTheory.ae_cond_of_forall_mem (Q.descendant P).measurable
intro y hy
refine ⟨?_,fun x hx => hpin z x hx y hy⟩
have hside : 0 ≤ (Q.descendant P).side := by rw [Q.descendant_side,hQ]; positivity
have hh := (Q.descendant P).dist_le hside hy hz
rw [dist_eq_norm,Q.descendant_side,hQ,hP] at hh
apply hh.trans
simpa only [mul_one_div] using div_le_div_of_nonneg_right hD
(by positivity : (0:ℝ) ≤ 2^g)
end Falconer.Regularization
end OAI
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