| import OAI.Analysis.CircleDomains.Topology.CompactArcReduction |
| import OAI.Analysis.CircleDomains.Selection.UncountableHitTail |
|
|
| namespace OAI |
|
|
| /-! The exact factor-two density and hit estimates yield the L² bounds |
| used in the compact-arc smallness statement. -/ |
|
|
| open Set Function Filter MeasureTheory |
| open scoped ENNReal Topology Classical |
|
|
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| theorem countingSquareNorm_twice {S : Type*} (m : S → ℝ≥0∞) : |
| countingSquareNorm (fun s => 2 * m s) = 2 * countingSquareNorm m := by |
| simp only [countingSquareNorm, mul_pow] |
| rw [ENNReal.tsum_mul_left, ENNReal.mul_rpow_of_nonneg _ _ (by norm_num)] |
| have htwo : ((2 : ℝ≥0∞) ^ 2) ^ (1 / 2 : ℝ) = 2 := by |
| simpa only [Nat.cast_ofNat, one_div] using ENNReal.pow_rpow_inv_natCast (by decide : (2 : ℕ) ≠ 0) (2 : ℝ≥0∞) |
| rw [htwo] |
|
|
| theorem arc_density_squareNorm_le |
| {X : Type*} [MeasurableSpace X] (σ : Measure X) |
| (g f : X → ℝ≥0∞) (hg : Measurable g) (W : Set X) (hW : MeasurableSet W) |
| (hf : f ≤ᵐ[σ] fun x => 2 * W.indicator g x) : |
| (∫⁻ x, f x ^ 2 ∂σ) ^ (1 / 2 : ℝ) ≤ |
| 2 * (∫⁻ x in W, g x ^ 2 ∂σ) ^ (1 / 2 : ℝ) := by |
| have hbound : (∫⁻ x, f x ^ 2 ∂σ) ≤ (2 : ℝ≥0∞) ^ 2 * ∫⁻ x in W, g x ^ 2 ∂σ := by |
| calc |
| _ ≤ ∫⁻ x, (2 * W.indicator g x) ^ 2 ∂σ := |
| lintegral_mono_ae (hf.mono fun x hx => pow_le_pow_left' hx 2) |
| _ = _ := by |
| have he : (fun x => (2 * W.indicator g x) ^ 2) = |
| fun x => (2 : ℝ≥0∞) ^ 2 * W.indicator (fun x => g x ^ 2) x := by |
| funext x |
| by_cases hx : x ∈ W <;> simp [hx, mul_pow] |
| rw [he, lintegral_const_mul _ ((hg.pow_const 2).indicator hW), lintegral_indicator hW] |
| calc |
| _ ≤ ((2 : ℝ≥0∞) ^ 2 * ∫⁻ x in W, g x ^ 2 ∂σ) ^ (1 / 2 : ℝ) := |
| ENNReal.rpow_le_rpow hbound (by norm_num) |
| _ = _ := by |
| rw [ENNReal.mul_rpow_of_nonneg _ _ (by norm_num)] |
| have htwo : ((2 : ℝ≥0∞) ^ 2) ^ (1 / 2 : ℝ) = 2 := by |
| simpa only [Nat.cast_ofNat, one_div] using ENNReal.pow_rpow_inv_natCast (by decide : (2 : ℕ) ≠ 0) (2 : ℝ≥0∞) |
| rw [htwo] |
|
|
| theorem arc_hit_squareNorm_le {S : Type*} (m h : S → ℝ≥0∞) (p : S) (W : Set S) |
| (hle : ∀ s, h s ≤ 2 * m s) (hout : ∀ s ∉ W, h s = 0) : |
| countingSquareNorm (fun s => if s ≠ p then h s else 0) ≤ |
| 2 * countingSquareNorm (fun s => if s ∈ W ∧ s ≠ p then m s else 0) := by |
| calc |
| _ ≤ countingSquareNorm (fun s => 2 * (if s ∈ W ∧ s ≠ p then m s else 0)) := by |
| apply countingSquareNorm_mono |
| intro s |
| by_cases hp : s = p |
| · simp [hp] |
| · by_cases hs : s ∈ W |
| · simpa only [hp, ne_eq, not_false_eq_true, hs, and_self, ite_true] using hle s |
| · simp [hp, hs, hout s hs] |
| _ = _ := countingSquareNorm_twice _ |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
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|
