| import Mathlib |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphDegrees |
|
|
| namespace OAI |
|
|
| /-! Recursive policies and quantitative depth estimates. -/ |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace Attempt4Near |
| open Set MeasureTheory |
| open scoped ComplexConjugate |
| variable {V : Type*} [MeasurableSpace V] |
|
|
| lemma sourceAngularMomentND_le (μ : Measure V) [IsFiniteMeasure μ] |
| {U W : ℝ × V → ℂ} (hW : Measurable W) {C : ℝ} |
| (hb : ∀ r, ∀ᵐ w ∂μ, ‖W (r,w)‖ ≤ C) (r : ℝ) |
| (hUW : ∀ᵐ w ∂μ, ‖U (r,w)‖ ≤ ‖W (r,w)‖) : |
| sourceAngularMomentND μ U r ≤ sourceAngularMomentND μ W r := by |
| apply integral_mono_of_nonneg (Filter.Eventually.of_forall (fun _ => sq_nonneg _)) |
| (angular_square_integrableND μ (hW.comp (measurable_const.prodMk measurable_id)) (hb r)) |
| filter_upwards [hUW] with w hw |
| exact pow_le_pow_left₀ (norm_nonneg _) hw 2 |
|
|
| lemma sourceAngularMomentND_cutoff_le (μ : Measure V) [IsFiniteMeasure μ] |
| {U : ℝ × V → ℂ} (hU : Measurable U) {C : ℝ} |
| (hb : ∀ r, ∀ᵐ w ∂μ, ‖U (r,w)‖ ≤ C) (ψ : V → ℂ) |
| (hψ : ∀ᵐ w ∂μ, ‖ψ w‖ ≤ 1) (r : ℝ) : |
| sourceAngularMomentND μ (fun z => ψ z.2*U z) r ≤ sourceAngularMomentND μ U r := by |
| apply sourceAngularMomentND_le μ hU hb |
| filter_upwards [hψ] with w hw |
| simpa only [norm_mul,one_mul] using mul_le_mul_of_nonneg_right hw (norm_nonneg (U (r,w))) |
|
|
| lemma sourceAngularMomentND_smul (μ : Measure V) (lam : ℝ) (U : ℝ × V → ℂ) (r : ℝ) : |
| sourceAngularMomentND μ (fun z => lam • U z) r = lam^2*sourceAngularMomentND μ U r := by |
| simp only [sourceAngularMomentND,norm_smul,Real.norm_eq_abs,mul_pow,sq_abs,integral_const_mul] |
|
|
| lemma sourceAngularHND_nonneg (d : ℕ) (μ : Measure V) (a v : ℝ) |
| {Ψ : ℝ → ℝ} (hΨ : ∀ r, 0 ≤ Ψ r) (U W : ℝ × V → ℂ) : |
| 0 ≤ sourceAngularHND d μ a v Ψ U W := by |
| apply mul_nonneg (by positivity) |
| exact integral_nonneg (fun r => mul_nonneg |
| (mul_nonneg (hΨ _) (sourceAngularMomentND_nonneg μ U r)) (sourceAngularMomentND_nonneg μ W r)) |
|
|
| lemma sourceAngularHFND_nonneg (d : ℕ) (μ : Measure V) (a v : ℝ) |
| {Ψ : ℝ → ℝ} (hΨ : ∀ r, 0 ≤ Ψ r) (ψ : V → ℂ) (U W : ℝ × V → ℂ) : |
| 0 ≤ sourceAngularHFND d μ a v Ψ ψ U W := by |
| apply mul_nonneg (by positivity) |
| exact integral_nonneg (fun r => mul_nonneg (hΨ _) (sq_nonneg _)) |
|
|
| lemma sourceAngularHND_smul (d : ℕ) (μ : Measure V) (a v : ℝ) |
| (Ψ : ℝ → ℝ) (U W : ℝ × V → ℂ) (lam₁ lam₂ : ℝ) : |
| sourceAngularHND d μ a v Ψ (fun z => lam₁ • U z) (fun z => lam₂ • W z) = |
| lam₁^2*lam₂^2*sourceAngularHND d μ a v Ψ U W := by |
| simp only [sourceAngularHND,sourceAngularMomentND_smul] |
| have he : (fun r => Ψ (r/(2:ℝ)^v)*(lam₁^2*sourceAngularMomentND μ U r)* |
| (lam₂^2*sourceAngularMomentND μ W r)) = |
| fun r => (lam₁^2*lam₂^2)*(Ψ (r/(2:ℝ)^v)*sourceAngularMomentND μ U r*sourceAngularMomentND μ W r) := by |
| funext r |
| ring |
| rw [he,integral_const_mul] |
| ring |
|
|
| lemma sourceAngularHND_depth (d : ℕ) (μ : Measure V) (a p v : ℝ) |
| (Ψ : ℝ → ℝ) (U W : ℝ × V → ℂ) : |
| sourceAngularHND d μ a v Ψ U W = |
| (2:ℝ)^((d:ℝ)*(p-a))*sourceAngularHND d μ p v Ψ U W := by |
| dsimp [sourceAngularHND] |
| rw [← mul_assoc,← mul_assoc,← Real.rpow_add (by norm_num : (0:ℝ)<2)] |
| congr 3 |
| ring |
|
|
| lemma sourceAngularHFND_cutoff (d : ℕ) (μ : Measure V) (a v : ℝ) |
| (Ψ : ℝ → ℝ) (ψ : V → ℂ) (U W : ℝ × V → ℂ) : |
| sourceAngularHFND d μ a v Ψ ψ U W = |
| sourceAngularHFND d μ a v Ψ (fun _ => 1) (fun z => ψ z.2*U z) W := by |
| simp [sourceAngularHFND] |
|
|
| end Attempt4Near |
| end |
| end |
|
|
| end OAI |
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|
