openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Graphs /AngularMomentBoundedFactor.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFiniteHFBound2 | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.NearGrid | |
| namespace OAI | |
| /-! Graph energies and pair estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4Near | |
| open Set MeasureTheory | |
| variable {V α β : Type*} [MeasurableSpace V] | |
| lemma sourceAngularMomentND_bounded_factor (μ : Measure V) [IsFiniteMeasure μ] | |
| {U : ℝ × V → ℂ} (hU : Measurable U) {C B : ℝ} | |
| (hb : ∀ r, ∀ᵐ w ∂μ, ‖U (r,w)‖≤C) (ψ : V → ℂ) | |
| (hψ : ∀ᵐ w ∂μ, ‖ψ w‖≤B) (r : ℝ) : | |
| sourceAngularMomentND μ (fun p => ψ p.2*U p) r≤B^2*sourceAngularMomentND μ U r := by | |
| rw [sourceAngularMomentND,sourceAngularMomentND,← integral_const_mul] | |
| apply integral_mono_of_nonneg (Filter.Eventually.of_forall (fun _ => sq_nonneg _)) | |
| ((angular_square_integrableND μ (hU.comp (measurable_const.prodMk measurable_id)) (hb r)).const_mul _) | |
| filter_upwards [hψ] with w hw | |
| rw [norm_mul,mul_pow] | |
| exact mul_le_mul_of_nonneg_right (pow_le_pow_left₀ (norm_nonneg _) hw 2) (sq_nonneg _) | |
| theorem sourceAngularHND_bounded_factors (d : ℕ) (μ : Measure V) [IsFiniteMeasure μ] | |
| (a v : ℝ) {U W : ℝ × V → ℂ} (hU : Measurable U) (hW : Measurable W) | |
| {C E B₁ B₂ : ℝ} | |
| (hbU : ∀ r, ∀ᵐ w ∂μ, ‖U (r,w)‖≤C) | |
| (hbW : ∀ r, ∀ᵐ w ∂μ, ‖W (r,w)‖≤E) | |
| (ψ φ : V → ℂ) (hψ : ∀ᵐ w ∂μ, ‖ψ w‖≤B₁) (hφ : ∀ᵐ w ∂μ, ‖φ w‖≤B₂) | |
| {Ψ : ℝ → ℝ} (hΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v))) (hΨpos : ∀ r, 0≤Ψ r) : | |
| sourceAngularHND d μ a v Ψ (fun p => ψ p.2*U p) (fun p => φ p.2*W p)≤ | |
| B₁^2*B₂^2*sourceAngularHND d μ a v Ψ U W := by | |
| have hi := graph_weighted_moments_integrableND μ hU hW hbU hbW hΨ | |
| have hm : (∫ r, Ψ (r/(2:ℝ)^v)* | |
| sourceAngularMomentND μ (fun p => ψ p.2*U p) r* | |
| sourceAngularMomentND μ (fun p => φ p.2*W p) r) ≤ | |
| ∫ r, (B₁^2*B₂^2)*(Ψ (r/(2:ℝ)^v)*sourceAngularMomentND μ U r*sourceAngularMomentND μ W r) := by | |
| apply integral_mono_of_nonneg (Filter.Eventually.of_forall (fun r => | |
| mul_nonneg (mul_nonneg (hΨpos _) (sourceAngularMomentND_nonneg _ _ _)) | |
| (sourceAngularMomentND_nonneg _ _ _))) (hi.const_mul _) | |
| filter_upwards with r | |
| have h₁ := sourceAngularMomentND_bounded_factor μ hU hbU ψ hψ r | |
| have h₂ := sourceAngularMomentND_bounded_factor μ hW hbW φ hφ r | |
| have hh := mul_le_mul_of_nonneg_left | |
| (mul_le_mul h₁ h₂ (sourceAngularMomentND_nonneg _ _ _) | |
| (mul_nonneg (sq_nonneg _) (sourceAngularMomentND_nonneg _ _ _))) (hΨpos (r/(2:ℝ)^v)) | |
| simpa only [mul_assoc,mul_left_comm,mul_comm] using hh | |
| rw [integral_const_mul] at hm | |
| dsimp [sourceAngularHND] | |
| have hh := mul_le_mul_of_nonneg_left hm | |
| (show 0≤(2:ℝ)^((d:ℝ)*(v-a))*((2:ℝ)^v)⁻¹ by positivity) | |
| simpa only [mul_assoc,mul_left_comm,mul_comm] using hh | |
| theorem graphChildMaxND_bounded_factors (d : ℕ) (μ : Measure V) [IsFiniteMeasure μ] | |
| (p v : ℝ) (A : Finset α) (B : Finset β) (hAB : (A.product B).Nonempty) | |
| (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) | |
| (hU : ∀ i∈A, Measurable (U i)) (hW : ∀ j∈B, Measurable (W j)) | |
| (C : α → ℝ) (E : β → ℝ) | |
| (hbU : ∀ i∈A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖≤C i) | |
| (hbW : ∀ j∈B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖≤E j) | |
| {B₁ B₂ : ℝ} | |
| (ψ : α → V → ℂ) (φ : β → V → ℂ) | |
| (hψ : ∀ i∈A, ∀ᵐ w ∂μ, ‖ψ i w‖≤B₁) (hφ : ∀ j∈B, ∀ᵐ w ∂μ, ‖φ j w‖≤B₂) | |
| {Ψ : ℝ → ℝ} (hΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v))) (hΨpos : ∀ r, 0≤Ψ r) : | |
| graphChildMaxND d μ p v Ψ A B hAB (fun i q => ψ i q.2*U i q) (fun j q => φ j q.2*W j q)≤ | |
| B₁^2*B₂^2*graphChildMaxND d μ p v Ψ A B hAB U W := by | |
| apply Finset.sup'_le | |
| intro q hq | |
| obtain ⟨hq₁,hq₂⟩ := Finset.mem_product.mp hq | |
| apply (sourceAngularHND_bounded_factors d μ p v (hU q.1 hq₁) (hW q.2 hq₂) | |
| (hbU q.1 hq₁) (hbW q.2 hq₂) (ψ q.1) (φ q.2) | |
| (hψ q.1 hq₁) (hφ q.2 hq₂) hΨ hΨpos).trans | |
| exact mul_le_mul_of_nonneg_left (Finset.le_sup' | |
| (fun q => sourceAngularHND d μ p v Ψ (U q.1) (W q.2)) hq) (by positivity) | |
| end Attempt4Near | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |