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b224933 | 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 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | 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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