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d5b562f | 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 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import Mathlib.Analysis.Convolution
import Mathlib.MeasureTheory.Function.L2Space
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Haar.Unique
namespace OAI
/-! # The continuous-frequency Young estimate used by the homogeneous algebra
Only the nonnegative L¹–L² estimate needed for Fourier magnitudes is proved.
Weighted Cauchy–Schwarz gives the pointwise bound; the existing convolution
Fubini theorem integrates its majorant.
-/
open MeasureTheory
open scoped ENNReal Convolution
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
/-- Weighted Cauchy–Schwarz without introducing a probability normalization. -/
theorem continuousWeightedCauchySchwarz (f g : E → ℝ)
(hf : Integrable f) (hg : AEStronglyMeasurable g volume)
(hfp : ∀ x, 0 ≤ f x) (hgp : ∀ x, 0 ≤ g x)
(hfg : Integrable (fun x => f x * g x ^ 2)) :
(∫ x, f x * g x) ^ 2 ≤ (∫ x, f x) * ∫ x, f x * g x ^ 2 := by
let A : E → ℝ := fun x => Real.sqrt (f x)
let B : E → ℝ := fun x => Real.sqrt (f x) * g x
have hAm : AEStronglyMeasurable A volume :=
Real.continuous_sqrt.comp_aestronglyMeasurable hf.aestronglyMeasurable
have hBm : AEStronglyMeasurable B volume := hAm.mul hg
have hAs (x : E) : A x ^ 2 = f x := Real.sq_sqrt (hfp x)
have hBs (x : E) : B x ^ 2 = f x * g x ^ 2 := by
simp only [B, mul_pow, Real.sq_sqrt (hfp x)]
have hA : MemLp A 2 volume := (memLp_two_iff_integrable_sq hAm).mpr (by
simpa only [hAs] using hf)
have hB : MemLp B 2 volume := (memLp_two_iff_integrable_sq hBm).mpr (by
simpa only [hBs] using hfg)
have hAB (x : E) : A x * B x = f x * g x := by
change Real.sqrt (f x) * (Real.sqrt (f x) * g x) = _
rw [← mul_assoc, Real.mul_self_sqrt (hfp x)]
have hholder : (2 : ℝ).HolderConjugate 2 := by rw [Real.holderConjugate_iff]; norm_num
have hh := integral_mul_le_Lp_mul_Lq_of_nonneg hholder
(ae_of_all _ (fun x => Real.sqrt_nonneg (f x)))
(ae_of_all _ (fun x => mul_nonneg (Real.sqrt_nonneg _) (hgp x)))
(by simpa using hA) (by simpa using hB)
change (∫ x, A x * B x) ≤ (∫ x, A x ^ (2 : ℝ)) ^ (1 / 2 : ℝ) *
(∫ x, B x ^ (2 : ℝ)) ^ (1 / 2 : ℝ) at hh
simp only [Real.rpow_two, hAB, hAs, hBs, ← Real.sqrt_eq_rpow] at hh
have hleft : 0 ≤ ∫ x, f x * g x := integral_nonneg (fun x => mul_nonneg (hfp x) (hgp x))
have h1 : 0 ≤ ∫ x, f x := integral_nonneg hfp
have h2 : 0 ≤ ∫ x, f x * g x ^ 2 := integral_nonneg (fun x => mul_nonneg (hfp x) (sq_nonneg _))
have hs := (sq_le_sq₀ hleft (mul_nonneg (Real.sqrt_nonneg _) (Real.sqrt_nonneg _))).mpr hh
simpa only [mul_pow, Real.sq_sqrt h1, Real.sq_sqrt h2] using hs
noncomputable def positiveFrequencyConvolution (f g : E → ℝ) (ξ : E) : ℝ :=
∫ η, f η * g (ξ - η)
theorem positiveFrequencyConvolution_comm (f g : E → ℝ) (ξ : E) :
positiveFrequencyConvolution f g ξ = positiveFrequencyConvolution g f ξ := by
unfold positiveFrequencyConvolution
rw [← integral_sub_left_eq_self (fun η => f η * g (ξ - η)) volume ξ]
simp only [sub_sub_self, mul_comm]
/-- The exact L¹–L² Young bound, in its squared form. -/
theorem continuousYoung_nonnegative (f g : E → ℝ)
(hf : Integrable f) (hg : Continuous g)
(hfp : ∀ x, 0 ≤ f x) (hgp : ∀ x, 0 ≤ g x)
(hg2 : Integrable (fun x => g x ^ 2))
(K : ℝ) (_hK : 0 ≤ K) (hgb : ∀ x, g x ≤ K) :
Integrable (fun ξ => positiveFrequencyConvolution f g ξ ^ 2) ∧
(∫ ξ, positiveFrequencyConvolution f g ξ ^ 2) ≤
(∫ ξ, f ξ) ^ 2 * ∫ ξ, g ξ ^ 2 := by
let L := ContinuousLinearMap.mul ℝ ℝ
have hconv : Integrable (fun ξ => ∫ η, f η * g (ξ - η) ^ 2) := by
exact hf.integrable_convolution L hg2
have hmajor : Integrable (fun ξ => (∫ x, f x) * ∫ η, f η * g (ξ - η) ^ 2) :=
hconv.const_mul _
have hpoint (ξ : E) : positiveFrequencyConvolution f g ξ ^ 2 ≤
(∫ x, f x) * ∫ η, f η * g (ξ - η) ^ 2 := by
have hfg : Integrable (fun η => f η * g (ξ - η) ^ 2) := by
apply (hf.mul_const (K ^ 2)).mono'
(hf.aestronglyMeasurable.mul ((hg.comp (continuous_const.sub continuous_id)).pow 2).aestronglyMeasurable)
filter_upwards [] with η
change ‖f η * g (ξ - η) ^ 2‖ ≤ f η * K ^ 2
rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (hfp η) (sq_nonneg _))]
exact mul_le_mul_of_nonneg_left
(pow_le_pow_left₀ (hgp _) (hgb _) 2) (hfp η)
exact continuousWeightedCauchySchwarz f (fun η => g (ξ - η)) hf
(hg.comp (continuous_const.sub continuous_id)).aestronglyMeasurable hfp
(fun η => hgp (ξ - η)) hfg
have hmeas : AEStronglyMeasurable (positiveFrequencyConvolution f g) volume :=
hf.aestronglyMeasurable.convolution L hg.aestronglyMeasurable
have hint : Integrable (fun ξ => positiveFrequencyConvolution f g ξ ^ 2) := by
apply hmajor.mono' (hmeas.pow 2)
filter_upwards [] with ξ
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
exact hpoint ξ
refine ⟨hint, (integral_mono hint hmajor hpoint).trans_eq ?_⟩
rw [integral_const_mul]
have he := integral_convolution L hf hg2
change (∫ ξ, ∫ η, f η * g (ξ - η) ^ 2) = (∫ ξ, f ξ) * ∫ ξ, g ξ ^ 2 at he
rw [he]
ring
end DefocusingNLS
end OAI
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