| 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) |
|
|
| / |
| 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] |
|
|
| / |
| 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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|
