| import OAI.Analysis.FourierLLogL.Fourier.Endpoint |
| /-! # The Luxemburg gauge for the Fourier convergence class -/ |
| noncomputable section |
| open MeasureTheory Filter Set Function |
| open scoped ENNReal BigOperators Topology |
|
|
| namespace OAI |
|
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| namespace FourierLLogL |
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| def priceΦ (x : ℝ) := x*Real.log (2+x) |
| def modBill (f : Torus → ℂ) (b : ℝ) := ∫ x, priceΦ (‖f x‖/b) ∂circleMeasure |
| def luxSet (f : Torus → ℂ) := {b | 0 < b ∧ modBill f b ≤ 1} |
| def luxNorm (f : Torus → ℂ) := sInf (luxSet f) |
|
|
| namespace Weak |
| open Real |
| open scoped Classical |
|
|
| lemma p_nonneg (x : ℝ) (hx : 0 ≤ x) : 0 ≤ priceΦ x := |
| mul_nonneg hx (log_nonneg (by linarith)) |
| lemma p_zero : priceΦ 0=0 := by simp [priceΦ] |
| lemma p_mono {x y : ℝ} (hx : 0 ≤ x) (he : x ≤ y) : |
| priceΦ x ≤ priceΦ y := mul_le_mul he (log_le_log (by linarith) (by linarith)) |
| (log_nonneg (by linarith)) (hx.trans he) |
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|
| private def slopeΦ (x : ℝ) := log (2+x)+x/(2+x) |
| private lemma p_support (x y : ℝ) (hx : 0 ≤ x) (hy : 0 ≤ y) : |
| priceΦ y+slopeΦ y*(x-y) ≤ priceΦ x := by |
| let a := 2+x; let b := 2+y |
| have ha : 0 < a := by unfold a; linarith |
| have hb : 0 < b := by unfold b; linarith |
| have H : (a-b)/a ≤ log a-log b := by |
| have hi := log_le_sub_one_of_pos (div_pos hb ha) |
| rw [log_div hb.ne' ha.ne'] at hi |
| rw [sub_div,div_self ha.ne']; linarith |
| have HH : y/b*(x-y) ≤ x*((a-b)/a) := by |
| have HE : 0 ≤ 2*(x-y)^2/(a*b) := by positivity |
| have HV : x*((a-b)/a)-y/b*(x-y)=2*(x-y)^2/(a*b) := by |
| field_simp; unfold a b; ring |
| linarith |
| change y*log b + (log b + y/b)*(x-y) ≤ x*log a |
| nlinarith |
|
|
| / |
| lemma price_convex {ι : Type*} [Fintype ι] [Nonempty ι] (f : ι → ℂ) : |
| let a : ℝ := Fintype.card ι |
| priceΦ ‖(a⁻¹:ℝ) • (∑ i, f i)‖ ≤ a⁻¹ * ∑ i,priceΦ ‖f i‖ := by |
| intro a |
| have ha : 0 < a := by unfold a; positivity |
| let k := (∑ i, ‖f i‖)/a |
| have hk : 0 ≤ k := by unfold k; positivity |
| apply le_trans (b:=priceΦ k) (p_mono (norm_nonneg _) ?_) |
| · rw [←div_eq_inv_mul] |
| apply (le_div_iff₀ ha).mpr |
| have HH : ∑ i : ι,(priceΦ k+slopeΦ k*(‖f i‖-k)) ≤ _ := |
| Finset.sum_le_sum fun (i : ι) (_ : i∈Finset.univ) => p_support ‖f i‖ k (norm_nonneg _) hk |
| rw [Finset.sum_add_distrib,← Finset.mul_sum,Finset.sum_sub_distrib] at HH |
| simp only [Finset.sum_const,nsmul_eq_mul,Finset.card_univ] at HH |
| have H : ∑ i : ι,‖f i‖ = a*k := by unfold k; field_simp |
| change a*priceΦ k+_ * (_-a*k) ≤ _ at HH |
| rw [H] at HH; linarith |
| rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg (inv_nonneg.mpr ha.le)] |
| apply (mul_le_mul_of_nonneg_left (norm_sum_le _ _) (inv_nonneg.mpr ha.le)).trans |
| exact (div_eq_inv_mul _ _).ge |
|
|
| / |
| lemma scale_admit (a : ℝ) (ha : 0 ≤ a) : |
| ∃ c : ℝ, 0 ≤ c ∧ ∀ x, 0 ≤ x → priceΦ (a*x) ≤ c*priceΦ x := by |
| rcases le_total a 1 with h|h |
| · refine ⟨1,by norm_num,fun x hx => ?_⟩; rw [one_mul] |
| apply p_mono (mul_nonneg ha hx); nlinarith |
| · refine ⟨a*(1+2*log a),by have := log_nonneg h; positivity,fun x hx => ?_⟩ |
| have HH : log (2+a*x) ≤ log a+log (2+x) := by |
| rw [← log_mul (by positivity) (by positivity)] |
| apply log_le_log (by positivity); nlinarith |
| have hi : 1/2 ≤ log (2+x) := |
| le_trans (by linarith [Real.log_two_gt_d9]) (log_le_log (by norm_num : 0 < (2:ℝ)) (by linarith)) |
| have HR : log a ≤ 2*log a * log (2+x) := by nlinarith [log_nonneg h] |
| unfold priceΦ; nlinarith [mul_le_mul_of_nonneg_left (le_trans HH (add_le_add_left HR _)) |
| (mul_nonneg ha hx)] |
|
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| def Young (f : Torus → ℂ) (x : Torus) := priceΦ ‖f x‖ |
| def inside (f : Torus → ℂ) := MemLLogL f ∧ modBill f 1 ≤ 1 |
|
|
| lemma young_meas {f : Torus → ℂ} (h : AEStronglyMeasurable f circleMeasure) : |
| AEStronglyMeasurable (Young f) circleMeasure := by |
| have hp := h.norm |
| unfold Young priceΦ |
| exact hp.mul ((aemeasurable_const.add hp.aemeasurable).log).aestronglyMeasurable |
|
|
| lemma mem_dilate (c : ℂ) {f : Torus → ℂ} (hp : MemLLogL f) : |
| MemLLogL (fun x => c*f x) := by |
| let g (x : Torus) := c*f x |
| have hi : AEStronglyMeasurable g circleMeasure := hp.1.const_mul _ |
| obtain ⟨v,hv,ht⟩ := scale_admit ‖c‖ (norm_nonneg _) |
| refine ⟨hi,?_⟩ |
| change Integrable (Young g) circleMeasure |
| apply ((show Integrable (Young f) _ from hp.2).const_mul v).mono' (young_meas hi) |
| apply ae_of_all |
| intro x |
| change ‖Young g x‖ ≤ _ |
| rw [Real.norm_eq_abs,abs_of_nonneg (show 0 ≤ Young g x from p_nonneg _ (norm_nonneg _))] |
| unfold Young g; rw [norm_mul]; apply ht; exact norm_nonneg _ |
|
|
| lemma mod_unit (f : Torus → ℂ) : |
| modBill f 1 = ∫ x, Young f x ∂circleMeasure := by simp only [modBill,div_one]; rfl |
|
|
| lemma sum_inside {ι : Type*} [Fintype ι] [Nonempty ι] |
| (f : ι → Torus → ℂ) (h : ∀ i, inside (f i)) : |
| let a : ℝ := Fintype.card ι |
| inside (fun x => a⁻¹ • ∑ i, f i x) := by |
| intro a |
| let g (x : Torus) : ℂ := a⁻¹ • ∑ i, f i x |
| let F (x : Torus) := a⁻¹ * ∑ i, Young (f i) x |
| have ha : 0 < a := by unfold a; positivity |
| have he (i) : Integrable (Young (f i)) circleMeasure := (h i).1.2 |
| have HH : AEStronglyMeasurable g circleMeasure := by |
| unfold g |
| have HH := Finset.aestronglyMeasurable_fun_sum Finset.univ (fun (i:ι) _ => (h i).1.1) |
| exact HH.const_smul (a⁻¹) |
| have HI : Integrable F circleMeasure := (integrable_finsetSum _ fun i _ => he i).const_mul _ |
| have HX (x : Torus) : Young g x ≤ F x := price_convex fun i => f i x |
| have HV : Integrable (Young g) circleMeasure := |
| HI.mono' (young_meas HH) (ae_of_all _ fun x => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (show 0 ≤ Young g x from p_nonneg _ (norm_nonneg _))] |
| exact HX x) |
| refine ⟨⟨HH,HV⟩,?_⟩ |
| rw [mod_unit] |
| apply (integral_mono HV HI HX).trans |
| unfold F; rw [integral_const_mul,integral_finsetSum _ (fun i _ => he i)] |
| rw [←div_eq_inv_mul] |
| apply (div_le_one ha).mpr |
| have Hz (i) : ∫ x,Young (f i) x ∂circleMeasure ≤ 1 := by rw [←mod_unit]; exact (h i).2 |
| exact le_trans (Finset.sum_le_sum (fun i _ => Hz i)) (by simp [a]) |
| end Weak |
| end FourierLLogL |
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| end OAI |
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