import Mathlib namespace OAI /-! Fourier extension from the unit sphere in the fourth-power range. -/ noncomputable section open MeasureTheory Set open scoped ENNReal NNReal namespace EllipticCapacity abbrev Space := EuclideanSpace ℝ (Fin 3) abbrev Sphere := {ω : Space // ω ∈ Metric.sphere (0 : Space) 1} def surfaceArea : Measure Sphere := (volume : Measure Space).toSphere def extension (g : Sphere → ℂ) (x : Space) : ℂ := ∫ ω, g ω * Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (inner ℝ x (ω : Space) : ℂ)) ∂surfaceArea def SphereRestriction : Prop := ∀ p : ℝ, 3 < p → ∃ C : ℝ≥0, ∀ g : Sphere → ℂ, Measurable g → (∃ M : ℝ, ∀ ω, ‖g ω‖ ≤ M) → eLpNorm (extension g) (ENNReal.ofReal p) volume ≤ (C : ℝ≥0∞) * eLpNorm g ∞ surfaceArea theorem surfaceArea_univ : surfaceArea Set.univ = ENNReal.ofReal (4 * Real.pi) := by unfold surfaceArea rw [Measure.toSphere_apply_univ] simp only [Space, finrank_euclideanSpace, Fintype.card_fin, EuclideanSpace.volume_ball_fin_three, ENNReal.ofReal_one, one_pow, one_mul] rw [← ENNReal.ofReal_natCast] norm_num only [Nat.cast_ofNat] rw [← ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 3)] congr 1 ring theorem kernel_norm (x : Space) (ω : Sphere) : ‖Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (inner ℝ x (ω : Space) : ℂ))‖ = 1 := by rw [Complex.norm_exp] simp instance surfaceArea_isFiniteMeasure : IsFiniteMeasure surfaceArea := by constructor rw [surfaceArea_univ] exact ENNReal.ofReal_lt_top theorem extension_integrable {g : Sphere → ℂ} (hg : Measurable g) {M : ℝ} (hM : ∀ ω, ‖g ω‖ ≤ M) (x : Space) : Integrable (fun ω => g ω * Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (inner ℝ x (ω : Space) : ℂ))) surfaceArea := by have hm : Measurable (fun ω : Sphere => g ω * Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (inner ℝ x (ω : Space) : ℂ))) := by fun_prop apply Integrable.of_bound hm.aestronglyMeasurable M filter_upwards [] with ω simpa only [norm_mul, kernel_norm, mul_one] using hM ω theorem extension_norm_le {g : Sphere → ℂ} {M : ℝ} (hM : ∀ ω, ‖g ω‖ ≤ M) (x : Space) : ‖extension g x‖ ≤ M * (4 * Real.pi) := by unfold extension have hmass : surfaceArea.real Set.univ = 4 * Real.pi := by rw [Measure.real, surfaceArea_univ, ENNReal.toReal_ofReal] positivity rw [← hmass] apply norm_integral_le_of_norm_le_const filter_upwards [] with ω simpa only [norm_mul, kernel_norm, mul_one] using hM ω theorem extension_continuous {g : Sphere → ℂ} (hg : Measurable g) {M : ℝ} (hM : ∀ ω, ‖g ω‖ ≤ M) : Continuous (extension g) := by apply continuous_of_dominated (bound := fun _ : Sphere => M) (fun x => (extension_integrable hg hM x).aestronglyMeasurable) · intro x filter_upwards [] with ω simpa only [norm_mul, kernel_norm, mul_one] using hM ω · exact integrable_const M · filter_upwards [] with ω fun_prop theorem norm_exp_mul_I_sub_le (a b : ℝ) : ‖Complex.exp ((a : ℂ) * Complex.I) - Complex.exp ((b : ℂ) * Complex.I)‖ ≤ |a - b| := by have hd : ∀ x : ℝ, HasDerivAt (fun y : ℝ => Complex.exp ((y : ℂ) * Complex.I)) (Complex.exp ((x : ℂ) * Complex.I) * Complex.I) x := by intro x simpa using (((hasDerivAt_id (x : ℂ)).comp_ofReal).mul_const Complex.I).cexp have hb : ∀ x : ℝ, ‖Complex.exp ((x : ℂ) * Complex.I) * Complex.I‖ ≤ 1 := by intro x simp [Complex.norm_exp] simpa using Convex.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x (_ : x ∈ (Set.univ : Set ℝ)) => (hd x).hasDerivWithinAt) (fun x _ => hb x) (convex_univ : Convex ℝ (Set.univ : Set ℝ)) (Set.mem_univ b) (Set.mem_univ a) theorem kernel_norm_sub_le (x y : Space) (ω : Sphere) : ‖Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (inner ℝ x (ω : Space) : ℂ)) - Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (inner ℝ y (ω : Space) : ℂ))‖ ≤ 2 * Real.pi * ‖x - y‖ := by have h := norm_exp_mul_I_sub_le (2 * Real.pi * inner ℝ x (ω : Space)) (2 * Real.pi * inner ℝ y (ω : Space)) have hω : ‖(ω : Space)‖ = 1 := mem_sphere_zero_iff_norm.mp ω.property have hi : |inner ℝ (x - y) (ω : Space)| ≤ ‖x - y‖ := by simpa only [Real.norm_eq_abs, hω, mul_one] using norm_inner_le_norm (𝕜 := ℝ) (x - y) (ω : Space) have he : ∀ a : ℝ, (2 * (Real.pi : ℂ) * Complex.I * (a : ℂ)) = ((2 * Real.pi * a : ℝ) : ℂ) * Complex.I := by intro a push_cast ring simp only [he] at ⊢ refine h.trans ?_ rw [← mul_sub, abs_mul, abs_of_pos (mul_pos (by norm_num) Real.pi_pos), ← inner_sub_left] exact mul_le_mul_of_nonneg_left hi (by positivity) theorem extension_norm_sub_le {g : Sphere → ℂ} (hg : Measurable g) {M : ℝ} (hM0 : 0 ≤ M) (hM : ∀ ω, ‖g ω‖ ≤ M) (x y : Space) : ‖extension g x - extension g y‖ ≤ (8 * Real.pi ^ 2 * M) * ‖x - y‖ := by unfold extension rw [← integral_sub (extension_integrable hg hM x) (extension_integrable hg hM y)] have hmass : surfaceArea.real Set.univ = 4 * Real.pi := by rw [Measure.real, surfaceArea_univ, ENNReal.toReal_ofReal] positivity have hbound := norm_integral_le_of_norm_le_const (μ := surfaceArea) (C := M * (2 * Real.pi * ‖x - y‖)) (f := fun ω : Sphere => g ω * Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (inner ℝ x (ω : Space) : ℂ)) - g ω * Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (inner ℝ y (ω : Space) : ℂ))) (by filter_upwards [] with ω rw [← mul_sub, norm_mul] exact mul_le_mul (hM ω) (kernel_norm_sub_le x y ω) (norm_nonneg _) hM0) rw [hmass] at hbound convert hbound using 1 ring open Filter def SphereFourthPower : Prop := ∀ g : Sphere → ℂ, Measurable g → (∃ M : ℝ, ∀ ω, ‖g ω‖ ≤ M) → (eLpNorm (extension g) 4 volume) ^ 4 ≤ ENNReal.ofReal (32 * Real.pi ^ 3) * (eLpNorm g ∞ surfaceArea) ^ 4 def character (x v : Space) : ℂ := Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (inner ℝ x v : ℂ)) lemma character_norm (x v : Space) : ‖character x v‖ = 1 := by simp [character, Complex.norm_exp] lemma character_add (x v w : Space) : character x (v + w) = character x v * character x w := by simp [character, inner_add_right, mul_add, Complex.exp_add] lemma character_neg (x v : Space) : character x (-v) = star (character x v) := by simp [character, inner_neg_right, ← Complex.exp_conj] rw [show (starRingEnd ℂ) (2 : ℂ) = 2 by change star (2 : ℂ) = _; simp] abbrev Quad := (Sphere × Sphere) × (Sphere × Sphere) def quadMeasure : Measure Quad := (surfaceArea.prod surfaceArea).prod (surfaceArea.prod surfaceArea) instance : IsFiniteMeasure quadMeasure := inferInstanceAs (IsFiniteMeasure ((surfaceArea.prod surfaceArea).prod (surfaceArea.prod surfaceArea))) def quadVector (q : Quad) : Space := (q.1.1 : Space) + q.1.2 - ((q.2.1 : Space) + q.2.2) def quadAmplitude (g : Sphere → ℂ) (q : Quad) : ℂ := g q.1.1 * g q.1.2 * star (g q.2.1 * g q.2.2) lemma quad_character (x : Space) (q : Quad) : character x (quadVector q) = character x q.1.1 * character x q.1.2 * star (character x q.2.1 * character x q.2.2) := by simp only [quadVector, sub_eq_add_neg, character_add, character_neg, star_mul] lemma fourth_power_expansion {g : Sphere → ℂ} (hg : Measurable g) {M : ℝ} (hM : ∀ ω, ‖g ω‖ ≤ M) (x : Space) : (‖extension g x‖ ^ 4 : ℝ) = (∫ q, quadAmplitude g q * character x (quadVector q) ∂quadMeasure : ℂ) := by let f : Sphere → ℂ := fun ω => g ω * character x ω have hf : Integrable f surfaceArea := extension_integrable hg hM x have hc : (∫ q : Sphere × Sphere, star (f q.1 * f q.2) ∂surfaceArea.prod surfaceArea) = star (extension g x ^ 2) := by change (∫ q : Sphere × Sphere, (starRingEnd ℂ) (f q.1 * f q.2) ∂surfaceArea.prod surfaceArea) = _ rw [integral_conj, integral_prod_mul] simp only [pow_two] rfl have hp := integral_prod_mul (μ := surfaceArea.prod surfaceArea) (ν := surfaceArea.prod surfaceArea) (fun q : Sphere × Sphere => f q.1 * f q.2) (fun q : Sphere × Sphere => star (f q.1 * f q.2)) have hq : (∫ q, quadAmplitude g q * character x (quadVector q) ∂quadMeasure) = extension g x ^ 2 * star (extension g x ^ 2) := by calc _ = ∫ q : Quad, f q.1.1 * f q.1.2 * star (f q.2.1 * f q.2.2) ∂quadMeasure := by apply integral_congr_ae filter_upwards [] with q simp only [quadAmplitude, quad_character, f, star_mul] ring _ = _ := by rw [quadMeasure, hp, hc, integral_prod_mul] simp only [pow_two] rfl rw [hq] change ((‖extension g x‖ ^ 4 : ℝ) : ℂ) = extension g x ^ 2 * (starRingEnd ℂ) (extension g x ^ 2) rw [RCLike.mul_conj] change (↑(‖extension g x‖ ^ 4) : ℂ) = (↑(‖extension g x ^ 2‖) : ℂ) ^ 2 rw [norm_pow, ← Complex.ofReal_pow, ← pow_mul] lemma quadAmplitude_bound {g : Sphere → ℂ} {M : ℝ} (hM0 : 0 ≤ M) (hM : ∀ ω, ‖g ω‖ ≤ M) (q : Quad) : ‖quadAmplitude g q‖ ≤ M ^ 4 := by unfold quadAmplitude simp only [norm_mul, norm_star] calc _ ≤ (M * M) * (M * M) := by gcongr <;> apply hM _ = _ := by ring def gaussian (b : ℝ) (x : Space) : ℝ := Real.exp (-b * ‖x‖ ^ 2) def heatKernel (b : ℝ) (v : Space) : ℝ := (Real.pi / b) ^ (3 / 2 : ℝ) * Real.exp (-Real.pi ^ 2 * ‖v‖ ^ 2 / b) lemma gaussian_pos (b : ℝ) (x : Space) : 0 < gaussian b x := Real.exp_pos _ lemma heatKernel_nonneg {b : ℝ} (hb : 0 < b) (v : Space) : 0 ≤ heatKernel b v := by unfold heatKernel positivity lemma gaussian_integrable {b : ℝ} (hb : 0 < b) : Integrable (gaussian b) := by have h := (GaussianFourier.integrable_cexp_neg_mul_sq_norm_add (V := Space) (b := (b : ℂ)) (by simpa using hb) 0 0).norm change Integrable (fun x : Space => Real.exp (-b * ‖x‖ ^ 2)) simpa [Complex.norm_exp, neg_mul, ← Complex.ofReal_pow] using h lemma gaussian_character_integral {b : ℝ} (hb : 0 < b) (v : Space) : (∫ x, (gaussian b x : ℂ) * character x v) = (heatKernel b v : ℂ) := by have heq : (fun x : Space => (gaussian b x : ℂ) * character x v) = (fun x : Space => Complex.exp (-(b : ℂ) * ‖x‖ ^ 2 + 2 * Real.pi * Complex.I * (inner ℝ v x : ℂ))) := by ext x rw [gaussian, character, Complex.ofReal_exp, ← Complex.exp_add, real_inner_comm x v] push_cast rfl rw [heq, GaussianFourier.integral_cexp_neg_mul_sq_norm_add (by simpa using hb)] simp only [Space, finrank_euclideanSpace, Fintype.card_fin] unfold heatKernel rw [Complex.ofReal_mul, Complex.ofReal_cpow (div_nonneg Real.pi_pos.le hb.le), Complex.ofReal_exp] push_cast congr 1 congr 1 rw [show (2 * (Real.pi : ℂ) * Complex.I) ^ 2 = -4 * (Real.pi : ℂ) ^ 2 by ring_nf; simp [Complex.I_sq]] ring lemma gaussian_quad_integrable {b M : ℝ} (hb : 0 < b) (hM0 : 0 ≤ M) {g : Sphere → ℂ} (hg : Measurable g) (hM : ∀ ω, ‖g ω‖ ≤ M) : Integrable (fun p : Space × Quad => (gaussian b p.1 : ℂ) * (quadAmplitude g p.2 * character p.1 (quadVector p.2))) (volume.prod quadMeasure) := by apply ((gaussian_integrable hb).mul_prod (integrable_const (M ^ 4) : Integrable (fun _ : Quad => M ^ 4) quadMeasure)).mono' · unfold gaussian quadAmplitude character quadVector fun_prop · filter_upwards [] with p simp only [norm_mul, character_norm, mul_one, Complex.norm_real, Real.norm_eq_abs, abs_of_pos (gaussian_pos _ _)] exact mul_le_mul_of_nonneg_left (quadAmplitude_bound hM0 hM _) (gaussian_pos _ _).le lemma gaussian_fourth_identity {b M : ℝ} (hb : 0 < b) (hM0 : 0 ≤ M) {g : Sphere → ℂ} (hg : Measurable g) (hM : ∀ ω, ‖g ω‖ ≤ M) : ((∫ x, gaussian b x * ‖extension g x‖ ^ 4 : ℝ) : ℂ) = ∫ q, quadAmplitude g q * (heatKernel b (quadVector q) : ℂ) ∂quadMeasure := by rw [← integral_complex_ofReal] simp only [Complex.ofReal_mul] simp_rw [fourth_power_expansion hg hM, ← integral_const_mul] rw [integral_integral_swap (gaussian_quad_integrable hb hM0 hg hM)] apply integral_congr_ae filter_upwards [] with q rw [show (fun x : Space => (gaussian b x : ℂ) * (quadAmplitude g q * character x (quadVector q))) = (fun x : Space => quadAmplitude g q * ((gaussian b x : ℂ) * character x (quadVector q))) by funext x; ring] rw [integral_const_mul, gaussian_character_integral hb] lemma heatKernel_integrable {b : ℝ} (hb : 0 < b) : Integrable (fun q : Quad => heatKernel b (quadVector q)) quadMeasure := by apply Integrable.of_bound (C := (Real.pi / b) ^ (3 / 2 : ℝ)) · unfold heatKernel quadVector fun_prop · filter_upwards [] with q rw [Real.norm_eq_abs, abs_of_nonneg (heatKernel_nonneg hb _)] unfold heatKernel apply mul_le_of_le_one_right (Real.rpow_nonneg (by positivity) _) apply Real.exp_le_one_iff.mpr apply div_nonpos_of_nonpos_of_nonneg _ hb.le exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (sq_nonneg _)) (sq_nonneg _) lemma gaussian_fourth_integrable {b M : ℝ} (hb : 0 < b) (_hM0 : 0 ≤ M) {g : Sphere → ℂ} (hg : Measurable g) (hM : ∀ ω, ‖g ω‖ ≤ M) : Integrable (fun x => gaussian b x * ‖extension g x‖ ^ 4) := by apply (gaussian_integrable hb).mul_bdd (c := (M * (4 * Real.pi)) ^ 4) ((extension_continuous hg hM).norm.pow 4).aestronglyMeasurable filter_upwards [] with x simp only [Pi.pow_apply, Real.norm_eq_abs, abs_of_nonneg (pow_nonneg (norm_nonneg _) 4)] exact pow_le_pow_left₀ (norm_nonneg _) (extension_norm_le hM x) 4 lemma gaussian_fourth_bound {b M : ℝ} (hb : 0 < b) (hM0 : 0 ≤ M) {g : Sphere → ℂ} (hg : Measurable g) (hM : ∀ ω, ‖g ω‖ ≤ M) : (∫ x, gaussian b x * ‖extension g x‖ ^ 4) ≤ M ^ 4 * ∫ x, gaussian b x * ‖extension (fun _ => 1) x‖ ^ 4 := by have hn (g : Sphere → ℂ) : 0 ≤ ∫ x, gaussian b x * ‖extension g x‖ ^ 4 := integral_nonneg (fun x => mul_nonneg (gaussian_pos _ _).le (by positivity)) have h1 := gaussian_fourth_identity (M := 1) hb (by norm_num) (g := fun _ => 1) measurable_const (fun _ => by simp) have h1' : (∫ x, gaussian b x * ‖extension (fun _ => 1) x‖ ^ 4) = ∫ q, heatKernel b (quadVector q) ∂quadMeasure := by have : ((∫ x, gaussian b x * ‖extension (fun _ => 1) x‖ ^ 4 : ℝ) : ℂ) = ((∫ q, heatKernel b (quadVector q) ∂quadMeasure : ℝ) : ℂ) := by simpa [quadAmplitude, integral_complex_ofReal] using h1 exact_mod_cast this rw [h1'] have hi := gaussian_fourth_identity hb hM0 hg hM calc _ = ‖((∫ x, gaussian b x * ‖extension g x‖ ^ 4 : ℝ) : ℂ)‖ := by rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (hn g)] _ = ‖∫ q, quadAmplitude g q * (heatKernel b (quadVector q) : ℂ) ∂quadMeasure‖ := by rw [hi] _ ≤ ∫ q, ‖quadAmplitude g q * (heatKernel b (quadVector q) : ℂ)‖ ∂quadMeasure := norm_integral_le_integral_norm _ _ ≤ ∫ q, M ^ 4 * heatKernel b (quadVector q) ∂quadMeasure := by apply integral_mono_of_nonneg · exact Eventually.of_forall (fun q => norm_nonneg _) · exact (heatKernel_integrable hb).const_mul (M ^ 4) · filter_upwards [] with q rw [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (heatKernel_nonneg hb _)] exact mul_le_mul_of_nonneg_right (quadAmplitude_bound hM0 hM _) (heatKernel_nonneg hb _) _ = _ := integral_const_mul _ _ lemma gaussian_le_one {b : ℝ} (hb : 0 ≤ b) (x : Space) : gaussian b x ≤ 1 := by unfold gaussian exact Real.exp_le_one_iff.mpr (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hb) (sq_nonneg _)) lemma lintegral_gaussian_fourth_bound {b M : ℝ} (hb : 0 < b) (hM0 : 0 ≤ M) {g : Sphere → ℂ} (hg : Measurable g) (hM : ∀ ω, ‖g ω‖ ≤ M) : (∫⁻ x, ENNReal.ofReal (gaussian b x * ‖extension g x‖ ^ 4)) ≤ ENNReal.ofReal (M ^ 4) * ∫⁻ x, ENNReal.ofReal (‖extension (fun _ => 1) x‖ ^ 4) := by rw [← ofReal_integral_eq_lintegral_ofReal (gaussian_fourth_integrable hb hM0 hg hM) (Filter.Eventually.of_forall (fun x => mul_nonneg (gaussian_pos b x).le (by positivity)))] refine (ENNReal.ofReal_le_ofReal (gaussian_fourth_bound hb hM0 hg hM)).trans ?_ rw [ENNReal.ofReal_mul (by positivity)] gcongr rw [ofReal_integral_eq_lintegral_ofReal (gaussian_fourth_integrable (M := 1) hb (by norm_num) measurable_const (fun _ => by simp)) (Filter.Eventually.of_forall (fun x => mul_nonneg (gaussian_pos b x).le (by positivity)))] apply lintegral_mono intro x apply ENNReal.ofReal_le_ofReal exact mul_le_of_le_one_left (by positivity) (gaussian_le_one hb.le x) lemma fourth_power_majorant {M : ℝ} (hM0 : 0 ≤ M) {g : Sphere → ℂ} (hg : Measurable g) (hM : ∀ ω, ‖g ω‖ ≤ M) : (∫⁻ x, ENNReal.ofReal (‖extension g x‖ ^ 4)) ≤ ENNReal.ofReal (M ^ 4) * ∫⁻ x, ENNReal.ofReal (‖extension (fun _ => 1) x‖ ^ 4) := by let F : ℕ → Space → ℝ≥0∞ := fun n x => ENNReal.ofReal (gaussian ((1 / 2 : ℝ) ^ n) x * ‖extension g x‖ ^ 4) have hm : ∀ n, Measurable (F n) := by intro n dsimp [F] have hc := extension_continuous hg hM unfold gaussian fun_prop have ht : ∀ x, Filter.Tendsto (fun n => F n x) Filter.atTop (nhds (ENNReal.ofReal (‖extension g x‖ ^ 4))) := by intro x have hp := tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2) (by norm_num : (1 / 2 : ℝ) < 1) have he := Real.continuous_exp.continuousAt.tendsto.comp (hp.neg.mul_const (‖x‖ ^ 2)) have h := ENNReal.continuous_ofReal.continuousAt.tendsto.comp (he.mul_const (‖extension g x‖ ^ 4)) simpa [F, gaussian, Function.comp_def] using h have hfat := lintegral_liminf_le (μ := (volume : Measure Space)) (u := Filter.atTop) hm simp_rw [(ht _).liminf_eq] at hfat apply hfat.trans apply Filter.liminf_le_of_frequently_le apply Filter.Eventually.frequently apply Filter.Eventually.of_forall intro n exact lintegral_gaussian_fourth_bound (by positivity) hM0 hg hM all_goals isBoundedDefault lemma integral_polar_three (f : Space → ℂ) (hf : Integrable f) : ∫ y, f y = ∫ r in Set.Ioi (0 : ℝ), r ^ 2 • ∫ ω : Sphere, f (r • (ω : Space)) ∂surfaceArea := by let h := homeomorphUnitSphereProd Space let μ := (volume : Measure Space) let g : Sphere × Set.Ioi (0 : ℝ) → ℂ := fun p => f (p.2.val • (p.1 : Space)) have heq : g ∘ h = f ∘ Subtype.val := by funext y dsimp [g] rw [← homeomorphUnitSphereProd_symm_apply_coe Space] simp [h] have hc : Integrable (f ∘ Subtype.val) (μ.comap (Subtype.val : ({0}ᶜ : Set Space) → Space)) := (integrableOn_iff_comap_subtypeVal (measurableSet_singleton (0 : Space)).compl).1 hf.integrableOn have hg : Integrable g (surfaceArea.prod (Measure.volumeIoiPow 2)) := by have hm := μ.measurePreserving_homeomorphUnitSphereProd.integrable_comp_emb (Homeomorph.measurableEmbedding h) (g := g) rw [heq] at hm simpa [Space, surfaceArea, μ] using hm.mp hc calc ∫ y, f y = ∫ y : ({0}ᶜ : Set Space), f y.val ∂μ.comap Subtype.val := by rw [integral_subtype_comap (measurableSet_singleton (0 : Space)).compl, restrict_compl_singleton] _ = ∫ p, g p ∂surfaceArea.prod (Measure.volumeIoiPow 2) := by have hi := μ.measurePreserving_homeomorphUnitSphereProd.integral_comp (Homeomorph.measurableEmbedding h) g change ∫ y, (g ∘ h) y ∂μ.comap Subtype.val = _ at hi rw [heq] at hi simpa [Space, surfaceArea, μ] using hi _ = ∫ r : Set.Ioi (0 : ℝ), ∫ ω : Sphere, f (r.val • (ω : Space)) ∂surfaceArea ∂Measure.volumeIoiPow 2 := integral_prod_symm g hg _ = ∫ r in Set.Ioi (0 : ℝ), r ^ 2 • ∫ ω : Sphere, f (r • (ω : Space)) ∂surfaceArea := by simp only [Measure.volumeIoiPow, ENNReal.ofReal] rw [integral_withDensity_eq_integral_smul] · rw [integral_subtype_comap measurableSet_Ioi (fun r : ℝ => Real.toNNReal (r ^ 2) • ∫ ω : Sphere, f (r • (ω : Space)) ∂surfaceArea)] apply setIntegral_congr_fun measurableSet_Ioi intro r hr dsimp only rw [NNReal.smul_def, Real.coe_toNNReal _ (sq_nonneg r)] · exact (measurable_subtype_coe.pow_const 2).real_toNNReal abbrev CrossPlane := EuclideanSpace ℝ (Fin 2) def sphereSplit : Space ≃ᵐ ℝ × CrossPlane := (((MeasurableEquiv.toLp 2 (Fin 3 → ℝ)).symm.trans (MeasurableEquiv.piFinSuccAbove (fun _ : Fin 3 => ℝ) 0)).trans ((MeasurableEquiv.refl ℝ).prodCongr (MeasurableEquiv.toLp 2 (Fin 2 → ℝ)))) lemma sphereSplit_measurePreserving : MeasurePreserving sphereSplit := by have h1 := EuclideanSpace.volume_preserving_symm_measurableEquiv_toLp (Fin 3) have h2 := volume_preserving_piFinSuccAbove (fun _ : Fin 3 => ℝ) 0 have h3 : MeasurePreserving ((MeasurableEquiv.refl ℝ).prodCongr (MeasurableEquiv.toLp 2 (Fin 2 → ℝ))) := by change MeasurePreserving (Prod.map id (WithLp.toLp 2)) _ _ simpa only [Measure.volume_eq_prod] using (MeasurePreserving.id (volume : Measure ℝ)).prod (PiLp.volume_preserving_toLp (Fin 2)) exact h3.comp (h2.comp h1) lemma sphereSplit_symm_apply (u : ℝ) (v : CrossPlane) : sphereSplit.symm (u, v) = WithLp.toLp 2 (Fin.cons u (WithLp.ofLp v)) := by ext i fin_cases i <;> rfl lemma sphereSplit_norm_sq (u : ℝ) (v : CrossPlane) : ‖sphereSplit.symm (u, v)‖ ^ 2 = u ^ 2 + ‖v‖ ^ 2 := by rw [sphereSplit_symm_apply, EuclideanSpace.real_norm_sq_eq, EuclideanSpace.real_norm_sq_eq] simp [Fin.sum_univ_succ] def sphereKernel (x y : Space) : ℂ := Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (inner ℝ x y : ℂ)) def ballFourier (x : Space) (R : ℝ) : ℂ := ∫ y in Metric.ball (0 : Space) R, sphereKernel x y lemma ballFourier_polar (x : Space) (R : ℝ) (hR : 0 ≤ R) : ballFourier x R = ∫ r in (0 : ℝ)..R, r ^ 2 • extension (fun _ => 1) (r • x) := by have hcont : Continuous (sphereKernel x) := by unfold sphereKernel; fun_prop have hint : Integrable ((Metric.ball (0 : Space) R).indicator (sphereKernel x)) := (integrable_indicator_iff measurableSet_ball).2 ((ContinuousOn.integrableOn_compact (isCompact_closedBall (0 : Space) R) hcont.continuousOn).mono_set Metric.ball_subset_closedBall) calc ballFourier x R = ∫ y, (Metric.ball (0 : Space) R).indicator (sphereKernel x) y := by rw [integral_indicator measurableSet_ball]; rfl _ = ∫ r in Set.Ioi (0 : ℝ), r ^ 2 • ∫ ω : Sphere, (Metric.ball (0 : Space) R).indicator (sphereKernel x) (r • (ω : Space)) ∂surfaceArea := integral_polar_three _ hint _ = ∫ r in Set.Ioi (0 : ℝ), (Set.Iio R).indicator (fun r => r ^ 2 • extension (fun _ => 1) (r • x)) r := by apply setIntegral_congr_fun measurableSet_Ioi intro r hr dsimp only have hr0 : 0 < r := hr have hn (ω : Sphere) : ‖r • (ω : Space)‖ = r := by rw [norm_smul, Real.norm_of_nonneg hr0.le, mem_sphere_zero_iff_norm.mp ω.property, mul_one] by_cases hrR : r < R · rw [Set.indicator_of_mem (show r ∈ Set.Iio R from hrR)] congr 1 unfold extension apply integral_congr_ae filter_upwards [] with ω rw [Set.indicator_of_mem (show r • (ω : Space) ∈ Metric.ball (0 : Space) R by simpa only [mem_ball_zero_iff, hn] using hrR)] simp [sphereKernel, inner_smul_right, real_inner_smul_left] · rw [Set.indicator_of_notMem (show r ∉ Set.Iio R from hrR)] have he : ∀ ω : Sphere, (Metric.ball (0 : Space) R).indicator (sphereKernel x) (r • (ω : Space)) = 0 := by intro ω apply Set.indicator_of_notMem simpa only [mem_ball_zero_iff, hn] using hrR simp only [he, integral_zero, smul_zero] _ = ∫ r in (0 : ℝ)..R, r ^ 2 • extension (fun _ => 1) (r • x) := by rw [setIntegral_indicator measurableSet_Iio, intervalIntegral.integral_of_le hR, integral_Ioc_eq_integral_Ioo] congr 1 lemma sphereSplit_symm_first (u : ℝ) (v : CrossPlane) : (sphereSplit.symm (u, v)) 0 = u := by rw [sphereSplit_symm_apply] rfl lemma sphereSplit_ball_slice (R : ℝ) (hR : 0 ≤ R) (u : ℝ) : {v : CrossPlane | sphereSplit.symm (u, v) ∈ Metric.ball (0 : Space) R} = Metric.ball (0 : CrossPlane) (Real.sqrt (R ^ 2 - u ^ 2)) := by ext v simp only [Set.mem_ofPred_eq, mem_ball_zero_iff, Real.lt_sqrt (norm_nonneg v)] have hn := sphereSplit_norm_sq u v constructor <;> intro hv · nlinarith [norm_nonneg (sphereSplit.symm (u, v))] · nlinarith [norm_nonneg (sphereSplit.symm (u, v))] lemma plane_slice_volume (R u : ℝ) (hR : 0 ≤ R) : (volume : Measure CrossPlane).real (Metric.ball 0 (Real.sqrt (R ^ 2 - u ^ 2))) = (Set.Ioo (-R) R).indicator (fun u => Real.pi * (R ^ 2 - u ^ 2)) u := by rw [Measure.real, EuclideanSpace.volume_ball_fin_two, ENNReal.toReal_mul, ENNReal.toReal_pow, ENNReal.toReal_ofReal (Real.sqrt_nonneg _), ENNReal.toReal_ofReal Real.pi_pos.le] by_cases hu : u ∈ Set.Ioo (-R) R · rw [Set.indicator_of_mem hu, Real.sq_sqrt] · ring · have := hu.1 have := hu.2 nlinarith · rw [Set.indicator_of_notMem hu, Real.sqrt_eq_zero_of_nonpos] · ring · simp only [mem_Ioo, not_and_or, not_lt] at hu rcases hu with h | h <;> nlinarith lemma integral_ball_head (f : ℝ → ℂ) (hf : Continuous f) (R : ℝ) (hR : 0 ≤ R) : ∫ y in Metric.ball (0 : Space) R, f (y 0) = ∫ u in (-R)..R, (Real.pi * (R ^ 2 - u ^ 2)) • f u := by let g : Space → ℂ := (Metric.ball (0 : Space) R).indicator (fun y => f (y 0)) have hc : Continuous (fun y : Space => f (y 0)) := by fun_prop have hg : Integrable g := (integrable_indicator_iff measurableSet_ball).2 ((ContinuousOn.integrableOn_compact (isCompact_closedBall (0 : Space) R) hc.continuousOn).mono_set Metric.ball_subset_closedBall) have hg' : Integrable (g ∘ sphereSplit.symm) ((volume : Measure ℝ).prod (volume : Measure CrossPlane)) := by have h := (sphereSplit_measurePreserving.symm.integrable_comp_emb sphereSplit.symm.measurableEmbedding).2 hg simpa only [Measure.volume_eq_prod] using h have hinner (u : ℝ) : ∫ v : CrossPlane, g (sphereSplit.symm (u, v)) = (Set.Ioo (-R) R).indicator (fun u => Real.pi * (R ^ 2 - u ^ 2)) u • f u := by have he : (fun v : CrossPlane => g (sphereSplit.symm (u, v))) = (Metric.ball (0 : CrossPlane) (Real.sqrt (R ^ 2 - u ^ 2))).indicator (fun _ => f u) := by funext v have ht : sphereSplit.symm (u, v) ∈ Metric.ball (0 : Space) R ↔ v ∈ Metric.ball (0 : CrossPlane) (Real.sqrt (R ^ 2 - u ^ 2)) := by change v ∈ {v : CrossPlane | sphereSplit.symm (u, v) ∈ Metric.ball (0 : Space) R} ↔ _ rw [sphereSplit_ball_slice R hR u] by_cases hv : v ∈ Metric.ball (0 : CrossPlane) (Real.sqrt (R ^ 2 - u ^ 2)) · rw [Set.indicator_of_mem hv] change (Metric.ball (0 : Space) R).indicator (fun y => f (y 0)) (sphereSplit.symm (u, v)) = _ rw [Set.indicator_of_mem (ht.mpr hv), sphereSplit_symm_first] · rw [Set.indicator_of_notMem hv] exact Set.indicator_of_notMem (fun h => hv (ht.mp h)) _ rw [he, integral_indicator measurableSet_ball, setIntegral_const, plane_slice_volume R u hR] calc ∫ y in Metric.ball (0 : Space) R, f (y 0) = ∫ y, g y := (integral_indicator measurableSet_ball).symm _ = ∫ p : ℝ × CrossPlane, g (sphereSplit.symm p) := (sphereSplit_measurePreserving.symm.integral_comp' g).symm _ = ∫ u : ℝ, ∫ v : CrossPlane, g (sphereSplit.symm (u, v)) := by rw [Measure.volume_eq_prod] exact integral_prod _ hg' _ = ∫ u : ℝ, (Set.Ioo (-R) R).indicator (fun u => (Real.pi * (R ^ 2 - u ^ 2)) • f u) u := by apply integral_congr_ae filter_upwards [] with u rw [hinner] by_cases hu : u ∈ Set.Ioo (-R) R <;> simp [Set.indicator, hu] _ = ∫ u in (-R)..R, (Real.pi * (R ^ 2 - u ^ 2)) • f u := by rw [integral_indicator measurableSet_Ioo, intervalIntegral.integral_of_le (by linarith), integral_Ioc_eq_integral_Ioo] lemma hasDerivAt_symmetric_integral (f : ℝ → ℂ) (hf : Continuous f) (R : ℝ) : HasDerivAt (fun r => ∫ u in (-r)..r, f u) (f R + f (-R)) R := by have hD (r : ℝ) : HasDerivAt (fun r => ∫ u in (0 : ℝ)..r, f u) (f r) r := intervalIntegral.integral_hasDerivAt_right (hf.intervalIntegrable _ _) hf.aestronglyMeasurable.stronglyMeasurableAtFilter hf.continuousAt have he (r : ℝ) : ∫ u in (-r)..r, f u = (∫ u in (0 : ℝ)..r, f u) - ∫ u in (0 : ℝ)..(-r), f u := by have h := intervalIntegral.integral_add_adjacent_intervals (hf.intervalIntegrable (μ := volume) 0 (-r)) (hf.intervalIntegrable (-r) r) exact eq_sub_iff_add_eq.mpr (by simpa only [add_comm] using h) simp_rw [he] simpa only [Pi.sub_def, Function.comp_def, id_eq, neg_one_smul, sub_neg_eq_add] using (hD R).sub ((hD (-R)).scomp R (hasDerivAt_id R).neg) lemma hasDerivAt_weighted_slice (f : ℝ → ℂ) (hf : Continuous f) (R : ℝ) : HasDerivAt (fun r => ∫ u in (-r)..r, (Real.pi * (r ^ 2 - u ^ 2)) • f u) ((2 * Real.pi * R) • ∫ u in (-R)..R, f u) R := by have hcont : Continuous (fun u : ℝ => u ^ 2 • f u) := by fun_prop have he (r : ℝ) : (∫ u in (-r)..r, (Real.pi * (r ^ 2 - u ^ 2)) • f u) = Real.pi • (r ^ 2 • (∫ u in (-r)..r, f u) - ∫ u in (-r)..r, u ^ 2 • f u) := by simp_rw [mul_smul, sub_smul] rw [intervalIntegral.integral_smul, intervalIntegral.integral_sub] · rw [intervalIntegral.integral_smul] · exact (hf.intervalIntegrable _ _).smul _ · exact hcont.intervalIntegrable _ _ simp_rw [he] convert! ((((hasDerivAt_id R).pow 2).smul (hasDerivAt_symmetric_integral f hf R)).sub (hasDerivAt_symmetric_integral _ hcont R)).const_smul Real.pi using 1 simp only [Pi.pow_apply, id_eq, Nat.cast_ofNat, Nat.reduceSub, pow_one, mul_one, neg_sq, smul_add] module lemma ballFourier_hasDerivAt_one (x : Space) : HasDerivAt (ballFourier x) (extension (fun _ => 1) x) 1 := by have hec : Continuous (extension (fun _ => (1 : ℂ))) := extension_continuous measurable_const (M := 1) (by simp) have hc : Continuous (fun r : ℝ => r ^ 2 • extension (fun _ => 1) (r • x)) := by fun_prop have hd := intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable (0 : ℝ) 1) hc.aestronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt have he : ballFourier x =ᶠ[nhds (1 : ℝ)] (fun R => ∫ r in (0 : ℝ)..R, r ^ 2 • extension (fun _ => 1) (r • x)) := by filter_upwards [eventually_gt_nhds (show (0 : ℝ) < 1 by norm_num)] with R hR exact ballFourier_polar x R hR.le simpa using hd.congr_of_eventuallyEq he lemma ballFourier_isometry (e : Space ≃ₗᵢ[ℝ] Space) (x : Space) (R : ℝ) : ballFourier (e x) R = ballFourier x R := by let g : Space → ℂ := (Metric.ball (0 : Space) R).indicator (sphereKernel (e x)) have he : (fun y => g (e y)) = (Metric.ball (0 : Space) R).indicator (sphereKernel x) := by funext y dsimp [g] by_cases hy : y ∈ Metric.ball (0 : Space) R · have hey : e y ∈ Metric.ball (0 : Space) R := by simpa only [mem_ball_zero_iff, e.norm_map] using hy rw [Set.indicator_of_mem hey, Set.indicator_of_mem hy] simp only [sphereKernel, e.inner_map_map] · have hey : e y ∉ Metric.ball (0 : Space) R := by simpa only [mem_ball_zero_iff, e.norm_map] using hy rw [Set.indicator_of_notMem hey, Set.indicator_of_notMem hy] have hm := e.measurePreserving.integral_comp e.toHomeomorph.measurableEmbedding g rw [he, integral_indicator measurableSet_ball] at hm dsimp only [g] at hm rw [integral_indicator measurableSet_ball] at hm exact hm.symm lemma extension_one_isometry (e : Space ≃ₗᵢ[ℝ] Space) (x : Space) : extension (fun _ => 1) (e x) = extension (fun _ => 1) x := by have he : ballFourier (e x) = ballFourier x := funext (ballFourier_isometry e x) have h := ballFourier_hasDerivAt_one (e x) rw [he] at h exact h.unique (ballFourier_hasDerivAt_one x) def spaceAxis (t : ℝ) : Space := PiLp.single 2 0 t lemma spaceAxis_norm (t : ℝ) : ‖spaceAxis t‖ = |t| := by simp [spaceAxis, PiLp.norm_single, Real.norm_eq_abs] lemma spaceAxis_inner (t : ℝ) (y : Space) : inner ℝ (spaceAxis t) y = t * y 0 := by simp [spaceAxis, EuclideanSpace.inner_single_left] lemma extension_one_axis (t : ℝ) : extension (fun _ => 1) (spaceAxis t) = (2 * Real.pi) • ∫ u in (-1 : ℝ)..1, Complex.exp (((2 * Real.pi * t * u : ℝ) : ℂ) * Complex.I) := by let f : ℝ → ℂ := fun u => Complex.exp (((2 * Real.pi * t * u : ℝ) : ℂ) * Complex.I) have hf : Continuous f := by fun_prop have he (R : ℝ) (hR : 0 ≤ R) : ballFourier (spaceAxis t) R = ∫ u in (-R)..R, (Real.pi * (R ^ 2 - u ^ 2)) • f u := by rw [ballFourier] have hk : sphereKernel (spaceAxis t) = fun y => f (y 0) := by funext y dsimp only [sphereKernel, f] rw [spaceAxis_inner] congr 1 push_cast ring rw [hk, integral_ball_head f hf R hR] have hEq : ballFourier (spaceAxis t) =ᶠ[nhds (1 : ℝ)] (fun R => ∫ u in (-R)..R, (Real.pi * (R ^ 2 - u ^ 2)) • f u) := by filter_upwards [eventually_gt_nhds (show (0 : ℝ) < 1 by norm_num)] with R hR exact he R hR.le have hD := (hasDerivAt_weighted_slice f hf 1).congr_of_eventuallyEq hEq simpa only [mul_one] using (ballFourier_hasDerivAt_one (spaceAxis t)).unique hD lemma extension_one_axis_formula (t : ℝ) (ht : t ≠ 0) : extension (fun _ => 1) (spaceAxis t) = ((2 * Real.sin (2 * Real.pi * t) / t : ℝ) : ℂ) := by rw [extension_one_axis] have hc : 2 * Real.pi * t ≠ 0 := mul_ne_zero (mul_ne_zero (by norm_num) Real.pi_ne_zero) ht have he := intervalIntegral.integral_comp_mul_left (fun u : ℝ => Complex.exp ((u : ℂ) * Complex.I)) (a := -1) (b := 1) hc simp only [mul_neg_one, mul_one] at he rw [he, integral_exp_mul_I_eq_sin] simp only [Complex.real_smul, Complex.ofReal_mul, Complex.ofReal_ofNat, Complex.ofReal_inv, Complex.ofReal_div] field_simp theorem extension_one_formula (x : Space) (hx : x ≠ 0) : extension (fun _ => 1) x = ((2 * Real.sin (2 * Real.pi * ‖x‖) / ‖x‖ : ℝ) : ℂ) := by have hn : ‖x‖ = ‖spaceAxis ‖x‖‖ := by rw [spaceAxis_norm, abs_of_nonneg (norm_nonneg x)] let e : Space ≃ₗᵢ[ℝ] Space := (ℝ ∙ (x - spaceAxis ‖x‖))ᗮ.reflection have he : e x = spaceAxis ‖x‖ := Submodule.reflection_sub hn have hi := extension_one_isometry e x rw [he] at hi rw [← hi] exact extension_one_axis_formula _ (norm_ne_zero_iff.mpr hx) open FourierTransform Convolution def box : ℝ → ℂ := (Icc (-1 : ℝ) 1).indicator (fun _ => 1) lemma box_measurable : Measurable box := measurable_const.indicator measurableSet_Icc lemma box_bound (x : ℝ) : ‖box x‖ ≤ 1 := by by_cases h : x ∈ Icc (-1 : ℝ) 1 <;> simp [box, h] lemma box_integrable : Integrable box := by apply (integrable_indicator_iff measurableSet_Icc).2 exact integrableOn_const (C := (1 : ℂ)) (ne_of_lt (measure_Icc_lt_top : volume (Icc (-1 : ℝ) 1) < ∞)) lemma box_integral : ∫ x, box x = (2 : ℂ) := by rw [box, integral_indicator measurableSet_Icc, integral_const] simp norm_num lemma box_neg (x : ℝ) : box (-x) = box x := by have h : -x ∈ Icc (-1 : ℝ) 1 ↔ x ∈ Icc (-1 : ℝ) 1 := by simp only [mem_Icc] constructor <;> intro h <;> constructor <;> linarith [h.1, h.2] simp only [box, indicator, h] lemma box_mul_self (x : ℝ) : box x * box x = box x := by by_cases h : x ∈ Icc (-1 : ℝ) 1 <;> simp [box, h] lemma box_fourier {r : ℝ} (hr : r ≠ 0) : 𝓕 box r = (Real.sin (2 * Real.pi * r) / (Real.pi * r) : ℝ) := by rw [Real.fourier_real_eq_integral_exp_smul] simp only [smul_eq_mul, box] simp_rw [← indicator_mul_right (Icc (-1 : ℝ) 1) (fun v : ℝ => Complex.exp (↑(-2 * Real.pi * v * r) * Complex.I)) (fun _ => 1)] rw [integral_indicator measurableSet_Icc, integral_Icc_eq_integral_Ioc, ← intervalIntegral.integral_of_le (by norm_num : (-1 : ℝ) ≤ 1)] simp only [mul_one] have he : (fun t : ℝ => Complex.exp ((↑(-2 * Real.pi * t * r) : ℂ) * Complex.I)) = (fun t : ℝ => Complex.exp ((-2 * Real.pi * r * Complex.I : ℂ) * t)) := by ext t congr 1 push_cast ring rw [he, integral_exp_mul_complex (by simp [hr, Real.pi_ne_zero])] simp only [Complex.ofReal_one, Complex.ofReal_neg, mul_one, mul_neg] rw [show (-2 * (Real.pi : ℂ) * r * Complex.I) = (-(2 * Real.pi * r : ℝ) : ℂ) * Complex.I by push_cast; ring] simp only [neg_mul, neg_neg] rw [← neg_mul] simp only [Complex.exp_mul_I, Complex.cos_neg, Complex.sin_neg] push_cast field_simp ring lemma box_fourier_continuous : Continuous (𝓕 box) := VectorFourier.fourierIntegral_continuous Real.continuous_fourierChar (by change Continuous (fun p : ℝ × ℝ => inner ℝ p.1 p.2); fun_prop) box_integrable lemma box_norm_integral : ∫ x, ‖box x‖ = (2 : ℝ) := by simp only [box, norm_indicator_eq_indicator_norm, norm_one] rw [integral_indicator measurableSet_Icc, integral_const] norm_num lemma box_fourier_bound (r : ℝ) : ‖𝓕 box r‖ ≤ 2 := by exact (VectorFourier.norm_fourierIntegral_le_integral_norm Real.fourierChar volume (innerₗ ℝ) box r).trans_eq box_norm_integral lemma box_fourier_decay {r : ℝ} (hr : 1 ≤ |r|) : ‖𝓕 box r‖ ≤ 1 / |r| := by have hr0 : 0 < |r| := by linarith rw [box_fourier (abs_pos.mp hr0), Complex.norm_real, Real.norm_eq_abs, abs_div, abs_mul, abs_of_pos Real.pi_pos] apply (div_le_div_of_nonneg_right (Real.abs_sin_le_one _) (by positivity)).trans apply one_div_le_one_div_of_le hr0 nlinarith [Real.pi_gt_three] lemma box_fourier_sq_integrable : Integrable (fun r => (𝓕 box r) ^ 2) := by apply (integrable_inv_one_add_sq.const_mul 8).mono' (box_fourier_continuous.pow 2).aestronglyMeasurable filter_upwards [] with r change ‖𝓕 box r ^ 2‖ ≤ 8 * (1 + r ^ 2)⁻¹ rw [norm_pow] have hd : 0 < 1 + r ^ 2 := by positivity rw [← div_eq_mul_inv, le_div_iff₀ hd] by_cases hr : |r| ≤ 1 · have hn := box_fourier_bound r have hsq : r ^ 2 ≤ 1 := by nlinarith [sq_abs r, abs_nonneg r] nlinarith [sq_nonneg ‖𝓕 box r‖, norm_nonneg (𝓕 box r)] · have hr' : 1 ≤ |r| := le_of_not_ge hr have hb := box_fourier_decay hr' have hr0 : 0 < |r| := lt_of_lt_of_le zero_lt_one hr' have hsq : 1 ≤ r ^ 2 := by nlinarith [sq_abs r] have hb' : ‖𝓕 box r‖ * |r| ≤ 1 := (le_div_iff₀ hr0).mp hb have hs : ‖𝓕 box r‖ ^ 2 * r ^ 2 ≤ 1 := by nlinarith [sq_nonneg (‖𝓕 box r‖ * |r|), sq_abs r, mul_self_le_mul_self (mul_nonneg (norm_nonneg _) (abs_nonneg r)) hb'] have hn : ‖𝓕 box r‖ ^ 2 ≤ 1 := by nlinarith [sq_nonneg ‖𝓕 box r‖] nlinarith noncomputable def tent : ℝ → ℂ := box ⋆[ContinuousLinearMap.mul ℂ ℂ] box lemma tent_integrable : Integrable tent := box_integrable.integrable_convolution (ContinuousLinearMap.mul ℂ ℂ) box_integrable lemma tent_fourier (r : ℝ) : 𝓕 tent r = (𝓕 box r) ^ 2 := by rw [tent, Real.fourier_mul_convolution_eq box_integrable box_integrable, pow_two] lemma tent_fourier_integrable : Integrable (𝓕 tent) := by have he : 𝓕 tent = fun r => (𝓕 box r) ^ 2 := funext tent_fourier rw [he] exact box_fourier_sq_integrable lemma tent_zero : tent 0 = 2 := by change (∫ t, box t * box (0 - t)) = 2 simpa only [zero_sub, box_neg, box_mul_self] using box_integral lemma tent_continuousAt_zero : ContinuousAt tent 0 := by change ContinuousAt (fun x => ∫ t, box t * box (x - t)) 0 apply continuousAt_of_dominated (bound := fun t => ‖box t‖) · exact Eventually.of_forall (fun x => (box_measurable.mul (box_measurable.comp (measurable_const.sub measurable_id))).aestronglyMeasurable) · apply Eventually.of_forall intro x filter_upwards [] with t rw [norm_mul] exact mul_le_of_le_one_right (norm_nonneg _) (box_bound _) · exact box_integrable.norm · filter_upwards [(volume : Measure ℝ).ae_ne (-1), (volume : Measure ℝ).ae_ne 1] with t ht1 ht2 have hct : ContinuousAt box (-t) := by apply continuous_const.continuousOn.continuousAt_indicator rw [frontier_Icc (by norm_num : (-1 : ℝ) ≤ 1)] simp only [mem_insert_iff, mem_singleton_iff, not_or] constructor · intro h; apply ht2; linarith · intro h; apply ht1; linarith have hc : ContinuousAt box (0 - t) := by simpa using hct have hg : ContinuousAt (fun x : ℝ => box (x - t)) 0 := hc.comp (f := fun x : ℝ => x - t) (continuous_id.sub continuous_const).continuousAt exact continuousAt_const.mul hg lemma integral_box_fourier_sq : ∫ r, (𝓕 box r) ^ 2 = (2 : ℂ) := by have h := tent_integrable.fourierInv_fourier_eq tent_fourier_integrable tent_continuousAt_zero simpa [Real.fourierInv_eq, tent_fourier, tent_zero] using h def sineSq (r : ℝ) : ℝ := Real.sin (2 * Real.pi * r) ^ 2 / r ^ 2 def sineFourth (r : ℝ) : ℝ := Real.sin (2 * Real.pi * r) ^ 4 / r ^ 2 lemma sineSq_fourier_ae : (fun r => (sineSq r : ℂ)) =ᵐ[volume] (fun r => (Real.pi : ℂ) ^ 2 * (𝓕 box r) ^ 2) := by filter_upwards [(volume : Measure ℝ).ae_ne 0] with r hr rw [box_fourier hr] simp only [sineSq] push_cast field_simp lemma sineSq_integrable : Integrable sineSq := by have h : Integrable (fun r => (sineSq r : ℂ)) := (box_fourier_sq_integrable.const_mul ((Real.pi : ℂ) ^ 2)).congr sineSq_fourier_ae.symm simpa using h.re lemma sineSq_integral : ∫ r, sineSq r = 2 * Real.pi ^ 2 := by have h : ((∫ r, sineSq r : ℝ) : ℂ) = (2 * Real.pi ^ 2 : ℝ) := by rw [← integral_complex_ofReal, integral_congr_ae sineSq_fourier_ae, integral_const_mul, integral_box_fourier_sq] push_cast ring exact_mod_cast h lemma sineFourth_identity (r : ℝ) : sineFourth r = sineSq r - sineSq (2 * r) := by by_cases hr : r = 0 · simp [hr, sineFourth, sineSq] unfold sineFourth sineSq rw [show 2 * Real.pi * (2 * r) = 2 * (2 * Real.pi * r) by ring, Real.sin_two_mul] field_simp nlinarith [Real.sin_sq_add_cos_sq (2 * Real.pi * r)] lemma sineFourth_integrable : Integrable sineFourth := by apply (sineSq_integrable.sub ((integrable_comp_mul_left_iff sineSq (by norm_num : (2 : ℝ) ≠ 0)).2 sineSq_integrable)).congr exact Eventually.of_forall (fun r => (sineFourth_identity r).symm) lemma sineFourth_integral : ∫ r, sineFourth r = Real.pi ^ 2 := by simp_rw [sineFourth_identity] rw [integral_sub sineSq_integrable ((integrable_comp_mul_left_iff sineSq (by norm_num : (2 : ℝ) ≠ 0)).2 sineSq_integrable), Measure.integral_comp_mul_left, sineSq_integral] norm_num ring lemma sineFourth_abs (r : ℝ) : sineFourth |r| = sineFourth r := by rcases le_total 0 r with hr | hr · rw [abs_of_nonneg hr] · simp only [abs_of_nonpos hr, sineFourth, mul_neg, Real.sin_neg] ring lemma sineFourth_integral_Ioi : ∫ r in Ioi (0 : ℝ), sineFourth r = Real.pi ^ 2 / 2 := by have h := integral_comp_abs (f := sineFourth) simp only [sineFourth_abs, sineFourth_integral] at h linarith lemma fourth_radial_density {r : ℝ} (hr : r ≠ 0) : r ^ 2 * (2 * Real.sin (2 * Real.pi * r) / r) ^ 4 = 16 * sineFourth r := by unfold sineFourth field_simp ring lemma extension_one_fourth_ae : (fun x : Space => ‖extension (fun _ => 1) x‖ ^ 4) =ᵐ[volume] (fun x : Space => (2 * Real.sin (2 * Real.pi * ‖x‖) / ‖x‖) ^ 4) := by filter_upwards [(volume : Measure Space).ae_ne 0] with x hx rw [extension_one_formula x hx, Complex.norm_real, Real.norm_eq_abs] rw [← abs_pow, abs_of_nonneg (by positivity)] lemma extension_one_fourth_integrable : Integrable (fun x : Space => ‖extension (fun _ => 1) x‖ ^ 4) := by have hp : IntegrableOn (fun r : ℝ => r ^ 2 * (2 * Real.sin (2 * Real.pi * r) / r) ^ 4) (Ioi 0) := by apply (sineFourth_integrable.const_mul 16).integrableOn.congr filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr exact (fourth_radial_density (ne_of_gt hr)).symm have hi := (integrable_fun_norm_addHaar (volume : Measure Space) (f := fun r : ℝ => (2 * Real.sin (2 * Real.pi * r) / r) ^ 4)).2 (by simpa only [Space, finrank_euclideanSpace, Fintype.card_fin, Nat.reduceSub, smul_eq_mul] using hp) exact hi.congr extension_one_fourth_ae.symm lemma extension_one_fourth_integral : ∫ x : Space, ‖extension (fun _ => 1) x‖ ^ 4 = 32 * Real.pi ^ 3 := by rw [integral_congr_ae extension_one_fourth_ae, integral_fun_norm_addHaar (volume : Measure Space) (fun r : ℝ => (2 * Real.sin (2 * Real.pi * r) / r) ^ 4)] simp only [Space, finrank_euclideanSpace, Fintype.card_fin, Nat.reduceSub, smul_eq_mul, nsmul_eq_mul, Nat.cast_ofNat] rw [setIntegral_congr_fun measurableSet_Ioi (fun r hr => fourth_radial_density (ne_of_gt hr)), integral_const_mul, sineFourth_integral_Ioi] rw [Measure.real, EuclideanSpace.volume_ball_fin_three] norm_num rw [ENNReal.toReal_ofReal (by positivity)] ring lemma extension_one_fourth_lintegral : (∫⁻ x : Space, ENNReal.ofReal (‖extension (fun _ => 1) x‖ ^ 4)) = ENNReal.ofReal (32 * Real.pi ^ 3) := by rw [← ofReal_integral_eq_lintegral_ofReal extension_one_fourth_integrable (Eventually.of_forall (fun x => by positivity)), extension_one_fourth_integral] lemma extension_congr_ae {g h : Sphere → ℂ} (he : g =ᵐ[surfaceArea] h) : extension g = extension h := by funext x apply integral_congr_ae filter_upwards [he] with ω hω rw [hω] lemma eLpNorm_fourth_power (f : Space → ℂ) (hf : AEStronglyMeasurable f volume) : (eLpNorm f 4 volume) ^ 4 = ∫⁻ x, ENNReal.ofReal (‖f x‖ ^ 4) := by have h := eLpNorm_nnreal_pow_eq_lintegral (μ := (volume : Measure Space)) (f := f) (p := (4 : ℝ≥0)) (by norm_num) hf have hn (x : ℝ≥0∞) : x ^ (4 : ℝ) = x ^ (4 : ℕ) := by exact ENNReal.rpow_natCast x 4 norm_num only [NNReal.coe_ofNat, ENNReal.coe_ofNat] at h simp only [hn] at h rw [h] apply lintegral_congr intro x rw [ENNReal.ofReal_pow (norm_nonneg _), ofReal_norm] theorem sphere_fourth_power : SphereFourthPower := by classical intro g hg ⟨B, hB⟩ let M : ℝ := (eLpNorm g ∞ surfaceArea).toReal have hM0 : 0 ≤ M := ENNReal.toReal_nonneg have hfin : eLpNorm g ∞ surfaceArea ≠ ∞ := by rw [eLpNorm_exponent_top hg.aestronglyMeasurable] exact (eLpNormEssSup_lt_top_of_ae_bound (Eventually.of_forall hB)).ne have hM : ∀ᵐ ω ∂surfaceArea, ‖g ω‖ ≤ M := by filter_upwards [ae_le_eLpNormEssSup (f := g) (μ := surfaceArea)] with ω hω have ht := ENNReal.toReal_mono (by simpa only [eLpNorm_exponent_top hg.aestronglyMeasurable] using hfin) hω simpa [M, eLpNorm_exponent_top hg.aestronglyMeasurable, enorm_eq_nnnorm] using ht let h : Sphere → ℂ := fun ω => if ‖g ω‖ ≤ M then g ω else 0 have hh : Measurable h := Measurable.ite (measurableSet_le hg.norm measurable_const) hg measurable_const have hbound : ∀ ω, ‖h ω‖ ≤ M := by intro ω dsimp [h] split_ifs with hw · exact hw · simpa using hM0 have he : h =ᵐ[surfaceArea] g := by filter_upwards [hM] with ω hω simp [h, hω] rw [← extension_congr_ae he, eLpNorm_fourth_power _ (extension_continuous hh hbound).aestronglyMeasurable] calc _ ≤ ENNReal.ofReal (M ^ 4) * ∫⁻ x, ENNReal.ofReal (‖extension (fun _ => 1) x‖ ^ 4) := fourth_power_majorant hM0 hh hbound _ = ENNReal.ofReal (32 * Real.pi ^ 3) * (eLpNorm g ∞ surfaceArea) ^ 4 := by rw [extension_one_fourth_lintegral, ENNReal.ofReal_pow hM0] dsimp [M] rw [ENNReal.ofReal_toReal hfin, mul_comm] lemma extension_enorm_le_essSup (g : Sphere → ℂ) (x : Space) : ‖extension g x‖ₑ ≤ ENNReal.ofReal (4 * Real.pi) * eLpNorm g ∞ surfaceArea := by unfold extension calc _ ≤ ∫⁻ ω, ‖g ω * Complex.exp (2 * (Real.pi : ℂ) * Complex.I * (inner ℝ x (ω : Space) : ℂ))‖ₑ ∂surfaceArea := enorm_integral_le_lintegral_enorm _ _ ≤ ∫⁻ _ : Sphere, eLpNorm g ∞ surfaceArea ∂surfaceArea := by apply lintegral_mono_ae filter_upwards [ae_le_eLpNormEssSup (f := g) (μ := surfaceArea)] with ω hω simpa only [← ofReal_norm, norm_mul, kernel_norm, mul_one] using hω.trans eLpNormEssSup_le_eLpNorm_top _ = _ := by rw [lintegral_const, surfaceArea_univ, mul_comm] theorem sphere_restriction_ge_four : ∀ p : ℝ, 4 ≤ p → ∃ C : ℝ≥0, ∀ g : Sphere → ℂ, Measurable g → (∃ M : ℝ, ∀ ω, ‖g ω‖ ≤ M) → eLpNorm (extension g) (ENNReal.ofReal p) volume ≤ (C : ℝ≥0∞) * eLpNorm g ∞ surfaceArea := by intro p hp have hp0 : 0 < p := lt_of_lt_of_le (by norm_num) hp have hdiff : 0 ≤ p - 4 := sub_nonneg.mpr hp have hinv : 0 ≤ p⁻¹ := le_of_lt (inv_pos.mpr hp0) let A : ℝ≥0∞ := ENNReal.ofReal (4 * Real.pi) let B : ℝ≥0∞ := ENNReal.ofReal (32 * Real.pi ^ 3) let C : ℝ≥0∞ := (A ^ (p - 4) * B) ^ p⁻¹ have hA : A ≠ ∞ := ENNReal.ofReal_ne_top have hB : B ≠ ∞ := ENNReal.ofReal_ne_top have hC : C ≠ ∞ := by dsimp [C] finiteness refine ⟨C.toNNReal, ?_⟩ intro g hg hbdd have hext : AEStronglyMeasurable (extension g) volume := (extension_continuous hg hbdd.choose_spec).aestronglyMeasurable rw [ENNReal.coe_toNNReal hC] let G := eLpNorm g ∞ surfaceArea have hG : G ≠ ∞ := by rcases hbdd with ⟨M, hM⟩ simpa only [G, eLpNorm_exponent_top hg.aestronglyMeasurable] using (eLpNormEssSup_lt_top_of_ae_bound (μ := surfaceArea) (Filter.Eventually.of_forall hM)).ne have hpoint : ∀ x, ‖extension g x‖ₑ ≤ A * G := extension_enorm_le_essSup g have hfour : (∫⁻ x, ‖extension g x‖ₑ ^ (4 : ℕ)) ≤ B * G ^ (4 : ℕ) := by have h := sphere_fourth_power g hg hbdd rw [eLpNorm_fourth_power _ hext] at h simpa only [ENNReal.ofReal_pow (norm_nonneg _), ofReal_norm] using h have hsplit : ∀ x, ‖extension g x‖ₑ ^ p ≤ (A * G) ^ (p - 4) * ‖extension g x‖ₑ ^ (4 : ℕ) := by intro x calc ‖extension g x‖ₑ ^ p = ‖extension g x‖ₑ ^ (p - 4) * ‖extension g x‖ₑ ^ (4 : ℝ) := by rw [← ENNReal.rpow_add_of_nonneg _ _ hdiff (by norm_num), sub_add_cancel] _ ≤ (A * G) ^ (p - 4) * ‖extension g x‖ₑ ^ (4 : ℝ) := mul_le_mul_left (ENNReal.rpow_le_rpow (hpoint x) hdiff) _ _ = _ := by rw [ENNReal.rpow_ofNat] have integral_bound : (∫⁻ x, ‖extension g x‖ₑ ^ p) ≤ (A ^ (p - 4) * B) * G ^ p := by calc _ ≤ ∫⁻ x, (A * G) ^ (p - 4) * ‖extension g x‖ₑ ^ (4 : ℕ) := lintegral_mono hsplit _ = (A * G) ^ (p - 4) * (∫⁻ x, ‖extension g x‖ₑ ^ (4 : ℕ)) := lintegral_const_mul' _ _ (by finiteness) _ ≤ (A * G) ^ (p - 4) * (B * G ^ (4 : ℕ)) := mul_le_mul_right hfour _ _ = (A ^ (p - 4) * B) * (G ^ (p - 4) * G ^ (4 : ℝ)) := by rw [ENNReal.mul_rpow_of_nonneg _ _ hdiff, ENNReal.rpow_ofNat] ring _ = _ := by rw [← ENNReal.rpow_add_of_nonneg _ _ hdiff (by norm_num), sub_add_cancel] rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (ne_of_gt (ENNReal.ofReal_pos.mpr hp0)) ENNReal.ofReal_ne_top hext, ENNReal.toReal_ofReal hp0.le, one_div] calc _ ≤ ((A ^ (p - 4) * B) * G ^ p) ^ p⁻¹ := ENNReal.rpow_le_rpow integral_bound hinv _ = C * G := by rw [ENNReal.mul_rpow_of_nonneg _ _ hinv, ← ENNReal.rpow_mul, mul_inv_cancel₀ hp0.ne', ENNReal.rpow_one] end EllipticCapacity end end OAI