Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Control approximation effect

control_approximation_effect'

harmonic analysis Fourier analysis measure theory control approximation effect continuous

Plain-language statement

For every $\delta,\varepsilon>0$ , there is an explicit positive uniform bound $C(\delta,\varepsilon)$ such that, if a measurable $2\pi$ -periodic function $g$ satisfies $\lVert g(x)\rVert\le C(\delta,\varepsilon)$ for every $x$ , then the set where $\sup_N\lVert S_Ng(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$ .

Exact Lean statement

lemma control_approximation_effect' {δ ε : ℝ≥0} (δpos : 0 < δ) (εpos : 0 < ε)
  {g : ℝ → ℂ} (g_measurable : Measurable g)
  (g_periodic : g.Periodic (2 * π))
  (g_bound : ∀ x, ‖g x‖ ≤ C_control_approximation_effect' δ ε) :
    distribution (fun x ↦ ⨆ N, ‖S_ N g x‖ₑ) δ (volume.restrict (Set.Ioc 0 (2 * π))) ≤ ε

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma control_approximation_effect' {δ ε : ℝ≥0} (δpos : 0 < δ) (εpos : 0 < ε)  {g : ℝ → ℂ} (g_measurable : Measurable g)  (g_periodic : g.Periodic (2 * π))  (g_bound : ∀ x, ‖g x‖ ≤ C_control_approximation_effect' δ ε) :    distribution (fun x ↦ ⨆ N, ‖S_ N g x‖ₑ) δ (volume.restrict (Set.Ioc 0 (2 * π))) ≤ ε := by  calc _    _ ≤ distribution (operatorBound g) δ (volume.restrict (Set.Ioc 0 (2 * π))) := by      apply distribution_mono_left      rw [ae_restrict_iff' (measurableSet_Ioc)]      filter_upwards with x hx      simp only [enorm_eq_self, iSup_le_iff]      intro N      apply partialFourierSum_bound g_periodic _ (Set.Ioc_subset_Icc_self hx)      exact intervalIntegrable_of_bdd g_measurable g_bound    _ = distribution (operatorBound g) (δ / 2 + δ / 2) (volume.restrict (Set.Ioc 0 (2 * π))) := by      congr      simp    _ ≤ distribution (fun x ↦ (T g x + T (conj ∘ g) x) / ENNReal.ofReal (2 * π)) (δ / 2) (volume.restrict (Set.Ioc 0 (2 * π)))          + distribution (fun x ↦ eLpNorm g 1 (volume.restrict (Set.Ioc 0 (2 * π))) / 2) (δ / 2) (volume.restrict (Set.Ioc 0 (2 * π))) := by      apply distribution_add_le    _ ≤ ε + 0 := by      gcongr      · rw [← distribution_mul (by left; exact ENNReal.ofReal_ne_top) (by left; simp [Real.pi_pos])]        calc _          _ ≤ distribution (T g) (ENNReal.ofReal (2 * π) * (↑δ / 2) / 2) (volume.restrict (Set.Ioc 0 (2 * π)))                + distribution (T (conj ∘ g)) (ENNReal.ofReal (2 * π) * (↑δ / 2) / 2) (volume.restrict (Set.Ioc 0 (2 * π))) := by            apply distribution_add_le.trans'            gcongr            · simp            rw [← two_mul, ENNReal.mul_div_cancel (by simp) (by simp)]          _ ≤ ENNReal.ofNNReal (ε / 2) + ENNReal.ofNNReal  (ε / 2) := by            have : ENNReal.ofReal (2 * π) * (↑δ / 2) / 2 = ENNReal.ofReal ((2 * π) * (↑δ / 2) / 2) := by              rw [ENNReal.ofReal_div_of_pos (by simp), ENNReal.ofReal_mul (by simp),                ENNReal.ofReal_mul two_pi_pos.le, ENNReal.ofReal_mul (by simp),                ENNReal.ofReal_ofNat, ENNReal.ofReal_div_of_pos (by simp),                ENNReal.ofReal_ofNat]              simp            rw [this]            gcongr            · apply distribution_carlesonOperatorReal_le' (by positivity) (by positivity)                g_measurable              intro x              exact (g_bound x).trans C_control_approximation_effect'_le            · have conj_g_periodic : (conj ∘ g).Periodic (2 * π) := by                intro x                simp only [Function.comp_apply]                congr 1                apply g_periodic              have conj_g_measurable : Measurable (conj ∘ g) := by fun_prop              have conj_g_bound :  ∀ (x : ℝ), ‖(conj ∘ g) x‖ ≤ ↑(C_control_approximation_effect' δ ε) := by                simpa              apply distribution_carlesonOperatorReal_le' (by positivity) (by positivity)                conj_g_measurable              intro x              exact (conj_g_bound x).trans C_control_approximation_effect'_le          _ = ε := by simp      · rw [← distribution_mul (by simp) (by simp)]        simp only [nonpos_iff_eq_zero]        rw [Function.const_def, distribution_const, Set.indicator_of_notMem]        simp only [enorm_eq_self, Set.mem_Iio, not_lt]        rw [eLpNorm_one_eq_lintegral_enorm]        calc _          _ ≤ ∫⁻ (x : ℝ) in Set.Ioc 0 (2 * π), ↑(C_control_approximation_effect' δ ε) := by            apply setLIntegral_mono measurable_const            intro x _            rw [← ofReal_norm, ENNReal.ofReal_le_coe]            exact g_bound x          _ ≤ ↑(C_control_approximation_effect' δ ε) * ENNReal.ofReal (2 * π) := by            rw [setLIntegral_const, Real.volume_Ioc, sub_zero]          _ ≤ δ := C_control_approximation_effect'_property          _ = 2 * (δ / 2) := by            rw [ENNReal.mul_div_cancel (by simp) (by simp)]    _ = ε := by simp

Project: Carleson formalization
License: Apache-2.0
Commit: 74ef907d6bdb
Source: Carleson/Classical/ControlApproximationEffectContinuous.lean:177-249

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Project-declaredLean 4.32.0

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antichain_operator

Plain-language statement

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