Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Is Cancellative of norm integral exp le

isCancellative_of_norm_integral_exp_le

harmonic analysis Fourier analysis measure theory doubling measure

Plain-language statement

Suppose the compatible phase system satisfies the following oscillatory cancellation estimate on every ball $B(x,r)$ : for every Lipschitz amplitude $\varphi$ supported in the ball and every pair of phases $f,g$ ,

$\left\lVert\int_{B(x,r)}e^{i(f-g)}\varphi\right\rVert\le A\,\mu(B(x,r))\,\lVert\varphi\rVert_{\mathrm{Lip}}\,(1+d_{x,r}(f,g))^{-\tau}.$

Then the metric phase space satisfies the project’s IsCancellative property with exponent $\tau$ .

Exact Lean statement

lemma isCancellative_of_norm_integral_exp_le (τ : ℝ) [CompatibleFunctions ℝ X A]
    (h : ∀ {x : X} {r : ℝ} {φ : X → ℂ} (_hr : 0 < r) (_h1 : iLipENorm φ x r ≠ ∞)
    (_h2 : support φ ⊆ ball x r) {f g : Θ X},
      ‖∫ x in ball x r, exp (I * (f x - g x)) * φ x‖ ≤
      A * volume.real (ball x r) * iLipNNNorm φ x r * (1 + dist_{x, r} f g) ^ (-τ)) :
    IsCancellative X τ

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma isCancellative_of_norm_integral_exp_le (τ : ℝ) [CompatibleFunctions ℝ X A]    (h : ∀ {x : X} {r : ℝ} {φ : X → ℂ} (_hr : 0 < r) (_h1 : iLipENorm φ x r ≠ ∞)    (_h2 : support φ ⊆ ball x r) {f g : Θ X},      ‖∫ x in ball x r, exp (I * (f x - g x)) * φ x‖ ≤      A * volume.real (ball x r) * iLipNNNorm φ x r * (1 + dist_{x, r} f g) ^ (-τ)) :    IsCancellative X τ := by  constructor  intro x r φ hr h1 h2 f g  convert ENNReal.ofReal_le_ofReal (h (x := x) (r := r) (φ := φ) hr h1 h2 (f := f) (g := g))  · rw [ofReal_norm]    congr 1    rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun y hy ↦ ?_)]    have : φ y = 0 := by      apply notMem_support.1      contrapose! hy      exact h2 hy    simp [this]  · rw [ENNReal.ofReal_mul (by positivity), ENNReal.ofReal_mul (by positivity),      ENNReal.ofReal_mul (by positivity)]    congr    · simp    · simp only [Measure.real, ofReal_toReal measure_ball_ne_top]    · simp [iLipNNNorm, coe_toNNReal h1]    · rw [← ENNReal.ofReal_rpow_of_pos (by positivity)]      congr      rw [ENNReal.ofReal_add zero_le_one dist_nonneg]      simp [edist_dist]

Project: Carleson formalization
License: Apache-2.0
Commit: 74ef907d6bdb
Source: Carleson/DoublingMeasure.lean:262-288

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Related declarations

Project-declaredLean 4.32.0

Adjoint Carleson adjoint

adjointCarleson_adjoint

Plain-language statement

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Source project: Carleson formalization

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Project-declaredLean 4.32.0

Ae tendsto zero of distribution le

ae_tendsto_zero_of_distribution_le

Plain-language statement

Suppose that, for every error threshold $\delta>0$ and every measure tolerance $\varepsilon>0$ , one can choose $N_0$ so that the set where $\sup_{N>N_0}\lVert f(x)-F_N(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$ . Then $F_N(x)$ converges to $f(x)$ for almost every $x$ .

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Project-declaredLean 4.32.0

Antichain operator

antichain_operator

Plain-language statement

For an antichain $\mathfrak{A}$ of pairwise incomparable tiles, and measurable functions $f$ and $g$ bounded by the indicators of $F$ and $G$ , the pairing of $g$ with the Carleson sum over $\mathfrak{A}$ is controlled by the $L^2$ norms of $f$ and $g$ and by positive powers of the two tile-density parameters. Concretely, the bound is

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harmonic analysisFourier analysismeasure theory

Source project: Carleson formalization

Person-level attribution pending.

View proof record