Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Finitary carleson

finitary_carleson

harmonic analysis Fourier analysis measure theory finitary carleson

Plain-language statement

There is a measurable exceptional set $G' \subseteq G$ with $2\mu(G') \le \mu(G)$ such that, for every measurable $f$ bounded by $\mathbf{1}_F$ , the integral over $G \setminus G'$ of the finitary oscillatory singular integral, summed only over the scales from $\sigma_1(x)$ to $\sigma_2(x)$ , is at most $C(a,q)\mu(G)^{1-1/q}\mu(F)^{1/q}$ .

Exact Lean statement

theorem finitary_carleson : ∃ G', MeasurableSet G' ∧ 2 * volume G' ≤ volume G ∧
    ∀ f : X → ℂ, Measurable f → (∀ x, ‖f x‖ ≤ F.indicator 1 x) →
    ∫⁻ x in G \ G', ‖∑ s ∈ Icc (σ₁ x) (σ₂ x), ∫ y, Ks s x y * f y * exp (I * Q x y)‖ₑ ≤
    C2_0_1 a nnq * (volume G) ^ (1 - q⁻¹) * (volume F) ^ q⁻¹

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem finitary_carleson : ∃ G', MeasurableSet G' ∧ 2 * volume G' ≤ volume G ∧    ∀ f : X → ℂ, Measurable f → (∀ x, ‖f x‖ ≤ F.indicator 1 x) →    ∫⁻ x in G \ G', ‖∑ s ∈ Icc (σ₁ x) (σ₂ x), ∫ y, Ks s x y * f y * exp (I * Q x y)‖ₑ ≤    C2_0_1 a nnq * (volume G) ^ (1 - q⁻¹) * (volume F) ^ q⁻¹ := by  have g : GridStructure X D κ S o := grid_existence X  have t : TileStructure Q D κ S o := tile_existence X  clear g  rcases discrete_carleson X with ⟨G', hG', h2G', hfG'⟩  refine ⟨G', hG', h2G', fun f meas_f h2f ↦ le_of_eq_of_le ?_ (hfG' f meas_f h2f)⟩  refine setLIntegral_congr_fun (measurableSet_G.diff hG') fun x hx ↦ ?_  simp_rw [carlesonSum, mem_univ, Finset.filter_true, tile_sum_operator hx, mul_sub, exp_sub,    mul_div, div_eq_mul_inv,    ← smul_eq_mul, integral_smul_const, ← Finset.sum_smul, _root_.enorm_smul]  suffices ‖(cexp (I • ((Q x) x : ℂ)))⁻¹‖ₑ = 1 by rw [this, mul_one]  simp [mul_comm I, enorm_eq_nnnorm]

Project: Carleson formalization
License: Apache-2.0
Commit: 74ef907d6bdb
Source: Carleson/FinitaryCarleson.lean:110-124

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Related declarations

Project-declaredLean 4.32.0

Adjoint Carleson adjoint

adjointCarleson_adjoint

Plain-language statement

adjointCarleson is the adjoint of carlesonOn.

harmonic analysisFourier analysismeasure theory

Source project: Carleson formalization

Person-level attribution pending.

View proof record

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 $x$ .

harmonic analysisFourier analysismeasure theory

Source project: Carleson formalization

Person-level attribution pending.

View proof record

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

$C(a,q)\,\mathrm{dens}_1(\mathfrak{A})^{(q-1)/(8a^4)}\,\mathrm{dens}_2(\mathfrak{A})^{1/q-1/2}\,\lVert f\rVert_2\lVert g\rVert_2.$

harmonic analysisFourier analysismeasure theory

Source project: Carleson formalization

Person-level attribution pending.

View proof record