Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Rcarleson general

rcarleson_general

harmonic analysis Fourier analysis measure theory carleson on the real line continuous

Plain-language statement

Let $1<q\le2$ and let $q'$ be its Hölder conjugate. For measurable sets $F,G\subseteq\mathbb{R}$ and measurable $f$ with $\lVert f(x)\rVert\le\mathbf{1}_F(x)$ , the real-line Carleson operator satisfies

$\int_G^+ T f(x)\,dx \le C(q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

Exact Lean statement

lemma rcarleson_general {q q' : ℝ≥0} (hq : q ∈ Set.Ioc 1 2) (hqq' : q.HolderConjugate q')
    {F G : Set ℝ} (hF : MeasurableSet F) (hG : MeasurableSet G)
    (f : ℝ → ℂ) (hmf : Measurable f) (hf : ∀ x, ‖f x‖ ≤ F.indicator 1 x) :
    ∫⁻ x in G, T f x ≤ C10_0_1 4 q * (volume G) ^ (q' : ℝ)⁻¹ * (volume F) ^ (q : ℝ)⁻¹

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma rcarleson_general {q q' : ℝ≥0} (hq : q ∈ Set.Ioc 1 2) (hqq' : q.HolderConjugate q')    {F G : Set ℝ} (hF : MeasurableSet F) (hG : MeasurableSet G)    (f : ℝ → ℂ) (hmf : Measurable f) (hf : ∀ x, ‖f x‖ ≤ F.indicator 1 x) :    ∫⁻ x in G, T f x ≤ C10_0_1 4 q * (volume G) ^ (q' : ℝ)⁻¹ * (volume F) ^ (q : ℝ)⁻¹ := by  calc ∫⁻ x in G, T f x    _ ≤ ∫⁻ x in G, carlesonOperator K f x :=      lintegral_mono (carlesonOperatorReal_le_carlesonOperator _)    _ ≤ C10_0_1 4 q * (volume G) ^ (q' : ℝ)⁻¹ * (volume F) ^ (q : ℝ)⁻¹ :=      two_sided_metric_carleson (a := 4) (by norm_num) hq hqq' hF hG        Hilbert_strong_2_2 hmf hf

Project: Carleson formalization
License: Apache-2.0
Commit: 74ef907d6bdb
Source: Carleson/Classical/CarlesonOnTheRealLineContinuous.lean:17-26

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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.

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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 $x$ .

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