Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Linearized metric carleson

linearized_metric_carleson

harmonic analysis Fourier analysis measure theory linearized

Plain-language statement

Let $1 < q \le 2$ and let $q'$ be its Hölder conjugate. If every phase-linearized nontangential operator has the required uniform $L^2$ bound, then for measurable $F,G$ and measurable $f$ with $\lVert f(x)\rVert \le \mathbf{1}_F(x)$ , the linearized Carleson operator satisfies

$\int_G^+ \mathcal{C}^{\mathrm{lin}}_{Q,K}f(x)\,dx \le C(a,q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

Exact Lean statement

theorem linearized_metric_carleson [IsCancellative X (defaultτ a)]
    (hq : q ∈ Ioc 1 2) (hqq' : q.HolderConjugate q') (mF : MeasurableSet F) (mG : MeasurableSet G)
    (mf : Measurable f) (nf : (‖f ·‖) ≤ F.indicator 1)
    (hT : ∀ θ : Θ X, HasBoundedStrongType (linearizedNontangentialOperator Q θ K · ·)
      2 2 volume volume (C_Ts a)) :
    ∫⁻ x in G, linearizedCarlesonOperator Q K f x ≤
    C1_0_2 a q * volume G ^ (q' : ℝ)⁻¹ * volume F ^ (q : ℝ)⁻¹

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem linearized_metric_carleson [IsCancellative X (defaultτ a)]    (hq : q ∈ Ioc 1 2) (hqq' : q.HolderConjugate q') (mF : MeasurableSet F) (mG : MeasurableSet G)    (mf : Measurable f) (nf : (‖f ·‖) ≤ F.indicator 1)    (hT : ∀ θ : Θ X, HasBoundedStrongType (linearizedNontangentialOperator Q θ K · ·)      2 2 volume volume (C_Ts a)) :    ∫⁻ x in G, linearizedCarlesonOperator Q K f x ≤    C1_0_2 a q * volume G ^ (q' : ℝ)⁻¹ * volume F ^ (q : ℝ)⁻¹ := by  have nf' : (‖f ·‖) ≤ 1 := nf.trans (indicator_le_self' (by simp))  calc    _ = ∫⁻ x, ⨆ n, G.indicator (lcoConvergent K Q f n) x := by      rw [← lintegral_indicator mG]; congr! 2 with x      rw [← iSup_apply, iSup_indicator rfl monotone_lcoConvergent monotone_const, iUnion_const,        iSup_lcoConvergent]    _ = ⨆ n, ∫⁻ x, G.indicator (lcoConvergent K Q f n) x :=      lintegral_iSup (fun _ ↦ (measurable_lcoConvergent mf nf').indicator mG)        (fun _ _ hl ↦ indicator_mono (monotone_lcoConvergent hl))    _ ≤ _ := by      refine iSup_le fun n ↦ ?_      unfold lcoConvergent; rw [lintegral_indicator mG]      exact R_truncation hq hqq' mF mG mf nf rfl hT

Project: Carleson formalization
License: Apache-2.0
Commit: 74ef907d6bdb
Source: Carleson/MetricCarleson/Linearized.lean:109-128

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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