Transference of weak type bounds of multiparameter ergodic and geometric maximal operators

• Autorzy: Paul Hagelstein , Alexander Stokolos
• Języki publikacji: EN

Abstrakty

EN

Let $U₁, ..., U_{d}$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $ℤ₊^{d}$ and 𝓑 the associated collection of rectangular parallelepipeds in $ℝ^d$ with sides parallel to the axes and dimensions of the form $n₁ × ⋯ × n_d$ with $(n₁,...,n_d) ∈ Γ.$ The associated multiparameter geometric and ergodic maximal operators $M_{𝓑}$ and $M_{Γ}$ are defined respectively on $L¹(ℝ^{d})$ and L¹(Ω) by
$M_{𝓑}g(x) = sup_{x ∈ R ∈ 𝓑} 1/|R| ∫_{R}|g(y)|dy$
and
$M_{Γ}f(ω) = sup_{(n₁, ..., n_{d}) ∈ Γ} 1/{n₁⋯ n_{d}} ∑_{j₁=0}^{n₁-1} ⋯ ∑_{j_{d}=0}^{n_{d}-1} |f(U₁^{j₁} ⋯ U_{d}^{j_{d}}ω)|$.
Given a Young function Φ, it is shown that $M_{𝓑}$ satisfies the weak type estimate
$|{x ∈ ℝ^d : M_{𝓑}g(x) > α}| ≤ C_{𝓑}∫_{ℝ^d} Φ(c_{𝓑}|g|/α)$
for a pair of positive constants $C_{𝓑}$, $c_{𝓑}$ if and only if $M_{Γ}$ satisfies a corresponding weak type estimate
$μ{ω ∈ Ω : M_{Γ} f(ω) > α} ≤ C_{Γ}∫_{Ω} Φ(c_{Γ}|f|/α)$.
for a pair of positive constants $C_{Γ}$, $c_{Γ}$. Applications of this transference principle regarding the a.e. convergence of multiparameter ergodic averages associated to rare bases are given.

Kategorie tematyczne

• 42B25: Maximal functions, Littlewood-Paley theory
• 37A45: Relations with number theory and harmonic analysis

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2012
• Tom: 218
• Numer: 3
• Strony: 269-283

Daty

wydano

2012

Twórcy

autor

Paul Hagelstein

• Department of Mathematics, Baylor University, Waco, TX 76798, U.S.A.

autor

Alexander Stokolos

• Department of Mathematical Sciences, Georgia Southern University, 203 Georgia Avenue, Statesboro, GA 30460-8093, U.S.A.

Identyfikatory

DOI

10.4064/fm218-3-4