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

Paul Hagelstein ; Alexander Stokolos

Fundamenta Mathematicae, Tome 219 (2012), p. 269-283 / Harvested from The Polish Digital Mathematics Library

Résumé

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 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 $M_{}$ and $M_{Γ}$ are defined respectively on $L¹(ℝ^{d})$ and L¹(Ω) by $M_{}g(x) = sup_{x ∈ R ∈ } 1/|R| ∫_{R}|g(y)|dy$ 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 $M_{}$ satisfies the weak type estimate $|{x ∈ ℝ^d : M_{}g(x) > α}| ≤ C_{}∫_{ℝ^d} Φ(c_{}|g|/α)$ for a pair of positive constants $C_{}$, $c_{}$ 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.

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Publié le : 2012-01-01

Zbl 1257.37011

EUDML-ID : urn:eudml:doc:282848