Transference of Weak Type Bounds of Multiparameter Ergodic and Geometric Maximal Operators

Document Type

Article

Publication Date

2012

Publication Title

Fundamenta Mathematicae

DOI

10.4064/fm218-3-4

ISSN

1730-6329

Abstract

Let U1,…,Ud 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 Zd+ and B the associated collection of rectangular parallelepipeds in Rd with sides parallel to the axes and dimensions of the form n1×⋯×nd with (n1,…,nd)∈Γ. The associated multiparameter geometric and ergodic maximal operators MB and MΓ are defined respectively on L1(Rd) and L1(Ω) by

MBg(x)=supx∈R∈B1/|R| ∫R |g(y)|dy

and

MΓf(ω)=sup(n1,…,nd)∈Γ1/n1⋯nd ∑j1=0n1−1 ⋯ ∑jd=0nd−1 |f (Uj11 ⋯ Ujddω)|.

Given a Young function Φ, it is shown that MB satisfies the weak type estimate

|{x∈Rd:MBg(x) > α}| ≤ CB∫Rd Φ (cB|g|/α)

for a pair of positive constants CB, cB if and only if MΓ satisfies a corresponding weak type estimate

μ {ω∈Ω:MΓf(ω) > α} ≤ CΓ∫ΩΦ(cΓ|f|/α).

for a pair of positive constants CΓ, cΓ. Applications of this transference principle regarding the a.e. convergence of multiparameter ergodic averages associated to rare bases are given.

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