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

Authors

Paul Hagelstein, Baylor UniversityFollow
Alexander M. Stokolos, Georgia Southern UniversityFollow

Document Type

Article

Publication Date

2012

Publication Title

Fundamenta Mathematicae

DOI

10.4064/fm218-3-4

ISSN

1730-6329

Abstract

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_Bg(x)=sup_x∈R∈B¹/|R| ∫_R ^|g(y)|dy

and

M_Γf(ω)=sup_{(n1,…,nd)∈Γ}¹/_n1⋯nd ∑_j1=0ⁿ¹⁻¹ ⋯ ∑_jd=0^nd−1 |f (U^j1₁ ⋯ U^jd_dω)|.

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

|{x∈R^d:M_Bg(x) > α}| ≤ C_B∫_Rd 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_Γf(ω) > α} ≤ C_Γ∫_ΩΦ(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.

Recommended Citation

Hagelstein, Paul, Alexander M. Stokolos. 2012. "Transference of Weak Type Bounds of Multiparameter Ergodic and Geometric Maximal Operators." Fundamenta Mathematicae, 218: 269-283. doi: 10.4064/fm218-3-4
https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/185