On Templeman averages and variation functions

Let S be a countable semigroup acting in a measure-preserving fashion (g ↦ Tg) on a measure space (Ω, A, µ). For a finite subset A of S, let |A| denote its cardinality. Let (Ak)k=1∞ be a sequence of subsets of S satisfying conditions related to those in the ergodic theorem for semi-group actions of A. A. Tempelman. For A-measureable functions f on the measure space (Ω, A, μ) we form for k ≥ 1 the Templeman averages \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _k (f)(x) = \left| {A_k } \right|^{ - 1} \sum\nolimits_{g \in A_k } {T_g f(x)}$$\end{document} and set Vqf(x) = (Σk≥1|πk+1(f)(x) − πk(f)(x)|q)1/q when q ∈ (1, 2]. We show that there exists C > 0 such that for all f in L1(Ω, A, µ) we have µ({x ∈ Ω: Vqf(x) > λ}) ≤ C(∫Ω | f | dµ/λ). Finally, some concrete examples are constructed.