On the integrability of a maximal square function in ergodic theory (Q2711128)

Given a bounded linear operator \(A\) in \(L^2(\Omega)\), \(f \in L^2\) and an increasing sequence of integers \((n_k)\), let \(A_n f = n^{-1} \sum_{k=1}^n A^{k-1} f\), \(n \in \mathbb{N}\), and \(S^{*} f = (\sum_{k=1}^{\infty} \max_{n_k < n \leq n_{k+1}} |A_n f - A_{n_k} f|^2)^{1/2}\). If \(S^{*} f \in L^2\) for any \(f\) and each unitary \(A\), \((n_k)\) is called admissible for the maximal square function of unitary operators. That admissibility holds if and only if \(\sup_k \frac{n_{k+1}}{n_k} = Q < \infty\) and then \(\|S^{*} f\|\leq C(Q) \|f\|\). This sharpens the previous result by the author [ibid. 22, 286-310 (1977), resp. ibid. 22, 295-319 (1977; Zbl 0377.60033)] and answers a question of \textit{R. L. Jones, I. V. Ostrovskij} and \textit{J. M. Rosenblatt} [Ergodic Theory Dyn. Syst. 16, No. 2, 267-305 (1996; Zbl 0854.28007)]. Extensions to arbitrary contractions, multiparameter unitary groups in \(L^2\) (in terms of wide-sense stationary random fields) and power-bounded operators in \(L^p\), \(p > 1\), are considered.