On variation functions for subsequence ergodic averages (Q1807622)

For a sequence \((a_n)\) write \(A_Nf(x)=(1/N)\sum_{n=1}^{N} f(T^{a_n}x)\) for the ergodic averages. Here certain relationships between the maximal function \(Mf=\sup_{N\geq 1}|A_Nf|\) and the \(q\)-variation function \(V_qf=\left( \sum_{N=1}^{\infty}|A_{N+1}f-A_Nf|^q\right)^{1/q}\) for \(q\geq 1\) are found. For \(q,p>1\), it is shown that \(\|Mf\|_p\leq C_p\|f\|_p\) implies that there is a \(C_p''>0\) with \(\|V_qf\|_p\leq C_p''\|f \|_p\). In contrast, this is shown to fail for \(q=1\). For certain sequences \((a_n)\) (polynomial-like and certain random constructions) and \((N_k)\) of exponential growth, it is shown that \(\|Sf\|_2\leq C\|f\|_2\) for a certain positive constant \(C\), where \(Sf=\left( \sum_{k=1}^{\infty}|A_{N_{k+1}}f-A_{N_k}f|^2\right)^{1/2}\).