Modulated ( C , α ) -ergodic theorems with non-integral orders for Dunford–Schwartz operators

Abstract

The definition of the usual $p$th Weyl semi-norm for sequences is extended to the case of $(C,\alpha )$ averages for $0<\alpha \le 1$ and the $(p,\alpha )$-Besicovitch sequences are defined similarly to the classical case $\alpha =1$. We study the effects of $(p,\alpha )$-Besicovitch sequences with non-integral orders as good modulators. The major finding is the almost everywhere convergence of $(C,\alpha )$ ergodic averages with discrete and continuous $(p,\alpha )$-Besicovitch modulators for Dunford–Schwartz operators. The results have the additional advantage that they are sufficiently general to give as corollaries a (new) weighted Abelian ergodic theorem and the a.e. convergence of random $(C,\alpha )$-ergodic averages for Dunford–Schwartz operators.