The definition of the usual th Weyl semi-norm for sequences is extended to the case of averages for and the -Besicovitch sequences are defined similarly to the classical case . We study the effects of -Besicovitch sequences with non-integral orders as good modulators. The major finding is the almost everywhere convergence of ergodic averages with discrete and continuous -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 -ergodic averages for Dunford–Schwartz operators.
"Modulated -ergodic theorems with non-integral orders for Dunford–Schwartz operators." Illinois J. Math. 64 (4) 613 - 644, December 2020. https://doi.org/10.1215/00192082-8746145