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 , α ) averages for 0 < α ≤ 1 and the ( p , α ) -Besicovitch sequences are defined similarly to the classical case α = 1 . We study the effects of ( p , α ) -Besicovitch sequences with non-integral orders as good modulators. The major finding is the almost everywhere convergence of ( C , α ) ergodic averages with discrete and continuous ( p , α ) -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 , α ) -ergodic averages for Dunford–Schwartz operators.