Inequalities for the ergodic maximal function and convergence of the averages in weighted $L^ p$-spaces

Abstract:

This paper is concerned with the characterization of those positive functions $w$ such that Hopf’s averages associated to an invertible measure preserving transformation $T$ and a positive function $g$ converge almost everywhere for every $f \in {L^p}(w d\mu )$. We also study mean convergence when $g$ satisfies a "doubling condition" over orbits. In order to do this, we first characterize the pairs of positive functions $(u,w)$ such that the ergodic maximal operator associated to $T$ and $g$ is of weak or strong type with respect to the measures $w d\mu$ and $u d\mu$.