Quantitative ergodic theorems for actions of groups of polynomial growth

TL;DR

This paper advances ergodic theorems for polynomial growth groups by establishing sharp jump inequalities and exponential decay in upcrossing, combining probability, harmonic analysis, and geometric methods.

Contribution

It introduces the strongest form of maximal ergodic theorems with jump inequalities for polynomial growth groups, extending previous results with new techniques.

Findings

Established sharp jump inequalities for polynomial growth group actions.

Proved exponential decay in upcrossing inequalities.

Unified probabilistic and geometric approach to ergodic theorems.

Abstract

We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As a consequence, we deduce in this setting the quantitative ergodic theorem, in particular, the upcrossing inequalities with exponential decay. The ideas or techniques involve probability theory, non-doubling Calder\'on-Zygmund theory, almost orthogonality argument and some delicate geometric argument involving the balls and the cubes on the group equipped with a not necessarily doubling measure.