Quantitative ergodic theorems for actions of groups of polynomial growth

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 Calderón-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.

arXiv comment43pages

References in corpus (8)

• Local stability of ergodic averages
• Variation estimates for averages along primes and polynomials
• Dyadic harmonic analysis beyond doubling measures
• Upcrossing inequalities for stationary sequences and applications
• A bootstrapping approach to jump inequalities and their applications
• Jump inequalities via real interpolation
• Maximal polynomial modulations of singular integrals
• Noncommutative maximal ergodic inequalities associated with doubling conditions