Some remarks on oscillation inequalities

TL;DR

This paper establishes uniform oscillation estimates for polynomial ergodic averages on L^p spaces, addressing a longstanding problem, and compares oscillation inequalities with jump and variation inequalities.

Contribution

It provides new uniform oscillation bounds for ergodic averages and offers a different proof for existing martingale inequalities, clarifying the relationship between oscillations and variation endpoints.

Findings

Proves uniform oscillation estimates for polynomial ergodic averages.

Provides a new proof for oscillation inequalities of bounded martingales.

Shows oscillations cannot serve as an endpoint for r-variation inequalities.

Abstract

In this paper we establish uniform oscillation estimates on $L^p(X)$ with $p\in(1,\infty)$ for the polynomial ergodic averages. This result contributes to a certain problem about uniform oscillation bounds for ergodic averages formulated by Rosenblatt and Wierdl in the early 1990's. We also give a slightly different proof of the uniform oscillation inequality of Jones, Kaufman, Rosenblatt and Wierdl for bounded martingales. Finally, we show that oscillations, in contrast to jump inequalities, cannot be seen as an endpoint for $r$-variation inequalities.