Oscillation inequalities in ergodic theory and analysis: one-parameter and multi-parameter perspectives

TL;DR

This survey reviews oscillation inequalities in ergodic theory and analysis, highlighting their role in pointwise convergence, and introduces new inequalities and elementary proofs for existing results in both single and multi-parameter contexts.

Contribution

The paper provides a comprehensive review of oscillation inequalities, introduces new inequalities, and offers elementary proofs for known results in ergodic theory and analysis.

Findings

New oscillation inequalities established

Elementary proofs for existing results provided

Enhanced understanding of pointwise convergence phenomena

Abstract

In this survey we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic theory and probability. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of oscillation estimates in both the single and several parameter settings. We establish a number of new oscillation inequalities and give new proofs for known results with elementary arguments.