Discrete maximal functions in higher dimensions and applications to ergodic theory

TL;DR

This paper extends Bourgain's pointwise ergodic theorem to higher dimensions using variational estimates for polynomial mappings, providing uniform bounds in coefficients for fixed degree polynomials.

Contribution

It introduces a higher-dimensional analogue of Bourgain's theorem with variational estimates on $L^p$ spaces, uniform in polynomial coefficients.

Findings

Established variational estimates $V_r$ on $L^p$ spaces for all $1<p< ext{and} > ext{max}\{p,p/(p-1) ",

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Abstract

We establish a higher dimensional counterpart of Bourgain's pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates $V_r$ on $L^p$ spaces for all $1<p<\infty$ and $r>\max\{p, p/(p-1)\}$. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.