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arXiv · 1804.07679

Dimension-free estimates for discrete Hardy-Littlewood averaging operators over the cubes in $\mathbb Z^d$

Jean Bourgain·Mariusz Mirek·Elias M. Stein·Błażej Wróbel
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Abstract

Dimension-free bounds will be provided in maximal and $r$-variational inequalities on $\ell^p(\mathbb Z^d)$ corresponding to the discrete Hardy-Littlewood averaging operators defined over the cubes in $\mathbb Z^d$. We will also construct an example of a symmetric convex body in $\mathbb Z^d$ for which maximal dimension-free bounds fail on $\ell^p(\mathbb Z^d)$ for all $p\in(1, \infty)$. Finally, some applications in ergodic theory will be discussed.

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Jean Bourgain, Mariusz Mirek, Elias M. Stein, Błażej Wróbel. 2018-04-20. Dimension-free estimates for discrete Hardy-Littlewood averaging operators over the cubes in $\mathbb Z^d$. https://arxiv.org/abs/1804.07679

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