Weak-type maximal function estimates on the infinite-dimensional torus

TL;DR

This paper establishes precise conditions for the weak-$L^p$ boundedness of a maximal operator on the infinite-dimensional torus, extending known results to higher dimensions and providing sharp quantitative bounds.

Contribution

It provides necessary and sufficient conditions for weak-$L^p$ boundedness of maximal operators on infinite-dimensional tori, including endpoint cases, with sharp quantitative estimates.

Findings

Characterization of weak-$L^p$ boundedness conditions

Extension of endpoint weak-type inequalities to infinite dimensions

Sharp quantitative bounds for the maximal operator

Abstract

We prove necessary and sufficient conditions for the weak-$L^p$ boundedness, for $p \in (1,\infty)$, of a maximal operator on the infinite-dimensional torus. In the endpoint case $p=1$ we obtain the same weak-type inequality enjoyed by the strong maximal function in dimension two. Our results are quantitatively sharp.