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

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.