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

Abstract

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