Sharp weak‐type estimates for maximal operators associated to rare bases

Abstract

AbstractLet denote a nonempty translation invariant collection of intervals in (which we regard as a rare basis), and define the associated geometric maximal operator by We provide a sufficient condition on so that the estimate is sharp. As a corollary, we obtain sharp weak‐type estimates for maximal operators associated to several classes of rare bases including unions of one‐parameter infinite families of Córdoba, Soria, and Zygmund‐type bases.