A boundedness criterion for general maximal operators

Abstract

We consider maximal operators MBMB with respect to a basis BB. In the case when MBMB satisfies a reversed weak type inequality, we obtain a boundedness criterion for MBMB on an arbitrary quasi-Banach function space XX. Being applied to specific BB and XX this criterion yields new and short proofs of a number of well-known results. Our principal application is related to an open problem on the boundedness of the two-dimensional one-sided maximal function M+M+ on LpwLwp.