A Boundedness Criterion for General Maximal Operators

Abstract

We consider maximal operators $M_{\mathcal B}$ with respect to a basis ${\mathcal B}$. In the case when $M_{\mathcal B}$ satisfies a reversed weak type inequality, we obtain a boundedness criterion for $M_{\mathcal B}$ on an arbitrary quasi-Banach function space $X$. Being applied to specific ${\mathcal B}$ and $X$ 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 $M^{+}$ on $L^p_w$.