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Abstract

Let $M_{\rm S}$ denote the strong maximal operator. Let $M_{x}$ and $M_{y}$ denote the one-dimensional Hardy–Littlewood maximal operators in the horizontal and vertical directions in ${\mathbb R}^{2}$. A function $h$ supported on the unit square $Q = [0,1] \times [0,1]$ is exhibited such that $\int _{Q}M_{y}M_{x}h < \infty $ but $\int _{Q}M_{x}M_{y}h = \infty $. It is shown that if $f$ is a function supported on $Q$ such that $\int _{Q}M_{y}M_{x}f < \infty $ but $\int _{Q}M_{x}M_{y}f = \infty $, then there exists a set $A$ of finite measure in ${\mathbb R}^{2}$ such that $\int _{A}M_{\rm S}f = \infty $.