Local integrability of strong and iterated maximal functions (Q2717718)

Let \(Q\) be the unit square in \({\mathbb R}^2\) and \(h\) a measurable nonnegative function on \({\mathbb R}^2\). Denote by \(M_xh(u,v):= \sup_{r<u<s}{1\over s-r}\int_r^s h(t,v) dt\) the horizontal Hardy-Littlewood maximal function, by \(M_yh\) the corresponding vertical maximal function and by \(M_S h(u,v):= \displaystyle\sup_{(u,v)\in R} {1\over |R|}\int_R h(s,t) ds dt\) (\(R\subset {\mathbb R}^2\) an interval) the strong maximal function. The main results of the paper under review state that, for some measurable function \(h\geq 0\) supported by \(Q\), \(\int_Q M_xM_yh=\infty\) and \(\int_Q M_yM_xh<\infty\), and that these facts imply \(\int_A M_Sh=\infty\) for some set \(A\) of finite measure in \({\mathbb R}^2\).