Strong maximal function

Summary

The strong maximal function of a function $f$ on $\R^n$ is a supremum of averages of $\abs{f}$. At a point $x$, it is taken over all rectangles that contain $x$ and have sides parallel to the axes. Unlike the Hardy–Littlewood maximal function, it allows rectangles of any shape, long and thin in some directions. It is bounded on $L^p(\R^n)$ for $1 < p \le \infty$ but is not of weak type $(1, 1)$ when $n \ge 2$. It controls multiparameter averages and is the basic maximal operator of product, or multiparameter, harmonic analysis.