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  "body": "\n## Summary\n\nThe strong maximal function of a function $f$ on $\\R^n$ is a supremum of averages of $\\abs{f}$.\nAt a point $x$, it is taken over all rectangles that contain $x$ and have sides parallel to the axes.\nUnlike the [[Hardy–Littlewood maximal function|VX1A56P5KQ]], it allows rectangles of any shape, long and thin in some directions.\nIt is bounded on $L^p(\\R^n)$ for $1 \u003c p \\le \\infty$ but is not of [[weak type|9534R6VMY0]] $(1, 1)$ when $n \\ge 2$.\nIt controls multiparameter averages and is the basic maximal operator of product, or multiparameter, harmonic analysis.\n\n## Explanations\n\n### L1 human\n\nThe usual maximal function looks at balls around a point, which have the same size in every direction.\nThe strong maximal function also looks at boxes that can be very long in one direction and very short in another.\nBecause there are many more boxes than balls, it can be much larger than the usual maximal function.\nStill, it is bounded on $L^p$ for $p \u003e 1$, because in each coordinate direction it is no worse than a one-dimensional maximal function.\nFor example, take the indicator of the unit square in the plane and the point $(a, 1/2)$ with $a \u003e 1$.\nThe rectangle $[0, a] \\times [0, 1]$ gives the average $1/a$.\nFor large $a$, balls centred at the point only give averages of order $1/a^2$.\n",
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