# Definition:Hardy-Littlewood Maximal Function

## Definition

The Hardy-Littlewood maximal operator takes a locally integrable function $f: \R^d \to \R$ and returns another function $M f$ that, at each point $x \in \R^d$ gives the maximum average value that $f$ can have on balls centered at that point.

More precisely,

$\ds \map {M f} x := \sup_{r \mathop > 0} \frac 1 {\size {\map B {x, r} } } \int_{\map B {x, r} } \size {\map f y} \rd y$

where:

$\map B {x, r}$ is the ball of radius $r$ centered at $x$
$\size E$ denotes the Lebesgue measure of $E \subset \R^d$.