\begin{definition}[Definition:Hardy-Littlewood Maximal Function]
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{{explain|presumably the "Hardy-Littlewood Maximal Function" is that function which is returned by the Hardy-Littlewood maximal operator? More precision needed here.}}
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 $E \subset \R^d$.
\end{definition}