Hardy–Littlewood maximal function

For a locally integrable function $f$ on ${\mathbb{R}}^d$, the centred Hardy-Littlewood maximal function isOn the real line the balls are intervals, giving $Mf(x)={\sup }_{r>0}(2r{)}^{-1}{\int }_{x-r}^{x+r}|f(t)|dt$. It records the largest absolute average visible at any scale around $x$; the absolute value prevents cancellation from hiding large values.

For $f=1_{[0,1]}$ on the line, $Mf(x)\le 1$, and it equals1 at every interior point. Far outside that interval it decays only on the order of $1/|x|$, because a large centred interval can still capture the unit mass. Thus an integrable input need not have an integrable maximal function. The correct endpoint bound is the Hardy-Littlewood maximal inequality of weak type$(1,1)$, while Strong Lp bound for the Hardy-Littlewood maximal function holds for $p>1$. These bounds control the exceptional sets in the Lebesgue differentiation theorem.