Definition:Hardy-Littlewood Maximal Function

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

More precisely,


 * $\displaystyle Mf(x):=\sup_{r>0} \frac{1}{|B(x, r)|}\int_{B(x, r)} |f(y)|\, dy $

where $B(x,r)$ is the ball of radius r centered at x, and $|E|$ denotes the  Lebesgue measure of $E\subset \mathbb{R}^{d}$.