The Hardy-Littlewood maximal operator in $\mathbb{R}^n$ is an operator defined on $L^1_{{loc}}(\mathbb{R}^n)$ (the space of locally integrable functions in $\mathbb{R}^n$ with the Lebesgue measure) which maps each locally integrable function $f$ to another function $Mf$ defined for each $x\in \mathbb{R}^n$ by $$Mf(x) = \sup_Q \frac{1}{m(Q)}\int_Q |f(y)|dy,$$ where the supremum is taken over all cubes $Q$ containing $x$ This function is lower semicontinuous (and hence measurable), and it is called the Hardy-Littlewood maximal function of $f$

The operator $M$ is sublinear, which means that $$M(af + bg) \leq |a|Mf + |b|Mg$$ for each pair of locally integrable functions $f,g$ and scalars $a,b$