Hardy-Littlewood maximal operator

The Hardy-Littlewood maximal operator in ${ℝ}^n$ is an operator defined on $L_{\text{loc}}^1({ℝ}^n)$ (the space of locally integrable functions in ${ℝ}^n$ with the Lebesgue measure) which maps each locally integrable function $f$ to another function $Mf$, defined for each $x\in {ℝ}^n$ by

$$Mf(x)=\underset{Q}{sup}\frac{1}{m(Q)}{\int }_Q|f(y)|𝑑y,$$

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

$$M(af+bg)\le |a|Mf+|b|Mg$$

for each pair of locally integrable functions $f,g$ and scalars $a,b$.