# Hardy-Littlewood maximal operator

The Hardy-Littlewood maximal operator in $\mathbb{R}^{n}$ is an operator defined on $L^{1}_{\textnormal{loc}}(\mathbb{R}^{n})$ (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 $a,b$.

Title Hardy-Littlewood maximal operator HardyLittlewoodMaximalOperator 2013-03-22 13:27:30 2013-03-22 13:27:30 azdbacks4234 (14155) azdbacks4234 (14155) 8 azdbacks4234 (14155) Definition msc 28A25 msc 28A15 HardyLittlewoodMaximalTheorem Hardy-Littlewood maximal function