Hardy and Littlewood

Theorem

For any dimension $n$ and any $p>1$, there is a constant $C(n,p)$ so that

$\Vert Mf{\Vert }_{L^p({\mathbb{R}}^n)}\le C(n,p)\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}$ where $Mf$ is the Hardy-Littlewood maximal function, that is $Mf={\sup }_{B\supset \{x\}}\frac{1}{m(B)}{\int }_B|f(y)|\text{dd}y.$