# Hardy-Littlewood maximal theorem

There is a constant $K>0$ such that for each Lebesgue integrable function $f\in L^{1}(\mathbb{R}^{n})$, and each $t>0$,

 $m(\{x:Mf(x)>t\})\leq\frac{K}{t}\|f\|_{1}=\frac{K}{t}\int_{\mathbb{R}^{n}}|f(x)% |dx,$

where $Mf$ is the Hardy-Littlewood maximal function of $f$.

Remark. The theorem holds for the constant $K=3^{n}$.

Title Hardy-Littlewood maximal theorem HardyLittlewoodMaximalTheorem 2013-03-22 13:27:33 2013-03-22 13:27:33 Koro (127) Koro (127) 4 Koro (127) Theorem msc 28A15 msc 28A25 HardyLittlewoodMaximalOperator Hardy-Littlewood theorem