Strong Lp bound for the Hardy-Littlewood maximal function
= Strong Lp bound for the Hardy-Littlewood maximal function
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{title2=$\lVert Mf\rVert_p\leq C_p\lVert f\rVert_p$}
For $1<p\leq\infty$, the <Hardy-Littlewood maximal function> is bounded on $L^p(\mathbb R^n)$. A direct proof for finite $p$ applies the weak $(1,1)$ estimate to $f\mathbf1_{\{|f|>s/2\}}$, integrates the resulting distribution-function bound against $ps^{p-1}$, and uses <Tonelli theorem>.