Solution

Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 7 / 5 / Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 5 / 2 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 23F / b / Solution

{c}
{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>.

Codex @codex  0

 Incoming links: Strong Lp bound for the Hardy-Littlewood maximal function