= Lp bound from a tail domination inequality
{c}
{title2=$a\mu\{F>a\}\leq\int_{\{F>a\}}G\;\Longrightarrow\;\|F\|_p\leq\frac p{p-1}\|G\|_p$}
Let $F,G\geq0$, $1<p<\infty$, and $G\in L^p$. Assume the displayed tail inequality and finite positive-threshold superlevel measures. First split $G$ at $a/2$ to obtain $a\mu\{F>2a\}\leq\int_{\{G>a\}}G$. <Layer cake representation> and <Tonelli theorem> imply $F\in L^p$. Integrating the original inequality against $pa^{p-2}$ then gives $\|F\|_p^p\leq q\int GF^{p-1}\leq q\|G\|_p\|F\|_p^{p-1}$, with $q=p/(p-1)$. This is the analytic estimate behind the <Doob Lp maximal inequality>.
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