= Strong Lp bound from weak L1 and L-infinity bounds
{title2=$\|Sf\|_p^p\leq2^p q C_1C_\infty^{p-1}\|f\|_p^p$}
Let $S$ be a <sublinear operator> with positive constants and $\mu\{Sf>a\}\leq C_1\|f\|_1/a$ and $\|Sf\|_\infty\leq C_\infty\|f\|_\infty$. Split $f$ into its part above $a/(2C_\infty)$ and its bounded remainder. The latter contributes at most $a/2$, so the weak bound controls the superlevel measure by $2C_1a^{-1}\int_{|f|>a/(2C_\infty)}|f|$. Integrating this estimate with <layer cake representation> proves the displayed bound for $1<p<\infty$ and $q=p/(p-1)$. In particular it gives the strong $L^p$ estimate for the <Hardy-Littlewood maximal function>.
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