Corollary 5.29.5.EACB00126847 Fix $p>1$. There is a constant $C_ p$ such that every nonnegative $F\in L^ p(X,\mathrm{d}V)$ satisfies
\[ \int _ E F\, \mathrm{d}V\leq C_ p\| F\| _ p\operatorname {Cap}(E)^2 \]
for every Borel set $E\subset X$.
Proof.
By Hölder's inequality and (5.29.3), the integral is at most
\[ C\| F\| _ p\exp (-a\operatorname {Cap}(E)^{-1/n}/p'), \qquad p'=p/(p-1). \]
The function $x\mapsto x^{-2}\exp (-ax^{-1/n}/p')$ is bounded on $0<x\leq \operatorname {Cap}(X)<\infty $, and sets of capacity zero have zero volume.
$\square$
Pome
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