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Corollary 17.5.256DAA97EFFAAF

Corollary 17.5.25.6DAA97EFFAAF Let $u\in \mathcal E^1(X,\theta ,\phi )$ with $u\leq \phi $, and let $A>0$ and $\delta >0$ be such that

\[ \mu _ u(E)\leq A\operatorname {Cap}_{\theta ,\phi }(E)^{1+\delta } \quad \text{for every Borel set }E\subset X. \]

If $L\geq 0$ and $E_\theta ^\phi (u)\geq -L$, then

\[ \phi -\left(2+(n+1)L(2A^{1/n})^{n/\delta } +\frac1{1-2^{-\delta }}\right)\leq u\leq \phi . \]

Proof. Apply theorem 17.5.20 with $a=0$ and $\nu =\mu _ u$. By the first inequality in (17.3.15), $I=\int _ X(\phi -u)\, \mu _ u\leq (n+1)\bigl (-E_\theta ^\phi (u)\bigr )\leq (n+1)L$, and (17.5.22) with $a=0$ gives the stated bound. $\square$

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