Theorem 15.5.1CA29182325B0

Proof. After adding a constant we may assume $-C_0\leq u\leq 0$. Choose

For small $\delta >0$ set

The curvature estimate of lemma 5.42.1 gives $v_\delta \in \mathrm{PSH}(X,\omega )$. Evaluating the infimum defining $\Phi _{c,\delta }$ at $t=\delta $ gives $\Phi _{c,\delta }\leq R_\delta u\leq 0$. On the other hand,

since $R_ tu+Kt^2\geq u$, $Kt\geq 0$ and $-c\log (t/\delta )\geq 0$ for $0<t\leq \delta $. For small $\delta $ this lower bound is at least $r$, and therefore

The Hermitian form $\omega $ is semipositive and big, so theorem 15.4.1 applies in the background $\omega $. Together with the $L^1$ estimate of lemma 5.42.2, it gives

Since $e\leq C\delta ^\alpha $ and $u-r\leq C_0+1$, it follows that

We now bound the minimizing radius from below. Fix $z$ and a radius $t=t(z)\in (0,\delta ]$ at which the infimum is attained. The lower estimate at this radius and (15.5.2) give

Thus $t(z)\geq a\delta $ with the constant $a=\mathrm{e}^{-C}>0$, which is independent of $z$ and $\delta $. Since $R_ su+Ks^2$ is increasing in $s$,

Combining this with (15.5.2) and the lower bound $R_ su-u\geq -Ks^2$, and setting $s=a\delta $, we obtain

for every sufficiently small $s>0$.

Finally, let $x,y$ be points at a small distance $d$. The gradient estimate $\| \nabla R_ su\| _\infty \leq C/s$ gives

With $s=d^{1/(1+\alpha )}$, both terms are bounded by a constant times $d^{\alpha /(1+\alpha )}$, and larger distances are controlled by the bound on the oscillation. This proves Hölder continuity with exponent $\alpha /(1+\alpha )$. Letting $\eta $ approach its upper endpoint gives every $\beta $ in the theorem; smaller exponents follow on compact $X$ by changing the constant. $\square$