Theorem 15.17.1D662421F3436

Proof. Since $u\geq V_\theta -C_0\geq \rho -C_0$, we may subtract a constant from the strict barrier so that $\rho \leq u-1$; it still satisfies $\theta +\mathrm{dd}^{\mathrm{c}}\rho \geq \kappa \omega $ for a fixed $\kappa >0$. Choose $0<\gamma <(p-1)/(np+p-1)$ and set $\alpha =2\gamma <1$. Apply lemma 5.42.10 with $a=u$, and let $A,K,C$ and $\Phi _{c,\delta }$ be the constants and transforms given there. For small $\delta >0$ set

By the curvature bound of lemma 5.42.10, $v_\delta \in \mathrm{PSH}(X,\theta )$. The lower bound for $\Phi _{c,\delta }$ in that lemma implies $\rho \leq \Phi _{c,\delta }$ when $\delta $ is small. Thus

In particular $v_\delta \leq V_\theta $, so theorem 15.15.1 applies with the reference $V_\theta $ and the comparison potential $v_\delta $. Together with the $L^1$ smoothing estimate, it gives

Fix a compact subset of $\Omega $ on which $\rho \geq -M$. Since $s\leq C\delta ^\alpha $ and $u\leq 0$, the identity

gives $\Phi _{c,\delta }-u\leq C_ M\delta ^\alpha $ there.

At a radius $t$ where the infimum defining $\Phi _{c,\delta }$ is attained, the monotone lower bound yields

Hence $t\geq a_ M\delta $ for a fixed $a_ M>0$. As in the proof of theorem 15.5.1, monotonicity of the smoothing now gives

on the chosen compact set. Since every small radius has the form $a_ M\delta $, we obtain $|R_ tu-u|\leq C_ Mt^\alpha $ there for all small $t>0$, with constants uniform on that compact set. We apply this to a slightly larger compact neighborhood, on which also $\| \nabla R_ tu\| \leq C_ M/t$. For nearby points $x,y$ at distance $d>0$, the smooth function $R_ tu$ and a short path in that neighborhood give

The choice $t=d^{1/(1+\alpha )}$ gives the exponent $\alpha /(1+\alpha )$, and letting $\gamma $ approach its upper endpoint gives the asserted range. Pairs of points farther apart are handled by the boundedness of $u$ on the compact neighborhood. $\square$