15.17 Local Hölder continuity in a big background1A5D83E43B0A

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. $\square$

Proof. By lemma 15.16.5, all $v_\lambda $ lie in a common reference interval, and their densities are bounded in $L^ p$ by $C\| f\| _ p$. We may therefore apply corollary 15.15.3 with $\eta =V_\theta $. Given any sequence $\lambda _ j\downarrow 0$, we pass to a subsequence with $v_{\lambda _ j}\to v$ uniformly in differences on $\Omega $ and in $L^1(X)$, and with $c_{\lambda _ j}\to c_*\in [m,C]$. The limit satisfies the same reference bounds and is normalized. Off the pole set,

as $\lambda \downarrow 0$. Multiplying by $f$ and using dominated convergence, we obtain $\mathrm{e}^{\lambda v_\lambda }f\to f$ in $L^ p$ for the whole family. The compactness corollary therefore identifies the limit equation as $\mathrm{MA}_\theta (v)=c_*f\, \mathrm{d}V$, and the uniqueness of the constant gives $c_*=c$. Since every subsequential limit of $c_\lambda $ as $\lambda \downarrow 0$ equals $c$, we conclude that $c_\lambda \to c$, and the stated convergence of the densities follows.

Finally, theorem 15.17.1 gives uniform local $C^\alpha $ bounds for every $\alpha $ below its endpoint, and the uniform limit inherits these bounds by pointwise convergence. For $0<\beta <\alpha $, uniform convergence and the elementary interpolation inequality on smaller coordinate balls give

where the Hölder bounds are taken on slightly larger compact neighborhoods. The uniformly convergent subsequence given by compactness thus converges in $C^\beta $ on every compact subset of $\Omega $ and for every exponent below the endpoint; no further extraction depending on the exponent is needed. $\square$

Proof. For $g\in G$, pullback by $g$ preserves $\theta $-plurisubharmonicity, and $V_\theta \circ g=V_\theta $ by the definition of the envelope. Pullback also preserves the positive-exponent equation, so its uniqueness gives $u_\lambda \circ g=u_\lambda $, and the normalization preserves this equality. Pullback by a fixed smooth biholomorphism is continuous on $L^1(X)$, since its Jacobian relative to the fixed smooth volume is bounded. Hence every cluster limit is invariant almost everywhere, and therefore everywhere for the canonical representatives. This argument does not require $g$ to preserve the barrier $\rho $ or the set $\Omega $.

If the invariant solution is unique, every cluster limit is the same potential. Relative compactness then implies convergence of the whole family for uniform convergence of differences on $\Omega $: a sequence staying at a fixed positive distance from that potential would have a convergent subsequence, a contradiction. The interpolation argument above upgrades this to the asserted local $C^\beta $ convergence. $\square$