Theorem 16.15.24A9F89B404AD6

Proof. We approximate $\min _ jf_ j$ uniformly by smooth obstacles to which (16.15.22) applies with uniform constants, and pass to the limit.

Step 1: smoothing the obstacles. By proposition 16.14.9, each $f_ j$ belongs to $W^{2,p}(X)$ for every finite $p$, and it is of class $C^1$; fix $p>2n$. Choose a finite smooth partition of unity $(\chi _\nu )$ with each $\chi _\nu $ supported in the interior of a coordinate chart. For small $\varepsilon >0$, let $f_{j,\nu ,\varepsilon }$ be the convolution of $f_ j$, written in the chart of $\chi _\nu $, with the mollifier $\rho _\varepsilon $ of section 2.5; it is defined on a neighborhood of the support of $\chi _\nu $. Put $f_{j,\varepsilon }=\sum _\nu \chi _\nu f_{j,\nu ,\varepsilon }$. These are smooth functions with $\| f_{j,\varepsilon }\| _\infty \leq \| f_ j\| _\infty $, and $f_{j,\varepsilon }\to f_ j$ in $C^1$, because the convolutions of the $C^1$ function $f_ j$ and of its first derivatives converge to them uniformly on compact sets (lemma 2.5.1(1)), and $\partial _\mu (f_ j*\rho _\varepsilon )=(\partial _\mu f_ j)*\rho _\varepsilon $.

We bound the Laplacians of the approximations. By (16.14.8), in the real coordinates $x_1,\ldots ,x_{2n}$ underlying a holomorphic chart, with $\partial _\mu =\partial /\partial x_\mu $, we have $D_ h=\sum _{\mu ,\sigma }c^{\mu \sigma }(x)\partial _\mu \partial _\sigma $ on real functions, with smooth real coefficients; there are no first-order terms. Since $f_ j\in W^{2,p}_{\mathrm{loc}}$ in the chart, by (16.14.3), and $\partial _\mu \partial _\sigma (f_ j*\rho _\varepsilon )=(\partial _\mu \partial _\sigma f_ j)*\rho _\varepsilon $, the commutator of $D_ h$ with convolution is

where $D_ hf_ j=-Lf_ j/(4\pi )$ is a bounded function. Since $|c^{\mu \sigma }(x)-c^{\mu \sigma }(y)|\leq C\varepsilon $ on the support of $\rho _\varepsilon (x-\cdot )$ and $\| \rho _\varepsilon \| _{L^{p'}}=\varepsilon ^{-2n/p}\| \rho \| _{L^{p'}}$ with $p'=p/(p-1)$, Hölder's inequality and (16.14.3) bound this commutator by $C\varepsilon ^{1-2n/p}\| f_ j\| _{W^{2,p}(X)}$, which tends to zero because $p>2n$. As $(D_ hf_ j)*\rho _\varepsilon $ lies between the essential infimum and the essential supremum of $D_ hf_ j$, since $\rho _\varepsilon \geq 0$ has integral one, we obtain on a neighborhood of the support of $\chi _\nu $

Next, $D_ hf_{j,\varepsilon }=\sum _\nu \chi _\nu D_ hf_{j,\nu ,\varepsilon }+\sum _\nu [D_ h,\chi _\nu ]f_{j,\nu ,\varepsilon }$, where each commutator $[D_ h,\chi _\nu ]$ is a first-order operator with coefficients supported in $\operatorname{Supp}\chi _\nu $. Since $\sum _\nu \chi _\nu =1$, we have $\sum _\nu [D_ h,\chi _\nu ]f_ j=D_ hf_ j-D_ hf_ j=0$, so the second sum equals $\sum _\nu [D_ h,\chi _\nu ](f_{j,\nu ,\varepsilon }-f_ j)$, which involves only $f_{j,\nu ,\varepsilon }-f_ j$ and its first derivatives and tends to zero uniformly. The first sum is a convex combination. Hence $|D_ hf_{j,\varepsilon }|\leq \| D_ hf_ j\| _{L^\infty }+1$ for all sufficiently small $\varepsilon $.

Step 2: smoothing the minimum. For $\delta >0$ set

By lemma 16.15.23,

Choose sequences $\varepsilon _ m,\delta _ m\to 0$ with $\varepsilon _ m$ small as in Step 1, and put $F_ m=F_{\varepsilon _ m,\delta _ m}$. Then $F_ m\to \min _ jf_ j$ uniformly on $X$, since $|\min _ jf_{j,\varepsilon _ m}-\min _ jf_ j|\leq \max _ j\| f_{j,\varepsilon _ m}-f_ j\| _\infty $, while $\| F_ m\| _\infty \leq \max _ j\| f_ j\| _\infty +\delta _ m\log r$ and $\sup _ XD_ hF_ m\leq \max _ j\| D_ hf_ j\| _{L^\infty }+1$ remain bounded.

Step 3: passage to the limit. By (16.15.22), applied to the smooth obstacles $F_ m$, we have $-\sup _ X\operatorname {tr}_ hk\leq D_ hP_\kappa (F_ m)\leq 2B_ m\mathrm e^{2A\| F_ m\| _\infty }$ in the sense of distributions, where $A$ depends only on $X$, $\omega $, $\kappa $, and $B_ m=\sup _ X\operatorname {tr}_ hk+\sup _ XD_ hF_ m$. By Step 2 these bounds are uniform in $m$ and depend only on the asserted data. By the stability of envelopes in proposition 16.11.4(4), $\| P_\kappa (F_ m)-u\| _\infty \leq \| F_ m-\min _ jf_ j\| _\infty \to 0$. Therefore $D_ hP_\kappa (F_ m)\to D_ hu$ in the sense of distributions, and $u$ satisfies (16.15.2) with $c_1=\sup _ X\operatorname {tr}_ hk$ and $c_2=\sup _ m2B_ m\mathrm e^{2A\| F_ m\| _\infty }$. By the remark following (16.15.2), $D_ hu$, and hence $Lu=-4\pi D_ hu$, is a bounded function with the asserted bound.

The function $u$ is continuous, as a uniform limit of the functions $P_\kappa (F_ m)$, which are continuous by proposition 16.15.21; so proposition 16.14.9 shows that $u\in C^{1,\alpha }(X)$ for every $0<\alpha <1$. Finally, $u$ is a continuous $\kappa $-psh function with bounded metric Laplacian, so by proposition 16.14.6, applied with $U=X$, $\theta _ j=\kappa $, and $u$ in place of both $u_ j$ and $f_ j$ for every $j$, the coefficients of $\kappa +\mathrm{dd}^{\mathrm{c}}u$ are bounded on every coordinate ball $W\Subset X$, hence on $X$. $\square$