Lemma 5.46.2EB97F93DAD74

Proof. Adding a smooth quadratic function and subtracting a constant, we obtain a negative plurisubharmonic function $h$ on $U$ with the same pole set as $v$. Put $h_ M=\max (h,-M)$ for $M\geq 1$, and choose $\psi \in C^\infty _ c(O)$ with $0\leq \psi \leq 1$ and $\psi =1$ near $K$. On a fixed compact neighborhood of $\operatorname {supp}\psi $, the positive measures $\Delta h_ M$ have uniformly bounded mass: choose a nonnegative smooth function $\chi $ with compact support in $U$, equal to one on this neighborhood, and use

Apply the integration-by-parts estimate of the preceding proof to mollifications of $h_ M$. Their absolute values are at most $M$, and their squares are at most $M$ times their absolute values. The uniform bounds for the mass of the Laplacians and for the local $L^1$ norms therefore give, after passing to the weak limit in $W^{1,2}$,

Here $C$ depends on the fixed cutoffs and the local $L^1$ norm of $h$, but not on $M$. The second inequality also uses $h_ M^2\leq M|h|$ in the terms containing $\psi $ and $\nabla \psi $.

The Sobolev function $-\psi h_ M/M$ takes values between zero and one. It equals one on a neighborhood of $K$, since $h$ is upper semicontinuous and therefore $\{ h<-M\} $ is an open neighborhood of $K$. Extend it by zero and mollify with a sufficiently small radius: the support stays in $O$, the function remains equal to one on a smaller neighborhood of $K$, and its Sobolev norm does not increase. Letting $M\to \infty $ gives the required smooth functions. $\square$