Corollary 6.5.833DEA3F4EDB8

Proof. We prove the second assertion; the first is the case $V=\{ a\} $, $r=n$ and $f=1$, whose slice norm with respect to the counting measure $\mathrm dV_0$ is $\mathrm e^{-p(a)}$. If $r=0$ we take $F=f$, and if $\Omega \cap V=\varnothing $ we take $F=0$.

Assume now that $r\ge 1$ and $\Omega \cap V\ne \varnothing $. Choose $b\in \Omega \cap V$ and unitary affine coordinates $w=(w_1,\ldots ,w_ n)$ on $\mathbb C^ n$ centered at $b$ with $V=\{ w_1=\cdots =w_ r=0\} $, and put $\Omega _ k=\Omega \cap \{ w_1=\cdots =w_ k=0\} $ for $0\le k\le r$, an open subset of a space isomorphic to $\mathbb C^{n-k}$. Unitary coordinates preserve Lebesgue measure. Each $\Omega _ k$ with $k<n$ is pseudoconvex: the restriction of a continuous plurisubharmonic exhaustion function of $\Omega $ to the closed subset $\Omega _ k$ of $\Omega $ is continuous and plurisubharmonic by proposition 5.9.3, applied to the linear embedding of $\{ w_1=\cdots =w_ k=0\} $, and its sublevel sets are compact. By the discussion following definition 6.3.1, every connected component of $\Omega _ k$ is pseudoconvex as well. Since $\Omega $ is open and has diameter at most $L$, we have $|z-b|<L$ for $z\in \Omega $ (a point $z\in \Omega $ with $|z-b|=L$ could be moved slightly away from $b$ inside $\Omega $), so the coordinate $w_ k$ takes its values in $\Delta _ L$ on $\Omega _{k-1}$.

Put $f_ r=f$, let $1\le k\le r$, and assume that $f_ k\in \mathcal O(\Omega _ k)$ has finite norm $\int _{\Omega _ k}|f_ k|^2\mathrm e^{-p}\, \mathrm dV_{2n-2k}$. Let $W$ be a connected component of $\Omega _{k-1}$; ordering the coordinates as $(w_{k+1},\ldots ,w_ n,w_ k)$, we regard $W$ as an open subset of $\mathbb C^{n-k}\times \Delta _ L$ whose slice by $\{ w_ k=0\} $ is the open subset $W\cap \Omega _ k$ of $\Omega _ k$. By proposition 5.9.3, the restriction of $p$ to $W$ is plurisubharmonic or identically $-\infty $. In the second case $\mathrm e^{-p}=\infty $ on $W\cap \Omega _ k$, so the finiteness of the norm of $f_ k$ forces $f_ k=0$ almost everywhere on $W\cap \Omega _ k$, hence everywhere there, since $f_ k$ is continuous, and we put $f_{k-1}=0$ on $W$. In the first case, theorem 6.5.4, applied to $W$ with normal coordinate $w_ k$, gives a holomorphic function $f_{k-1}$ on $W$ extending the restriction of $f_ k$ to $W\cap \Omega _ k$, with

The components $W$ are open, disjoint and cover $\Omega _{k-1}$, and the sets $W\cap \Omega _ k$ are disjoint and cover $\Omega _ k$. Hence these functions define $f_{k-1}\in \mathcal O(\Omega _{k-1})$ extending $f_ k$, and summing over $W$ gives

Proceeding in this way for $k=r,r-1,\ldots ,1$ and multiplying the $r$ estimates, we see that $F=f_0$ has the required properties. $\square$