Lemma 6.8.103B37857D5B64

Proof. We write $\mathcal H=\mathcal H_\varphi (\Omega ,\varrho )$ and $\| \cdot \| =\| \cdot \| _{\varphi ,\varrho }$. Choose balls $V_1\supset V_2\supset \cdots $ centered at $x$, with radii tending to zero, inside the neighborhood on which the $g_ i$ generate $\mathcal I(\varphi )$. For positive integers $j$ and $k$ let $S_{j,k}$ be the set of $F\in \mathcal H$ that can be written as $F=\sum _ ig_ ih_ i$ on $V_ j$ with $h_ i\in \mathcal O(V_ j)$ and $\sup _{V_ j}|h_ i|\leq k$. These sets cover $\mathcal H$: by theorem 6.8.9(3), the germ at $x$ of each $F\in \mathcal H$ lies in $\mathcal I(\varphi )_ x$, so $F=\sum _ ig_ ih_ i$ on some neighborhood of $x$ with holomorphic $h_ i$, and these coefficients are bounded on a smaller ball $V_ j$. We show that each $S_{j,k}$ is closed and then apply Baire's theorem.

Let $F_ l\in S_{j,k}$ converge to $F$ in $\mathcal H$, with coefficients $h_{i,l}$. Then $F_ l\to F$ locally uniformly on $\Omega $, by theorem 6.8.9(1). The coefficients are bounded by $k$ on $V_ j$. Let $K\subset V_ j$ be compact, and let $\varepsilon >0$ be such that the closed polydiscs of radius $2\varepsilon $ centered at points of $K$ lie in $V_ j$. For $y\in K$ and $y'\in B(y,\varepsilon )$, the closed polydisc of radius $\varepsilon $ centered at $y'$ then lies in $V_ j$, and the coefficient bound in (5.2.1) on this polydisc bounds all first partial derivatives of all $h_{i,l}$ at $y'$ by $k/\varepsilon $. So the $h_{i,l}$ are Lipschitz with a common constant on the balls of radius $\varepsilon $ centered at points of $K$, and they are equicontinuous on $K$. By proposition 2.7.4, applied on a compact exhaustion of $V_ j$, and a diagonal argument, a subsequence converges locally uniformly on $V_ j$ to continuous functions $h_ i$ with $|h_ i|\leq k$. Locally uniform convergence implies convergence in the sense of distributions, so $\bar\partial h_ i=0$, and the continuous functions $h_ i$ are holomorphic by lemma 6.4.10(1) (if $n=0$, every function on $V_ j$ is holomorphic). Passing to the limit, we obtain $F=\sum _ ig_ ih_ i$ on $V_ j$, that is, $F\in S_{j,k}$.

By Baire's theorem, a complete metric space is not a countable union of closed subsets with empty interior. (If all the sets had empty interior, we could choose closed balls of radii at most $2^{-l}$, each contained in the interior of the previous one and disjoint from the $l$th set, since the interior of a ball minus a closed set with empty interior is a nonempty open set; their centers would form a Cauchy sequence whose limit lies in every ball but in none of the sets.) Hence some $S_{j,k}$ contains a ball $\{ F:\| F-F_0\| <\delta \} $. If $\| F\| <\delta $, then $F_0$ and $F_0+F$ both lie in $S_{j,k}$, and subtracting their representations writes $F$ on $V_ j$ with coefficients bounded by $2k$. For $F\neq 0$ we apply this to $\delta F/(2\| F\| )$ and obtain the first assertion with $V=V_ j$ and $C=4k/\delta $; for $F=0$ we take zero coefficients.

For the second assertion, apply the first part with $\varrho =1$ to the weight $m\varphi $, and let $V$ and $C$ be the resulting neighborhood and constant. Let $a\in V$ with $\varphi (a)>-\infty $. By corollary 6.5.8, applied on the pseudoconvex set $\Omega $ to the plurisubharmonic function $m\varphi $, there is $F\in \mathcal H_{m\varphi }(\Omega )$ with $F(a)=1$ and $\| F\| _{m\varphi }^2\leq C_\Omega \mathrm e^{-m\varphi (a)}$, where $C_\Omega $ depends only on $n$ and the diameter of $\Omega $; if $n=0$, then $F=1$ has these properties with $C_\Omega =1$. Writing $F=\sum _ ig_ ih_ i$ on $V$ and using the Cauchy–Schwarz inequality, we obtain

Taking logarithms gives (6.8.11) at $a$ with $C_ m=m^{-1}\log (rC^2C_\Omega )$. At points where $\varphi =-\infty $ the left side is $-\infty $; by the last display, these points include all common zeros of the $g_ i$. $\square$