Pome

Lemma 6.15.1C2732E72E566

Lemma 6.15.1.C2732E72E566 Let $\Omega \subset \mathbb C^ n$, $n\geq 1$, be a bounded pseudoconvex open set, let $\psi \in \mathrm{PSH}(\Omega )$, and let $\varrho $ be a measurable function on $\Omega $ with $c_\varrho \leq \varrho \leq C_\varrho $ for positive constants $c_\varrho $ and $C_\varrho $. Put

\[ \mathcal H=\Bigl \{ F\in \mathcal O(\Omega ):\| F\| ^2=\int _\Omega |F|^2\mathrm e^{-\psi }\varrho \, \mathrm dV<\infty \Bigr \} ,\qquad \langle F,G\rangle =\int _\Omega F\bar G\, \mathrm e^{-\psi }\varrho \, \mathrm dV. \]

  1. For every compact set $K\subset \Omega $ there is a constant $C_ K$ such that $|F(z)|\leq C_ K\| F\| $ for all $F\in \mathcal H$ and $z\in K$. The space $\mathcal H$ is a separable Hilbert space, and it has an orthonormal basis, that is, an orthonormal family $(e_\ell )$, finite or countable, whose linear span is dense. For every such family and every $F\in \mathcal H$, the partial sums of $\sum _\ell \langle F,e_\ell \rangle e_\ell $ converge to $F$ in norm and locally uniformly on $\Omega $, and $\sum _\ell |\langle F,e_\ell \rangle |^2=\| F\| ^2$.

  2. The germs of the elements of $\mathcal H$ generate $\mathcal I(\psi )$ at every point of $\Omega $, and near each point finitely many elements of $\mathcal H$ already generate $\mathcal I(\psi )$.

  3. Let $x\in \Omega $, and let $g_1,\ldots ,g_ s\in \mathcal H$ generate $\mathcal I(\psi )$ on a neighborhood of $x$. Then there are a neighborhood $V$ of $x$ and a constant $C$ such that every $F\in \mathcal H$ can be written as $F=\sum _ ig_ ih_ i$ on $V$ with $h_ i\in \mathcal O(V)$ and $\sup _ V|h_ i|\leq C\| F\| $.

Proof. We regard $\Omega $ as a complex manifold, with the single chart given by the identity map. Then $\psi $ is a weight on $\Omega $ in the sense of section 6.8, the set $\Omega $ satisfies the hypotheses of theorem 6.8.9, and $\mathcal H$ is the space $\mathcal H_\psi (\Omega ,\varrho )$ of that theorem, with the same inner product. Hence (1) is contained in theorem 6.8.9(1) and (2), and (2) follows from theorem 6.8.9(3), because an initial segment of an enumerated orthonormal basis is a finite family of elements of $\mathcal H$. Finally, (3) is the first assertion of lemma 6.8.10, applied to the weight $\psi $ on $\Omega $. $\square$

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