In this question I'll be talking about the proof of Behnke-Stein theorem as given in B. V. Shabat's book "Introduction to Complex Analysis" (its English translation can be found here, but it seems like it has been quite noticably restructured; I am using a Polish translation). I believe there is a major gap in this proof, but I might be missing something relatively easy. I'll try to give a bit of detail about the proof because it might be difficult to access.
Theorem: An increasing union $D$ of a family $D_1\subseteq D_2\subseteq\dots$ of domains of homolorphy is a domain of holomorphy, as long as it's bounded.
The proof is split into four parts, but only the transition between the first and the second is relevant to my question.
First, it is shown that $D$ is an increasing union of analytic polyhedra. This is done roughly as follows: by passing to a suitable subsequence, we may assume that $D_i$ satisfy the condition $$\sup_{z\in\partial D_i}\varrho(z,\partial D_{i+1})<\varrho(D_{i-1},\partial D_i),$$ where $\varrho$ is the Euclidean metric (existence of such a subsequence is more of a technicality). The following characterizations of domains of holomorphy and $F$-convexity (analogue of holomorphic convexity with the family of all holomorphic functions replaced by a subfamily $F$) have been proven earlier:
- $G$ is a domain of holomorphy iff it's holomorphically convex,
- For "nice" (closed under differentiation and taking powers) families $F$, $G$ is $F$-convex iff for each $K\Subset G$, $\varrho(\hat K_F,\partial G)=\varrho(K,\partial G)$ ($\hat K_F$ is $F$-convex hull of $K$).
We use them to construct an analytic polyhedron $D_{i-1}\Subset\Pi_i\Subset D_i$ (I'm omitting the details; the functions $f_j$ involved are not constrained in any way apart from being holomorphic on $D_{i+1}$, which concludes the first step.
In the second step, we show that $H(D)$ (functions holomorphic on $D$) is dense in $H(\Pi_i)$. Simple argument shows that it's enough to show $H(\Pi_{i+1})$ is dense in $H(\Pi_i)$. For that, Weil decomposition is used (we take intial terms of the last formula in the linked page).
But here is the problem - Bergman-Weil formula was only proven for (and I suspect is only valid for) Weil polyhedra (which satisfy additional constraints on its boundary). This to me seems like a gap in the proof which cannot be easily salvaged. I have a few ideas on how the proof could be modified to fix this problem, but I can't think of a way how to prove any of these:
- $\Pi_i$ are in fact Weil domains. Since $f_j$ are about as general as they can be, even if that's true, I doubt it's easy to show.
- Weil decomposition works for arbitrary analytic polyhedra. I don't quite believe that, but Shabat claims to prove it for arbitrary analytic polyhedra, even though the proof uses Bergman-Weil representation, valid only for Weil domains
- The approximation result works for arbitrary analytic polyhedra. This is the most plausible explanation for me, proof of which would most likely require some slight work-around.
Does anyone have an idea on how to fix this argument?