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Condition R and holomorphic mappings of domains with generic corners. (English) Zbl 1307.32015
We say that a domain \(D\subset\mathbb C^n\) satisfies Condition R if the orthogonal Hilbert space projection \(P:L^2(D)\longrightarrow L^2(D)\cap\mathcal O(D)\) is such that \(P(L^2(D)\cap\mathcal C^\infty(\overline D))\subset\mathcal C^\infty(\overline D)\).
We say that a bounded domain \(\varOmega\subset\mathbb C^n\) is a domain with generic corners if \(\varOmega=\bigcup_{j=1}^N\varOmega_j\), where \(\varOmega_1,\dots,\varOmega_N\) are smooth domains, the intersection \(\partial\varOmega_j\) and \(\partial\varOmega_k\) is transversal for all \(j\neq k\), and for each set \(S\subset\{1,\dots,N\}\), if \(B_S:=\bigcap_{j\in S}\partial\varOmega_j\neq\emptyset\), then \(B_S\) is a CR-manifold of dimension \(n-\# S\). Assume that \(D\subset\mathbb C^n\) is a pseudoconvex domain with generic corners which satisfies Condition R. The following two theorems are the main results of the paper.
(1)
For any biholomorphic mapping \(f:D\longrightarrow G\) the following conditions are equivalent:
(i)
\(f\) extends to a \(\mathcal C^\infty\)-diffeomorphism \(\overline D\longrightarrow\overline G\);
(ii)
\(G\) is a domain with generic corners and satisfies Condition R.
(2)
Let \(G\subset\mathbb C^n\) is a bounded domain of the form \(G=G_1\times\dots\times G_k\), where \(G_1,\dots, G_k\) are smooth. Then every proper holomorphic mapping \(f:D\longrightarrow G\) extends continuously to \(\overline D\) in such a way that the extension is \(\mathcal C^\infty\)-smooth on \(\partial D^{\text{reg}}:=\{z\in\partial D:\) there exists at most one \(j\in\{1,\dots,N\}\) with \(z\in\partial\varOmega_j\}\).

MSC:
32H40 Boundary regularity of mappings in several complex variables
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