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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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##### References:
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