Extension of CR-forms and related problems
Let D be a bounded domain of \({\mathbb{C}}^ n\), \(n\geq 3\) and let \(\partial D=S\cup A\) when S is a real \(C^ 1\) hypersurface with boundary \(\partial S\) and A is a non empty piecewise \(C^ 1\) real hypersurface. Assume there exists a family \(\{V_ j\}_{j\in {\mathbb{N}}}\) of (n-3)-complete open subsets such that \(\bar V{}_{j+1}\subset V_ j\) and \((\cup_{j}V_ j)\cap \bar D=\bar A\) (and this is always true if A is contained in the zero-set of a pluriharmonic function). Then the following is proved: Every Lipschitz CR-form of type (p,0) on \(\overset\circ S\) can be extended on D by a (p,0)-form holomorphic on D and continuous on \(D\cup\overset\circ S\). For CR-forms of type (p,q) the author proves a jump theorem: assume that \(V_ j\) is (n-s-2)-complete; then every regular CR-form of type (p,q) where \(0\leq p\leq n\), \(0\leq q\leq s-1\) is the jump across S of two \(\partial\)-closed forms on D and \({\mathbb{C}}^ n\setminus (\bar D\cup (\cap_{j}\bar V_ j))\), respectively. When S is assumed to be strictly pseudoconvex, as an application of this result, the author obtains an extension theorem for CR-forms of type (p,q).
scientific article; zbMATH DE number 4015145
scientific article