Extension and decomposition with local boundary regularity properties in pseudoconvex domains
Let D be a strictly pseudoconvex domain in \({\mathbb{C}}^ n\). The author studies the following extension and decomposition problems. Let M' be a closed complex submanifold of \({\mathbb{C}}^ n\) which intersects \(\partial D\) transversally. Put \(M=M'\cap D\). Let E be an open subset of \(\partial M\), \(f\in {\mathcal O}(M)\cap C^ k(M\cup E).\) Is it possible to extend f to a function \(F\in {\mathcal O}(D)\cap C^ k(D\cup U),\) where U is a neighbourhood of E in \(\partial D?\) Let E as above, \(s\in D, f\in {\mathcal O}(D)\cap C^ k(D\cup E), f(s)=0\). Is it possible to find \(f_ 1,...,f_ n\in {\mathcal O}(D)\cap C^ k(D\cup E)\) such that \(f(z)=(z_ 1-s_ 1)f_ 1(z)+...+(z_ n-s_ n)f_ n(z),\) \(z\in D?\) The author obtains the complete solutions of the above two problems in the case \(k=\infty\) and some partial results if \(k<\infty\). He considers also the case when \(C^ k\)-regularity is substituted by some Lipschitz- type regularity conditions and presents some applications of his results to problems concerning the approximation of holomorphic functions.
scientific article; zbMATH DE number 3916711