Continuation of bounded holomorphic functions from certain subvarieties to weakly pseudoconvex domains (Q1073975)

Let D be a bounded pseudoconvex domain in \({\mathbb{C}}^ n\) with \(C^{\infty}\)-boundary. Let \(\tilde V\) be a subvariety in a neighborhood \(\tilde D\) of \(\bar D\) which intersects \(\partial D\) transversally. Let \(V=\tilde V\cap D\) and \(D=\{z\in \tilde D: \rho (z)<0\}.\) Suppose that \(\tilde V\) is written in the form \(\tilde V=\{z\in \tilde D: h_ 1(z)=...=h_ p(z)=0\},\) where \(h_ 1,...,h_ p\) are holomorphic in \(\tilde D\) and \(\partial h_ 1\wedge...\wedge \partial h_ p\neq 0\) on \(\partial D\cap \tilde V\). In addition, we assume that \(\partial V\) consists of strictly pseudoconvex boundary points of D. Then we have Theorem. There exists a continuous linear operator \(E: H^{\infty}(V)\to H^{\infty}(D)\) satisfying \(Ef|_ V=f\). Moreover Ef\(\in A(D)\) if \(f\in A(V).\) The above theorem is a generalization of the fundamental theorem of \textit{G. M. Henkin} [Izv. Akad. Nauk SSSR, Ser. mat. 36, 540-567 (1972; Zbl 0249.32009)].