Holomorphic extension of integrable CR-functions from part of the boundary of the domain (Q803336)

Let D be a bounded domain in \({\mathbb{C}}^ n\), \(n>1\); \(\bar D\) be the holomorphic convex compact domain, \(H(\bar D)\) be the set of holomorphic functions in \(\bar D.\) Let a compact \(K\subset \bar D\) be such that \[ K=\hat K_{\bar D}=\{z\in \bar D: | h(z)| \leq \max_{K}| h|,\quad h\in H(\bar D)\}, \] \(\partial D\setminus K\in C^ 2\) and \(\partial D\setminus K\) is connected. Theorem 1. Let \(\phi\) be a CR-function, \(\phi \in L_{loc}(\partial D\setminus K)\). Then there exists a holomorphic function F on \(D\setminus K\) such that its boundary values on \(\partial D\setminus K\) coincide with \(\phi\). With the help of this theorem the author obtains a result about the elimination of singularities of integrable CR-functions.