Integral representation of a CR-function and its holomorphic continuation (Q1363937)

The author addresses the problem of extendibility of a CR-function from a piece of a hypersurface in \(\mathbf C^n\) to a holomorphic function on one side of the surface. Consider a simply connected domain \(D\) in \(\mathbf C^n\) obtained by slicing the tip off a cone with a smooth hypersurface \(S\), and suppose \(f\) is a continuous CR-function given on \(S\). The author announces a necessary and sufficient condition, phrased in terms of a one-parameter family of Cauchy-Fantappiè integrals, for the existence of a holomorphic function \(F\) in \(D\) whose restriction to \(S\) equals \(f\). When the condition is satisfied, there is a Carleman-type formula that reconstructs \(F\) inside \(D\) from a limit of integrals of \(f\) over \(S\). The special case when the dimension \(n=1\) and the cone is a half-space recovers a theorem of \textit{V. A. Fok} and \textit{F. M. Kuni} [Dokl. Akad. Nauk SSSR 127, 1195-1198 (1959; Zbl 0092.45501); English translation in Sov. Phys., Dokl. 4, 871-874 (1960)].