Holomorphic Extension of Continuous Functions Along Finite Families of Complex Lines in a Ball

Abstract

This paper presents some results related to the holomorphic extension of functions f , defined on the boundary of a ball B ⊂ C, n > 1, in this ball. We deal with a functions with the one-dimensional holomorphic extension property along the complex lines. The first result related to our subject was received by M.L.Agranovsky and R.E.Valsky in [1], who studied the functions with a one-dimensional holomorphic continuation property on the boundary of a ball. The proof was based on the automorphism group properties of a sphere. E.L.Stout in [2] used complex Radon transformation to generalize the Agranovsky and Valsky theorem for an arbitrary bounded domain with a smooth boundary. An alternative proof of the Stout theorem was obtained by A.M.Kytmanov in [3] by applying the Bochner–Martinelli integral. The idea of using the integral representations (Bochner–Martinelli, Cauchy–Fantappie, multidimensional logarithmic residue) has been useful in the study the functions with onedimensional holomorphic continuation property (see review [4]). The problem of finding the different families of complex lines, sufficient for holomorphic extension was posed in [5]. Clearly, the family of complex lines passing through one point is not enough. As shown in [6], the family of complex lines passing through a finite number of points also, generally speaking, is not sufficient. In [6] we proved that the family of complex lines crossing the germ of generic manifold γ, is sufficient for the holomorphic extension. In [7] we consider a family of complex lines passing through the germ of a complex hypersurface. In particular, C, it can be any real analytic curve. Various other families are given in [8–11]. We note here the work [9, 11], where it is shown that a family of complex lines passing through somehow located a finite number of points is sufficient for holomorphic extension. But it is approved only for real-analytic or infinitely differentiable functions defined on the boundary. Then in C Agranovsky and Globevnik for real-analytic