Multidimensional boundary analog of the hartogs theorem in circular domains

This paper presents some results related to the holomorphic extension of functions, defined on the boundary of a domain D ⊂ C, n > 1, into this domain. We consider a functions with the one-dimensional holomorphic extension property along the complex lines. The first result related to our subject was obtained M.L.,Agranovsky and R.E.Valsky in [1], who studied functions with the one-dimensional holomorphic continuation property into a ball. The proof was based on the properties of the automorphism group of a sphere. E. L. Stout in [2] used the 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 using the Bochner–Martinelli integral. The idea of using the integral representations (Bochner–Martinelli, Cauchy–Fantappiè, logarithmic residue) has been useful in the study of functions with the one-dimensional holomorphic continuation property (see review [4]). The question of finding different families of complex lines sufficient for holomorphic extension was put in [5]. As shown in [6], a family of complex lines passing through a finite number of points, generally speaking, is not sufficient. Thus, a simple analog of the Hartogs theorem should be not expected. Various other families are given in [7–11]. In [12–16] it is shown that for holomorphic extension of continuous functions defined on the boundary of ball,there are enough n+ 1 points inside the bal, not lying on a complex hyperplane. This result was generalized by the authors n-circular domains.