On the Families of Complex Lines Sufficient for Holomorphic Continuation of Functions Defined on a Domain Boundary

This paper presents selected results on the study of the onedimensional property of holomorphic continuations of functions defined on the boundary of a bounded domain in Cn. This paper presents selected results on holomorphic continuation of functions defined on the boundary of a bounded domain D ⊂ C, n > 1, into this domain. We consider functions with a one-dimensional holomorphic continuation property along families of complex lines. On a complex plane C, results for functions with a one-dimensional holomorphic continuation property are trivial; therefore, results of this work are essentially multidimensional. The first result concerning the subject of our interest was obtained by M. L. Agranovsky and R. E. Val’ski in [2], who studied functions with a onedimensional holomorphic continuation property in a sphere. Their approach was based on the automorphism group properties of a sphere. E.L. Stout in [12] used complex Radon transformation to generalize the Agranovsky and Val’ski 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 integral representations (Bochner-Martinelli, Cauchy-Fantappiè, logarithmic residue) has turned out to be useful for studying of functions with one-dimensional holomorphic continuation property along complex lines and curves [7,8]. A review of the results on the subject under discussion can be found in [9]. Let D be a bounded domain in C, n > 1, with connected smooth boundary ∂D (of a class C). Let us formulate E.L. Stout’s result [12]. We will be concerned with one-dimensional complex lines l having the form (1) l = {ζ : ζj = zj + bjt, j = 1, . . . , n, t ∈ C}, and passing through a point z ∈ C in the direction of a vector b ∈ CPn−1 (direction b is defined accurate to multiplication by a complex number λ = 0). 2010 Mathematics Subject Classification. Primary 32A26, 32A40. The work of the authors was supported by RFBR, grant 11-01-00852, and NSH, grant 7347.2010.1. c ©2013 A. M. Kytmanov, S. G. Myslivets