Functions with the One-Dimensional Holomorphic Extension Property

Abstract

The first result related to our subject was obtained by Agranovskii and Val’sky in (Sib. Math. J. 12, 1–7, 1971), who studied functions with the one-dimensional holomorphic extension property in a ball. Their proof was based on the properties of the automorphism group of the ball. Stout (Duke Math. J. 44, 105–108, 1977) used the complex Radon transform to extend the Agranovskii–Val’sky theorem to arbitrary bounded domains with smooth boundaries. An alternative proof of the Stout theorem was suggested in Integral Representations and Residues in (Multidimensional Complex Analysis. AMS, Providence, 1983) by Kytmanov, who applied the Bochner–Martinelli integral. The idea of using integral representations (those of Bochner–Martinelli, Cauchy–Fantappie, and the logarithmic residue) turns out to be useful in studying functions with a one-dimensional holomorphic extension property along complex curves (Kytmanov and Myslivets, Sib. Math. J. 38, 302–311, 1997; Kytmanov and Myslivets, J. Math. Sci. 120, 1842–1867, 2004).