Higher-Dimensional Boundary Analogs of the Morera Theorem in Problems of Analytic Continuation of Functions

Abstract

This paper is a review of certain results related to the holomorphic continuation of functions given on the boundary of a bounded domain D ⊂ Cn, n > 1, to this domain. The subject is far from new, but here we speak of functions with a one-dimensional holomorphic continuation property along complex lines and curves and also of boundary higher-dimensional variants of the Morera theorem. On the complex plane C, the results on functions with the one-dimensional holomorphic continuation property are trivial, and there are no boundary Morera theorems. Therefore, most of the results of the paper are essentially higher-dimensional. We note that ordinary (nonboundary) Morera theorems in domains of the space Cn are well known (see, e.g., [10, 26]). The first result pertaining to our topic was obtained in [3] by M. L. Agranovskii and R. E. Val’skii, who studied functions with the one-dimensional holomorphic continuation property in a ball. The proof was based on the properties of the automorphism group of the ball. E. L. Stout [44] has used the complex Radon transform and extended the Agranovskii–Val’skii theorem to arbitrary bounded domains with smooth boundaries. An alternative proof of the Stout theorem was obtained in [16] by A. M. Kytmanov, who has applied the Bochner–Martinelli integral. The idea of using integral representations (of Bochner–Martinelli, Cauchy–Fantappie, and logarithmic residue) turns out to be useful in studying functions with the one-dimensional holomorphic continuation property along complex curves [17,18]. The so-called Morera property is weaker than the one-dimensional holomorphic continuation property. It consists of the vanishing of integrals of a given function over the intersection of the boundary of the domain considered with complex lines (complex planes). E. Grinberg [37] has studied functions with the Morera property in a ball (in fact, this result is contained in the work [3] of M. L. Agranovskii and R. E. Val’skii). J. Globevnik and E. L. Stout [33] have obtained the boundary Morera theorem for an arbitrary bounded domain. A local variant of the Morera theorem was considered by J. Globevnik in [32] and D. Govekar in [36]. In [20], functions with the Morera properties along complex curves were considered. In Sec. 1 of the review, we consider global boundary analogs of the Morera theorem along complex and real planes. In the second section, we present various results related to the Morera property along complex lines. In Sec. 3, we consider different variants of the Morera theorem in a ball. In Sec. 4, we present global analogs of the boundary Morera theorems along complex curves. In Sec. 5, we present local variants of the Morera theorems. Section 6 contains results on functions with the Morera property in classical domains. Section 7 is devoted to the Morera theorems in unbounded domains. In the last section, we formulate certain unsolved problems.