Karakalpak Scientific Journal


In the theory of functions of one complex variable, Morer's theorem is known, which is inverse in some sense to the classical Cauchy theorem. On the complex plane, results on functions with the one-dimensional property of holomorphic continuation are trivial, and Morer's boundary theorems are absent. We note that the ordinary (non-boundary) Morera theorems in domains of space are well known. The first result related to our topic was obtained by Agranovsky M.L. and Valsky R.E. [1], who studied functions with the one-dimensional property of holomorphic continuation in a ball. The proof was based on the properties of the automorphism group of a ball. By Stout E.L., who used the complex Radon transform, the Agranovsky and Walski theorem was carried over to arbitrary bounded domains with a smooth boundary [4]. An alternative proof of Stout's theorem was obtained by Kytmanov A.M. [6], who applied the Bochner – Martinelli integral. The idea of using integral representations (Bochner – Martinelli, Cauchy – Fantappier, logarithmic residue) proved to be useful in studying functions with the one-dimensional property of holomorphic continuation along complex curves. A weaker property than the property of one-dimensional holomorphic continuation is the so-called Morera property. It consists in the vanishing of the integrals of a given function over the intersection of the boundary of the region with complex lines (complex planes). Greenberg E. [3] studied functions with the Morera property in a ball. Globevnik I. and Stout E.L. [4] obtained Morer's boundary theorem for an arbitrary bounded domain. In this paper, we consider the Morera boundary theorem for one Siegel domain of the second kind defined in the space of complex rectangular matrices. The proof is based on the properties of the Poisson integral for the Siegel domain, and the Cayley transform is also used.

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