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Russia Krasnoyarsk
Year
2021
Volume
31
Issue
1
Pages
50-58
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Section Mathematics
Title On functions with the boundary Morera property in domains with piecewise-smooth boundary
Author(-s) Kytmanov A.M.a, Myslivets S.G.a
Affiliations Siberian Federal Universitya
Abstract The problem of holomorphic extension of functions defined on the boundary of a domain into this domain is actual in multidimensional complex analysis. It has a long history, starting with the proceedings of Poincaré and Hartogs. This paper considers continuous functions defined on the boundary of a bounded domain $ D $ in $ \mathbb C ^ n $, $ n> 1 $, with piecewise-smooth boundary, and having the generalized boundary Morera property along the family of complex lines that intersect the boundary of a domain. Morera property is that the integral of a given function is equal to zero over the intersection of the boundary of the domain with the complex line. It is shown that such functions extend holomorphically to the domain $ D $. For functions of one complex variable, the Morera property obviously does not imply a holomorphic extension. Therefore, this problem should be considered only in the multidimensional case $ (n> 1) $. The main method for studying such functions is the method of multidimensional integral representations, in particular, the Bochner-Martinelli integral representation.
Keywords bounded domain with piecewise-smooth boundary, continuous function, Morera property, Bochner-Martinelli integral representation
UDC 517.55
MSC 32A25, 32A40
DOI 10.35634/vm210104
Received 16 December 2020
Language English
Citation Kytmanov A.M., Myslivets S.G. On functions with the boundary Morera property in domains with piecewise-smooth boundary, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 1, pp. 50-58.
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