On functions with the boundary Morera property in domains with piecewise-smooth boundary
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 31 (2021) no. 1, pp. 50-58
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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.
@article{VUU_2021_31_1_a3,
author = {A. M. Kytmanov and S. G. Myslivets},
title = {On functions with the boundary {Morera} property in domains with piecewise-smooth boundary},
journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
pages = {50--58},
publisher = {mathdoc},
volume = {31},
number = {1},
year = {2021},
language = {en},
url = {http://geodesic.mathdoc.fr/item/VUU_2021_31_1_a3/}
}
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%0 Journal Article %A A. M. Kytmanov %A S. G. Myslivets %T On functions with the boundary Morera property in domains with piecewise-smooth boundary %J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki %D 2021 %P 50-58 %V 31 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/item/VUU_2021_31_1_a3/ %G en %F VUU_2021_31_1_a3
A. M. Kytmanov; S. G. Myslivets. On functions with the boundary Morera property in domains with piecewise-smooth boundary. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 31 (2021) no. 1, pp. 50-58. http://geodesic.mathdoc.fr/item/VUU_2021_31_1_a3/