Geodesic
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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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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/

[1] E. A. Grinberg, “A boundary analogue of Morera's theorem in the unit ball of $\mathbb{C}^n$”, Proceedings of the American Mathematical Society, 102:1 (1988), 114–116 | DOI | MR | Zbl

[2] Agranovskii M. L., Val'skii R. E., “Maximality of invariant algebras of functions”, Siberian Mathematical Journal, 12:1 (1971), 1–7 | DOI | MR

[3] J. Globevnik, E. L. Stout, “Boundary Morera theorems for holomorphic functions of several complex variables”, Duke Mathematical Journal, 64:3 (1991), 571–615 | DOI | MR | Zbl

[4] D. Govekar, “Morera conditions along real planes and a characterization of CR functions on boundaries of domains in $\mathbb{C}^n$”, Mathematische Zeitschrift, 216 (1994), 195–207 | DOI | MR | Zbl

[5] J. Globevnik, “A boundary Morera theorem”, The Journal of Geometric Analysis, 3:3 (1993), 269–277 | DOI | MR | Zbl

[6] D. Govekar-Leban, “Local boundary Morera theorems”, Mathematische Zeitschrift, 233:2 (2000), 265–286 | DOI | MR | Zbl

[7] Myslivets S. G., “One boundary version of Morera's theorem”, Siberian Mathematical Journal, 42:5 (2001), 952–966 | DOI | MR

[8] Kytmanov A. M., Myslivets S. G., “On families of complex lines sufficient for holomorphic extension”, Mathematical Notes, 83:3–4 (2008), 500–505 | DOI | MR | Zbl

[9] Kytmanov A. M., Myslivets S. G., “Some families of complex lines sufficient for holomorphic extension of functions”, Russian Mathematics, 55:4 (2011), 60–66 | DOI | MR | Zbl

[10] A. M. Kytmanov, S. G. Myslivets, Multidimensional integral representations. Problems of analytic continuation, Springer, Cham, 2015 | DOI | MR | Zbl

[11] Kytmanov A. M., Myslivets S. G., “On a certain boundary analog of Morera's theorem”, Siberian Mathematical Journal, 36:6 (1995), 1171–1174 | DOI | MR | Zbl

[12] A. M. Kytmanov, S. G. Myslivets, “Higher-dimensional boundary analogs of the Morera theorem in problems of analytic continuation of functions”, Journal of Mathematical Sciences, 120:6 (2004), 1842–1867 | DOI | MR | Zbl

[13] H. Federer, Geometric measure theory, Springer, Berlin–Heidelberg, 1969 | DOI | MR | Zbl

[14] E. Martinelli, “Sopra una dimonstrazione de R. Fueter per un teorema di Hartogs”, Commentarii Mathematici Helvetici, 15 (1942), 340–349 (in Italian) | DOI | MR

[15] Kytmanov A. M., The Bochner–Martilelli integral and its applications, Birkhauser, Basel, 1995 ; A. M. Kytmanov, Integral Bokhnera-Martinelli i ego primeneniya, Nauka, Novosibirsk, 1992 | DOI | MR

[16] Dzhumabaev D. Kh., “Holomorphic continuation of functions from closed hypersurfaces with singular edges”, Russian Mathematics, 56:1 (2012), 9–18 | DOI | MR | Zbl