Functions with the one-dimensional holomorphic extension property

Simona G. Myslivets

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 4, pp. 439-443.

Résumé

In this paper we consider different families of complex lines, sufficient for holomorphic extension the functions $f$, defined on the boundary of a domain $ D\subset\mathbb C^n $, $n>1$, into this domain, and possessing the one-dimensional holomorphic extension property along this complex lines.

Détail

Keywords: holomorphic extension.

Bibliographie

[1] M.L. Agranovsky, R.E. Valsky, “Maximality of invariant algebras of functions”, Sib. Mat. J., 12:1 (1971), 3–12 | MR

[2] E.L. Stout, “The boundary values of holomorphic functions of several complex variables”, Duke Math. J., 44:1 (1977), 105–108 | DOI | MR

[3] L.A. Aizenberg, A.P. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis, Translations of Mathematical Monographs, 58, American Mathematical Society, Providence, RI, 1983 | DOI | MR | Zbl

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

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

[6] A.M. Kytmanov, S.G. Myslivets, “On the families of complex lines, sufficient for holomorphic continuations”, Math. Notes, 83:4 (2008), 545–551 | MR | Zbl

[7] A.M. Kytmanov, S.G. Myslivets, V.I. Kuzovatov, “Families of complex lines of the minimal dimension, sufficient for holomorphic continuation of functions”, Sib. Math. J., 52:2 (2011), 256–266 | DOI | MR | Zbl

[8] M. Agranovsky, “Propagation of boundary $CR$-foliations and Morera type theorems for manifolds with attached analytic discs”, Advan. in Math., 211:1 (2007), 284–326 | DOI | MR | Zbl

[9] M. Agranovsky, “Analog of a theorem of Forelli for boundary values of holomorphic functions on the unit ball of $\mathbb C^n$”, J. d'Anal. Math., 13:1 (2011), 293–304 | DOI | MR

[10] L. Baracco, “Holomorphic extension from the sphere to the ball”, J. Math. Anal. Appl., 388:2 (2012), 760–762 | DOI | MR | Zbl

[11] J. Globevnik, “Small families of complex lines for testing holomorphic extendibility”, Am. J. Math., 134:6 (2012), 1473–1490 | DOI | MR | Zbl

[12] L. Baracco, “Separate holomorphic extension along lines and holomorphic extension from the sphere to the ball”, Am. J. Math., 135:2 (2013), 493–497 | DOI | MR | Zbl

[13] J. Globevnik, “Meromorphic extensions from small families of circles and holomorphic extensions from spheres”, Trans. Am. Math. Soc., 364:11 (2012), 5857–5880 | DOI | MR | Zbl

[14] A.M. Kytmanov, S.G. Myslivets, “Holomorphic extension of functions along finite families of complex linea in a ball”, J. Sib. Fed. Univ. Math. and Phys., 5:4 (2012), 547–557

[15] A.M. Kytmanov, S.G. Myslivets, “An analog of the Hartogs theorem in a ball of $\mathbb C^n$”, Math. Nahr., 288:2–3 (2015), 224–234 | MR | Zbl

[16] A.M. Kytmanov, S.G. Myslivets, Multidimensional Integral Representations, Springer Inter. Publ., Switzarland, 2015 | MR | Zbl

[17] A.M. Kytmanov, S.G. Myslivets, “Holomorphic extension of functions along finite families of complex linea in a $n$-circular domain”, Sib. Math. J., 57:4 (2016), 618–631 | DOI | MR | Zbl

[18] A.M. Kytmanov, S.G. Myslivets, “On functions with one-dimensional holomorphic extension property in circular domains”, Math. Nachr., 292:6 (2019), 1321–1332 | DOI | MR | Zbl

[19] J.J. Carmona, P.V. Paramonov, K.Yu. Fedorovskii, “On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions”, Sb. Math., 193 (2002), 1469–1492 | DOI | MR | Zbl