Geodesic
Fine-analytic functions in $\mathbb{C}^n$
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 4, pp. 444-448.

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In this paper we study class of fine-analytic functions in the multidimensional space $\mathbb{C}^n$. The definition of fine-analytic functions in the multidimensional case differs somewhat from the well-known definition of fine-analytic functions on the plane. We give a relationship between classical notion of fine-analyticity and fine-analyticity in $\mathbb{C}^n$.
Keywords: Gonchar class, finite order functions, rational approximation, fine-analytic functions
Mots-clés : pluripolar sets. 
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Azimbai Sadullaev. Fine-analytic functions in $\mathbb{C}^n$. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 4, pp. 444-448. http://geodesic.mathdoc.fr/item/JSFU_2019_12_4_a5/

[1] E. Bedford, B.A. Taylor, “Fine Topology, S̆ilov Boundary and ${{\left( d{{d}^{c}} \right)}^{n}}$”, Journal of Func. Analysis, 72:2 (1987), 225–251 | DOI | MR | Zbl

[2] M. Brelot, On Topologies and Boundaries in Potential Theory, Lecture Notes in Math., 175, Springer-Verlage, Berlin, 1971 | DOI | MR | Zbl

[3] A. Edigarian, J. Wiegerinck, “The pluripolar hull of the graph of a holomorphic function with polar singularities”, Indiana Univ. Math. J., 52 (2003), 1663–1680 | DOI | MR | Zbl

[4] A. Edigarian, J. Wiegerinck, “Graphs that are not complete pluripolar”, Proc. Amer. Math. Soc., 131 (2003), 2459–2465 | DOI | MR | Zbl

[5] A. Edigarian, S. El Marzguioui, J. Wiegerinck, “The image of a finely holomorphic map is pluripolar”, Ann. Polon. Math., 97 (2010), 137–149 | DOI | MR | Zbl

[6] T. Edlund, “Complete pluripolar curves and graphs”, Ann. Polon. Math., 84 (2004), 75–86 | DOI | MR | Zbl

[7] T. Edlund, B. Joricke, “The pluripolar hull of a graph and fine analytic continuation”, Ark. Mat., 44 (2006), 39–60 | DOI | MR | Zbl

[8] B. Fuglede, “Finely harmonic mappings and finely holomorphic functions”, Ann. Acad. Sci. Fennicæ, 2 (1976), 113–127 | DOI | MR | Zbl

[9] B. Fuglede, “Sur les functions finement holomorphes”, Ann. Inst. Fourier (Grenoble), 31:4 (1981), 57–88 | DOI | MR | Zbl

[10] B. Fuglede, “Finely holomorphic functions. A survey”, Rev. Roumaine Math. Pures Appl., 33 (1988), 283–295 | MR | Zbl

[11] N.S. Landkof, Foundations of Modern Potential theory, Springer–Verlage, 1972 | MR | Zbl

[12] N. Levenberg, G. Martin, E.A. Poletsky, “Analytic disks and pluripolar sets”, Indiana Univ. Math. J., 41 (1992), 515–532 | DOI | MR | Zbl

[13] N. Levenberg, E.A. Poletsky, “Pluripolar hulls”, Michigan Math. J., 46 (1999), 151–162 | DOI | MR | Zbl

[14] S. El Marzguioui, J. Wiegerinck, “Continuity properties of finely plurisubharmonic functions”, Indiana Univ Math J., 59:5 (2010), 1793–1800 | DOI | MR | Zbl

[15] A.S. Sadullaev, “Plurisubharmonic measures and capacities on complex manifolds”, Russian Math. Surveys, 36:4 (1981), 61–119 | DOI | MR | Zbl

[16] A.S. Sadullaev, Z.S. Ibragimov, “The class R and finely analytic functions”, Sbornik: Mathematics, 209:8 (2018), 1234–1247 | DOI | MR | Zbl

[17] A.S. Sadullaev, Z.S. Ibragimov, “Class R of Gonchar in ${{\mathbb{C}}^{n}}$”, Algebra, Complex Analysis, and Pluripotential, USUZCAMP 2017 (Urgench, Uzbekistan, August 8–12, 2017), Springer Proceedings in Mathematics and Statistics, 264, 191–205 | DOI | MR | Zbl

[18] J. Wiegerinck, “Pluri fine potential theory”, Ann. Polon. Math., 106 (2012), 275–292 | DOI | MR | Zbl