Geodesic
The Dirichlet problem for polyanalytic functions
Sbornik. Mathematics, Tome 200 (2009) no. 10, pp. 1473-1493 Cet article a éte moissonné depuis la source Math-Net.Ru

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Connections between the boundary behaviour of polyanalytic functions and the structure of the boundary are investigated. In particular, a Jordan domain with Lipschitz boundary is constructed which is regular for the Dirichlet problem in the class of bianalytic functions. Bibliography: 14 titles.
Keywords: polyanalytic functions, boundary properties, Luzin-Privalov construction, univalent functions, lacunary series, Rudin-Carleson theorem. 
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M. Ya. Mazalov. The Dirichlet problem for polyanalytic functions. Sbornik. Mathematics, Tome 200 (2009) no. 10, pp. 1473-1493. http://geodesic.mathdoc.fr/item/SM_2009_200_10_a3/

[1] M. B. Balk, “Polyanalytic functions and their generalizations”, Complex analysis I, Encyclopaedia Math. Sci., 85, Springer-Verlag, Berlin, 1997, 195–253 | MR | MR | Zbl | Zbl

[2] I. I. Privalov, Granichnye svoistva analiticheskikh funktsii, GITTL, M.–L., 1950 | MR | Zbl

[3] Ph. J. Davis, The Schwarz function and its applications, The Mathematical Association of America, Washington, 1974 | MR | Zbl

[4] M. Ya. Mazalov, “An example of a nonconstant bianalytic function vanishing everywhere on a nowhere analytic boundary”, Math. Notes, 62:4 (1997), 524–526 | DOI | MR | Zbl

[5] J. J. Carmona, P. V. Paramonov, K. Yu. Fedorovskii, “On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions”, Sb. Math., 193:10 (2002), 1469–1492 | DOI | MR | Zbl

[6] W. Rudin, Functional analysis, McGraw-Hill, New York–Düsseldorf–Johannesburg, 1973 | MR | MR | Zbl

[7] Ch. Pommerenke, Univalent functions, Vandenhoeck Ruprecht, Göttingen, 1975 | MR | Zbl

[8] K. Yu. Fedorovskii, “Uniform $n$-analytic polynomial approximations of functions on rectifiable contours in $\mathbb C$”, Math. Notes, 59:4 (1996), 435–439 | DOI | MR | Zbl

[9] W. Rudin, “Boundary values of continuous analytic functions”, Proc. Amer. Math. Soc., 7:5 (1956), 808–811 | DOI | MR | Zbl

[10] L. Carleson, “Representations of continuous functions”, Math. Z., 66:1 (1956), 447–451 | DOI | MR | Zbl

[11] P. Koosis, Introduction to $H^p$ spaces with an appendix on Wolff's proof of the corona theorem, London Math. Soc. Lecture Note Ser., 40, Cambridge Univ. Press, Cambridge–New York, 1980 | MR | MR | Zbl | Zbl

[12] G. M. Goluzin, Geometric theory of functions of a complex variable, Transl. Math. Monogr., 26, Amer. Math. Soc., Providence, RI, 1969 | MR | MR | Zbl | Zbl

[13] A. I. Markushevich, Teoriya analiticheskikh funktsii, t. 2. Dalneishee postroenie teorii, 2-e izd., Nauka, M., 1968 | Zbl

[14] J. B. Garnett, Bounded analytic functions, Pure Appl. Math., 96, Academic Press, New York–London, 1981 | MR | MR | Zbl | Zbl