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Concrete Operators

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Hardy spaces of generalized analytic functions and composition operators

Elodie Pozzi
Published Online: 2018-04-28 | DOI: https://doi.org/10.1515/conop-2018-0002


We present some recent results on Hardy spaces of generalized analytic functions on D specifying their link with the analytic Hardy spaces. Their definition can be extended to more general domains Ω . We discuss the way to extend such definitions to more general domains that depends on the regularity of the boundary of the domain ∂Ω. The generalization over general domains leads to the study of the invertibility of composition operators between Hardy spaces of generalized analytic functions; at the end of the paper, we discuss invertibility and Fredholm property of the composition operator C on Hardy spaces of generalized analytic functions on a simply connected Dini-smooth domain for an analytic symbol ∅.

Keywords: Hardy spaces; Generalized analyticity; Composition operators

MSC 2010: 30H10; 47B33; 30C62


  • [1] R. Adams, J. Fournier, Sobolev Spaces, Academic Press, 2003.Google Scholar

  • [2] K. Astala, L. Päivärinta, Calderón’s inverse conductivity problem in the plane, Ann. of Math. 2 (16), no. 1, 265-299, 2006.Google Scholar

  • [3] S. Axler, P. Bourdon, W. Ramey, Harmonic function theory, Second edition, Graduate Texts in Mathematics 137, Springer- Verlag, New York, 2001.Google Scholar

  • [4] L. Baratchart, A. Borichev, S. Chaabi, Pseudo-holomorphic functions at the critical exponent, Journal of the European Mathematical Society, 18, 1919-1960, 2016.CrossrefWeb of ScienceGoogle Scholar

  • [5] L. Baratchart, Y. Fischer, J. Leblond, Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation, Compl. Var. Elliptic Eq., 59 (4), 504-538, 2014.Google Scholar

  • [6] L. Baratchart, J. Leblond, S. Rigat and E. Russ, Hardy spaces of the conjugate Beltrami equation, J. Funct. Anal., 259 (2), 384-427, 2010.Google Scholar

  • [7] L. Baratchart, E. Pozzi, and E. Russ, Smirnov classes of pseudo-analytic functions, in Preparation, 2017.Google Scholar

  • [8] L. Bers, L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coeffcients and its applications, Conv. Int. EDP, Cremonese, Roma, 111-138, 1954.Google Scholar

  • [9] P. S. Bourdon, Fredholm multiplication and composition operators on the Hardy space, Integral Equations Operator Theory, 13, 607-610, 1990.Google Scholar

  • [10] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.Google Scholar

  • [11] C.C. Cowen, B.D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, 1995.Google Scholar

  • [12] P. L. Duren, Theory of Hp spaces, Pure and Applied Mathematics, Vol. 38 Academic Press, New York-London, 1970.Google Scholar

  • [13] M. Efendiev, E. Russ, Hardy spaces for the conjugated Beltrami equation in a doubly connected domain, J. Math. Anal. and Appl., 383, 439-450, 2011.Web of ScienceGoogle Scholar

  • [14] Y. Fischer, J. Leblond, J.R. Partington, E. Sincich, Bounded extremal problems in Hardy spaces for the conjugate Beltrami equation in simply connected domains, Appl. Comp. Harmo. Anal., 31, 264-285, 2011.Google Scholar

  • [15] O. Forster, Lectures on Riemann Surfaces, Grad. Texts in Math., vol. 81, Springer, 1981.Google Scholar

  • [16] J.B. Garnett, Bounded analytic functions, Pure and Applied Mathematics 96, 1981.Google Scholar

  • [17] I. Tadeusz, G. Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, 2001.Google Scholar

  • [18] S. B. Klimentov, The Riemann-Hilbert problem for generalized analytic functions in Smirnov classes, Vladikavkaz. Mat. Zh.,14(3), 63-73, 2012.Google Scholar

  • [19] V.V. Kravchenko, Applied pseudoanalytic function theory, Frontiers in Math., Birkhäuser Verlag, 2009.CrossrefGoogle Scholar

  • [20] J. Leblond, E. Pozzi, E. Russ, Composition operators on generalized Hardy spaces, Complex Analysis and Operator Theory, 9 (8), 1733-1757, 2015.Google Scholar

  • [21] B.D. MacCluer,Fredholm composition operators, Proceeding of the American Mathematical Society, 15 (1), 163-166, 1997. functions, Izv. Acad. Nauk Azerb. S.S.R., 2, 40-46, 1971 (in Russian).Google Scholar

  • [22] K.M. Musaev,Some classes of generalized analytic functions, Izv. Acad. Nauk Azerb. SSR, No. 2, 40-46, 1971 (in Russian). functions, Izv. Acad. Nauk Azerb. S.S.R., 2, 40-46, 1971 (in Russian).Google Scholar

  • [23] W. Rudin, Analytic functions of class Hp, Trans. Amer. Math. Soc. 78, 46-66, 1955.Google Scholar

  • [24] W. Rudin, Real and Complex analysis, Third, McGraw-Hill Book Co., New York, 1987.Google Scholar

  • [25] C. Pommerenke, Boundary behaviour of conformal maps, Springer Verlag, 1992.Google Scholar

  • [26] C. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, G´’ ottingen, 1975.Google Scholar

  • [27] J.H. Shapiro, W. Smith, Hardy spaces that support no compact composition operators, J. Funct. Anal., 205 (1), 62-89, 2003.Google Scholar

  • [28] I.N. Vekua, Generalized Analytic Functions, International Series of Monographs in Pure and Applied Mathematics, 1962.Google Scholar

  • [29] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970.Google Scholar

  • [30] W. P. Ziemer, Weakly differentiable functions, GTM 120, Springer, 1989.Google Scholar

About the article

Received: 2017-03-15

Accepted: 2018-03-01

Published Online: 2018-04-28

Citation Information: Concrete Operators, Volume 5, Issue 1, Pages 9–23, ISSN (Online) 2299-3282, DOI: https://doi.org/10.1515/conop-2018-0002.

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© 2018, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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