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Lacunary series in mixed norm spaces in the disc

  • Real and Complex Analysis
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Abstract

The paper establishes a necessary and sufficient condition under which a lacunary series belong to a mixed norm space of functions holomorphic in the unit disc. As a corollary, some sharp pointwise estimates are obtained for lacunary series.

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References

  1. P. Duren, Theory of H p Spaces, Academic Press, New York (1970).

    MATH  Google Scholar 

  2. H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, Springer-Verlag, New York / Berlin / Heidelberg (2000).

    MATH  Google Scholar 

  3. S. Gadbois, “Mixed-norm generalizations of Bergman spaces and duality”, Proc. Amer. Math. Soc., 104, 1171–1180 (1988).

    MATH  MathSciNet  Google Scholar 

  4. M. M. Djrbashian, “On the representability problem of analytic functions”, Soobshch. Inst. Matem. i Mekh. AN Arm. SSR, 2, 3–40 (1948).

    Google Scholar 

  5. M. Mateljević and M. Pavlović, “L p-behavior of power series with positive coefficients and Hardy spaces”, Proc. Amer. Math. Soc., 87, 309–316 (1983).

    MATH  MathSciNet  Google Scholar 

  6. A. Zygmund, Trigonometric series (Cambridge University Press, Cambridge, England, 1959).

    MATH  Google Scholar 

  7. R. Aulaskari, J. Xiao and R. Zhao, “On subspaces and subsets of BMOA and UBC”, Analysis, 15, 101–121 (1995).

    MATH  MathSciNet  Google Scholar 

  8. R. Aulaskari and G. Csordas, “Besov spaces and the Q q,0 classes”, Acta Sci. Math. (Szeged), 60, 31–48 (1995).

    MATH  MathSciNet  Google Scholar 

  9. D. Girela, M. Pavlović and J. A. Peláez, “Spaces of analytic functions of Hardy-Bloch type”, J. d’Analyse Math., 100, 53–83 (2006).

    Article  MATH  Google Scholar 

  10. D. Girela and J.A. Peláez, “Integralmeans of analytic functions”, Ann. Acad. Sci. Fenn., 29, 459–469 (2004).

    MATH  Google Scholar 

  11. D. Girela and J. A. Peláez, “Growth properties and sequences of zeros of analytic functions in spaces of Dirichlet type”, J. Austral.Math. Soc., 80, 397–418 (2006).

    Article  MATH  Google Scholar 

  12. D.Girelaand J. A. Peláez, “Carleson measures for spaces ofDirichlet type”, Integral Equations Oper. Theory, 55, 415–427 (2006).

    Article  MATH  Google Scholar 

  13. J. Miao, “A property of analytic functions with Hadamard gaps”, Bull. Austral. Math. Soc., 45, 105–112 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  14. S. Stević, “A generalization of a result of Choa on analytic functions with Hadamard gaps”, J.KoreanMath. Soc., 43, 579–591 (2006).

    MATH  Google Scholar 

  15. S. Stević, “On Bloch-type functions with Hadamard gaps”, Abstract Appl. Anal., 2007, Article ID 39176, 8 pages (2007).

  16. H. Wulan and K. Zhu, Bloch and BMO functions in the unit ball”, Complex Var. Elliptic Equ., 53, 1009–1019 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  17. K. Zhu, “A class of Möbius invariant function spaces”, Illinois J. Math., 51, 977–1002 (2007).

    MATH  MathSciNet  Google Scholar 

  18. K. L. Avetisyan, “Lacunary series and sharp estimates in weighted spaces of holomorphic functions”, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 2(2), 3–9 (2007).

    MathSciNet  Google Scholar 

  19. K. L. Avetisyan, “Hardy-Bloch type spaces and lacunary series on the polydisc”, Glasgow Math. J., 49, 345–356 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  20. S. G. Samko, A. A. Kilbas, O. I. Marichev, Integrals and Derivatives of fractional order and some of their applications (Nauka i Tekhnika, Minsk, 1987).

    MATH  Google Scholar 

  21. F. Beatrous and J. Burbea, “Holomorphic Sobolev spaces on the ball”, Diss. Math., 276, 1–57 (1989).

    MathSciNet  Google Scholar 

  22. S. Yamashita, “Gap series and α-Bloch functions”, Yokohama Math. J., 28, 31–36 (1980).

    MATH  MathSciNet  Google Scholar 

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Correspondence to K. L. Avetisyan.

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Original Russian Text © K.L.Avetisyan, 2010, published in Izvestiya NAN Armenii. Matematika, 2010, No. 5, pp. 9–18.

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Avetisyan, K.L. Lacunary series in mixed norm spaces in the disc. J. Contemp. Mathemat. Anal. 45, 258–265 (2010). https://doi.org/10.3103/S106836231005002X

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  • DOI: https://doi.org/10.3103/S106836231005002X

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