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Sequences of zeros of analytic function spaces and weighted superposition operators

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Abstract

We use properties of the sequences of zeros of certain spaces of analytic functions in the unit disc \({\mathbb {D}}\) to study the question of characterizing the weighted superposition operators which map one of these spaces into another. We also prove that for a large class of Banach spaces of analytic functions in \({\mathbb {D}}\), Y, we have that if the superposition operator \(S_\varphi \) associated to the entire function \(\varphi \) is a bounded operator from X, a certain Banach space of analytic functions in \(\mathbb D\), into Y, then the superposition operator \(S_{\varphi ^\prime }\) maps X into Y.

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References

  1. Alvarez, V., Márquez, M.A., Vukotić, D.: Superposition operators between the Bloch space and Bergman spaces. Ark. Mat. 42, 205–216 (2004)

    Article  MathSciNet  Google Scholar 

  2. Anderson, J.M., Clunie, J., Pommerenke, Ch.: On Bloch functions and normal functions. J. Reine Angew. Math. 270, 12–37 (1974)

    MathSciNet  MATH  Google Scholar 

  3. Bao, G., Ye, F.: The Bloch space and the dual space of a Luecking-type subspace of \(A^1\). Complex Var. Elliptic Equ. 63(10), 1438–1443 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bonet, J., Vukotić, D.: Superposition operators between weighted Banach spaces of analytic functions of controlled growth. Monatsh. Math. 170(3–4), 311–323 (2013)

    Article  MathSciNet  Google Scholar 

  5. Boyd, C., Rueda, P.: Superposition operators between weighted spaces of analytic functions. Quaest. Math. 36(3), 411–419 (2013)

    Article  MathSciNet  Google Scholar 

  6. Boyd, C., Rueda, P.: Holomorphic superposition operators between Banach function spaces. J. Aust. Math. Soc. 96(2), 186–197 (2014)

    Article  MathSciNet  Google Scholar 

  7. Buckley, S.M., Fernández, J.L., Vukotić, D.: Superposition operators on Dirichlet type spaces. In: Papers on Analysis: A Volume Dedicated to Olli Martio on the Occasion of his 60th Birthday, 41–61. Rep. University of Jyväskylä Department of Mathematics and Statistics, vol. 83, University of Jyväskylä, Jyväskylä (2001)

  8. Buckley, S.M., Vukotić, D.: Univalent interpolation in Besov spaces and superposition into Bergman Spaces. Potential Anal. 29(1), 1–16 (2008)

    Article  MathSciNet  Google Scholar 

  9. Cámera, G.A., Giménez, J.: The nonlinear superposition operator acting on Bergman spaces. Compos. Math. 93(1), 23–35 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Cámera, G.A.: Nonlinear superposition on spaces of analytic functions. In: Harmonic Analysis and Operator Theory (Caracas, 1994). Contemporary Mathematics, vol. 189, pp. 103–116. Amer. Math. Society, Providence (1995)

  11. Duren, P.L.: Theory of \(H^{p}\) Spaces. Academic Press, New York (1970). Reprint: Dover, Mineola-New York (2000)

  12. Duren, P.L., Schuster, A.P.: Bergman Spaces. Mathematical Surveys and Monographs, vol. 100. American Mathematical Society, Providence (2004)

    MATH  Google Scholar 

  13. Galanopoulos, P., Girela, D., Márquez, M.A.: Superposition operators, Hardy spaces, and Dirichlet type spaces. J. Math. Anal. Appl. 463(2), 659–680 (2018)

    Article  MathSciNet  Google Scholar 

  14. Girela, D., Márquez, M.A.: Superposition operators between \(Q_p\) spaces and Hardy spaces. J. Math. Anal. Appl. 364(2), 463–472 (2010)

    Article  MathSciNet  Google Scholar 

  15. Girela, D., Nowak, M., Waniurski, P.: On the zeros of Bloch functions. Math. Proc. Camb. Philos. Soc. 129(1), 117–128 (2000)

    Article  MathSciNet  Google Scholar 

  16. Girela, D., Peláez, J.A., Pérez-González, F., Rättyä, J.: Carleson measures for the Bloch space. Integral Equ. Oper. Theory 61(4), 511–547 (2008)

    Article  MathSciNet  Google Scholar 

  17. Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Graduate Texts in Mathematics, vol. 199. Springer, New York (2000)

    Book  Google Scholar 

  18. Hille, E.: Analytic Function Theory, vol. II. Ginn and Co., Boston (1962)

    MATH  Google Scholar 

  19. Horowitz, C.: Zeros of functions in the Bergman spaces. Duke Math. J. 41, 693–710 (1974)

    Article  MathSciNet  Google Scholar 

  20. Malavé-Malavé, R.J., Ramos-Fernández, J.C.: Superposition operators between logarithmic Bloch spaces. Rend. Circ. Mat. Palermo (2018). https://doi.org/10.1007/s12215-018-0345-y

    Article  MATH  Google Scholar 

  21. Nowak, M.: On zeros of normal functions. Ann. Acad. Sci. Fenn. Math. 27(2), 381–390 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Ramos-Fernández, J.C.: Bounded superposition operators between weighted Banach spaces of analytic functions. Appl. Math. Comput. 219(10), 4942–4949 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Sedletskii, A.M.: Zeros of analytic functions of the classes \(A^p_\alpha \). In: Current Problems in Function Theory (Russian) (Teberda, 1985). Rostov. Gos. Univ., Rostov-on-Don, vol. 177, pp. 24–29 (1987)

  24. Zhu, K.: Operator Theory in Function Spaces. Mathematical Surveys and Monographs, vol. 138, 2nd edn. American Mathematical Society, Providence (2007)

    Book  Google Scholar 

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Authors and Affiliations

  1. Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071, Málaga, Spain

    Salvador Domínguez & Daniel Girela

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  1. Salvador Domínguez
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  2. Daniel Girela
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Correspondence to Daniel Girela.

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Communicated by A. Constantin.

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This research is supported in part by a Grant from “El Ministerio de Economía y Competitividad”, Spain (MTM2014-52865-P) and by a Grant from la Junta de Andalucía FQM-210.

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Domínguez, S., Girela, D. Sequences of zeros of analytic function spaces and weighted superposition operators. Monatsh Math 190, 725–734 (2019). https://doi.org/10.1007/s00605-019-01279-5

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