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Hankel Measures for Hardy Spaces

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In this paper, we study the so-called Hankel measures on the open unit disk. We obtain several new characterizations for such measures and answer a question raised by J. Xiao in 2000.

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References

  1. Arcozzi, N., Rochberg, R., Sawyer, E., Wick, B.: Function spaces related to the Dirichlet space. J. Lond. Math. Soc. 83, 1–18 (2011)

    Article  MathSciNet  Google Scholar 

  2. Aulaskari, R., Xiao, J., Zhao, R.: On subspaces and subsets of \(BMOA\) and \(UBC\). Analysis 15, 101–121 (1995)

    Article  MathSciNet  Google Scholar 

  3. Baernstein, A.: Analytic functions of bounded mean oscillation. In: Brannan, D.A., Clunie, J.G. (eds.) Aspects of Contemporary Complex Analysis, pp. 3–36. Academic Press, London (1980)

    Google Scholar 

  4. Bao, G., Wulan, H.: Hankel matrices acting on Dirichlet spaces. J. Math. Anal. Appl. 409, 228–235 (2014)

    Article  MathSciNet  Google Scholar 

  5. Chatzifountas, C., Girela, D., Peláez, J.: A generalized Hilbert matrix acting on Hardy spaces. J. Math. Anal. Appl. 413, 154–168 (2014)

    Article  MathSciNet  Google Scholar 

  6. Choe, B., Ramey, W., Ullrich, D.: Bloch-to-BMOA pullbacks on the disk. Proc. Am. Math. Soc. 125, 2987–2996 (1997)

    Article  MathSciNet  Google Scholar 

  7. Diamantopoulos, E.: Operators induced by Hankel matrices on Dirichlet spaces. Analysis (Munich) 24, 345–360 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Duren, P.: Theory of \(H^p\) Spaces. Academic Press, New York (1970)

    MATH  Google Scholar 

  9. Flett, T.: The dual of an inequality of Hardy and Littlewood and some related inequalities. J. Math. Anal. Appl. 38, 746–765 (1972)

    Article  MathSciNet  Google Scholar 

  10. Galanopoulos, P., Peláez, J.: A Hankel matrix acting on Hardy and Bergman spaces. Studia Math. 200, 201–220 (2010)

    Article  MathSciNet  Google Scholar 

  11. Garnett, J.: Bounded Analytic Functions. Springer, New York (2007)

    MATH  Google Scholar 

  12. Girela, D.: Analytic functions of bounded mean oscillation, in Complex Function Spaces (Mekrijärvi, : edited by R. Aulaskari), Univ. Joensuu Dept. Math. Rep. Ser. 4. Univ. Joensuu, Joensuu 2001, pp. 61–170 (1999)

  13. Girela, D., Merchán, N.: A Hankel matrix acting on spaces of analytic functions. Integral Equations Operator Theory 89, 581–594 (2017)

    Article  MathSciNet  Google Scholar 

  14. Girela, D., Merchán, N.: A generalized Hilbert operator acting on conformally invariant spaces. Banach J. Math. Anal. 12, 374–398 (2018)

    Article  MathSciNet  Google Scholar 

  15. Girela, D., Merchán, N.: Hankel matrices acting on the Hardy space \(H^1\) and on Dirichlet spaces. Rev. Mat. Comput. 32, 799–822 (2019)

    Article  Google Scholar 

  16. Jevtić, M., Karapetrović, B.: Generalized Hilbert matrices acting on spaces that are close to the Hardy space \(H^1\) and to the Space BMOA. Complex Anal. Oper. Theory 13, 2357–2370 (2019)

    Article  MathSciNet  Google Scholar 

  17. Merchán, N.: Mean Lipschitz spaces and a generalized Hilbert operator. Collect. Math. 70, 59–69 (2019)

    Article  MathSciNet  Google Scholar 

  18. Power, S.: Vanishing Carleson measures. Bull. Lond. Math. Soc. 12, 207–210 (1980)

    Article  MathSciNet  Google Scholar 

  19. Widom, H.: Hankel matrices. Trans. Am. Math. Soc. 121, 1–35 (1966)

    Article  MathSciNet  Google Scholar 

  20. Xiao, J.: Hankel measures on Hardy space. Bull. Austral. Math. Soc. 62, 135–140 (2000)

    Article  MathSciNet  Google Scholar 

  21. Xiao, J.: Pseudo-Carleson measures for weighted Bergman spaces. Mich. Math. J. 47, 447–452 (2000)

    Article  MathSciNet  Google Scholar 

  22. Xiao, J.: Holomorphic \({\cal{Q}}\) Classes. In: LNM 1767. Springer, Berlin (2001)

  23. Xiao, J.: Geometric \({\cal{Q}}_p\) Functions. Birkhäuser Verlag, Basel/Boston/Berlin (2006)

    Google Scholar 

  24. Zhao, R.: Composition operators from Bloch type spaces into Hardy and Besov spaces. J. Math. Anal. Appl. 233, 749–766 (1999)

    Article  MathSciNet  Google Scholar 

  25. Zhu, K.: Translating certain inequalities between Hardy and Bergman spaces. Am. Math. Monthly 111, 520–525 (2004)

    Article  Google Scholar 

  26. Zhu, K.: Operator Theory in Function Spaces, 2nd edn. American Mathematical Society, New York (2007)

    Book  Google Scholar 

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Acknowledgements

We thank J. Xiao for several useful suggestions and conversations that helped us understand the topic better and present the material more clearly.

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Authors and Affiliations

  1. Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China

    Guanlong Bao

  2. Business School, Shantou University, Shantou, Guangdong, 515063, China

    Fangqin Ye

  3. Department of Mathematics and Statistics, State University of New York, Albany, NY, 12222, USA

    Kehe Zhu

Authors
  1. Guanlong Bao
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  2. Fangqin Ye
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  3. Kehe Zhu
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Correspondence to Kehe Zhu.

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The work was supported by NNSF of China (Nos. 11801347 and 11720101003), NSF of Guangdong Province (No. 2018A030313512), Guangdong Basic and Applied Basic Research Foundation (No. 2019A1515110178), Key Projects of Fundamental Research in Universities of Guangdong Province (No. 2018KZDXM034), and STU Scientific Research Foundation for Talents (Nos. NTF17009, NTF17020, and STF17005).

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Bao, G., Ye, F. & Zhu, K. Hankel Measures for Hardy Spaces. J Geom Anal 31, 5131–5145 (2021). https://doi.org/10.1007/s12220-020-00472-5

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  • DOI: https://doi.org/10.1007/s12220-020-00472-5

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