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Weak factorization and Hankel forms for weighted Bergman spaces on the unit ball

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Abstract

We establish weak factorizations for a weighted Bergman space \(A^p_{\alpha }\), with \(1<p<\infty \), into two weighted Bergman spaces on the unit ball of \(\mathbb {C}^n\). To obtain this result, we characterize bounded Hankel forms on weighted Bergman spaces on the unit ball of \(\mathbb {C}^n\).

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References

  1. Arcozzi, N., Rochberg, R., Sawyer, E., Wick, B.D.: Bilinear forms on the Dirichlet space. Anal. PDE 3, 21–47 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Attele, K.: Toeplitz and Hankel operators on Bergman one space. Hokkaido Math. J. 21, 279–293 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Blasi, D., Pau, J.: A characterization of Besov type spaces and applications to Hankel type operators. Mich. Math. J. 56, 401–417 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bonami, A., Luo, L.: On Hankel operators between Bergman spaces on the unit ball. Houst. J. Math. 31, 815–828 (2005)

    MATH  MathSciNet  Google Scholar 

  5. Coifman, R., Rochberg, R.: Representation theorems for holomorphic and harmonic functions in \(L^p\). Asterisque 77, 11–66 (1980)

    MATH  MathSciNet  Google Scholar 

  6. Coifman, R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 2(103), 611–635 (1976)

    Article  MathSciNet  Google Scholar 

  7. Doubtsov, E.: Carleson–Sobolev measures for weighted Bloch spaces. J. Funct. Anal. 258, 2801–2816 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

  9. Garnett, J., Latter, R.: The atomic decomposition for Hardy spaces in several complex variables. Duke Math. J. 45, 815–845 (1978)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Gowda, M.: Nonfactorization theorems in weighted Bergman and Hardy spaces on the unit ball of \(\mathbb{C}^n\). Trans. Am. Math. Soc. 277, 203–212 (1983)

    MATH  MathSciNet  Google Scholar 

  12. Horowitz, C.: Factorization theorems for functions in the Bergman spaces. Duke Math. J. 44, 201–213 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  13. Janson, S., Peetre, J., Rochberg, R.: Hankel forms and the Fock space. Rev. Mat. Iberoamericana 3, 61–138 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  14. Limperis, T.G.: ‘Embedding theorems for the Bloch space’. PhD thesis, University of Arkansas (1998)

  15. Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. Lecture notes in Math, vol. 338. Springer-Verlag, Berlin (1973)

  16. Luecking, D.H.: Representations and duality in weighted spaces of analytic functions. Indiana Univ. Math. J. 34, 319–336 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  17. Luecking, D.H.: Embedding theorems for spaces of analytic functions via Khinchine’s inequality. Mich. Math. J. 40, 333–358 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nehari, Z.: On bounded bilinear forms. Ann. Math. 2(65), 153–162 (1957)

    Article  MathSciNet  Google Scholar 

  19. Ortega, J.M., Fabrega, J.: Pointwise multipliers and corona type decomposition in \(BMOA\). Ann. Inst. Fourier (Grenoble) 46, 111–137 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rochberg, R.: Decomposition theorems for Bergman spaces and the IR applications. In: Power, S.C. (ed.) Operators and Function Theory. pp. 225–277. Springer, Netherlands (1985)

  21. Wojtaszczyk, P.: Banach Spaces for Analysts. Cambridge Studies in Advanced Mathematics 25. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  22. Zhao, R., Zhu, K.: Theory of Bergman spaces in the unit ball of \(\mathbb{C}^n\). Mem. Soc. Math. Fr. (N.S.), 115, vi+103 (2008)

  23. Zhu, K.: Hankel operators on the Bergman spaces of bounded symmetric domains. Trans. Am. Math. Soc. 324, 707–730 (1991)

    Article  MATH  Google Scholar 

  24. Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Springer-Verlag, New York (2005)

    MATH  Google Scholar 

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Acknowledgments

The authors would like to thank the referee for valuable comments that improved the final version of the paper.

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Authors and Affiliations

  1. Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Catalonia, Spain

    Jordi Pau

  2. Department of Mathematics, SUNY Brockport, Brockport, NY, 14420, USA

    Ruhan Zhao

Authors
  1. Jordi Pau
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  2. Ruhan Zhao
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Corresponding author

Correspondence to Jordi Pau.

Additional information

This work started when the second named author visited the University of Barcelona in 2013. He thanks the support given by the IMUB during his visit. The first author was supported by DGICYT Grant MTM\(2011\)-\(27932\)-\(C02\)-\(01\) (MCyT/MEC) and the Grant 2014SGR289 (Generalitat de Catalunya).

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Pau, J., Zhao, R. Weak factorization and Hankel forms for weighted Bergman spaces on the unit ball . Math. Ann. 363, 363–383 (2015). https://doi.org/10.1007/s00208-015-1176-1

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  • DOI: https://doi.org/10.1007/s00208-015-1176-1

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