Skip to main content
SpringerLink
Log in

Weighted Composition Operators between Bergman Spaces with Exponential Weights

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we study some properties of weighted composition operators on a class of weighted Bergman spaces \(A_\varphi^p\) with 0 < p ≤ ∞ and \(\varphi \in \mathcal{W}_0\). Also, we completely characterize the q-Carleson measure for \(A_\varphi^p\) in terms of the averaging function and the generalized Berezin transform with 0 < q < ∞. As applications, the boundedness and compactness of weighted composition operators acting from one Bergman space \(A_\varphi^p\) to another \(A_\varphi^q\) are equivalently described and the Schatten class property of the weighted composition operator acting on \(A_\varphi^2\) are given. Our main results are expressed in terms of certain Berezin type integral transforms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aron, R., Lindström, M.: Spectra of weighted composition operators on weighted Banach spaces of analytic functions. Israel J. Math., 141, 263–276 (2004)

    Article  MATH  Google Scholar 

  2. Arroussi, H., Park, I., Pau, J.: Schatten class Toeplitz operators acting on large weighted Bergman spaces. Studia Math., 229(3), 203–221 (2015)

    MATH  Google Scholar 

  3. Arroussi, H., Pau, J.: Reproducing kernel estimates, bounded projections and duality on large weighted Bergman spaces. J. Geom. Anal., 25(4), 2284–2312 (2015)

    Article  MATH  Google Scholar 

  4. Arroussi, H., Tong, C.: Weighted composition operators between large Fock spaces in several complex variables. J. Funct. Anal., 277(10), 3436–3466 (2019)

    Article  MATH  Google Scholar 

  5. Asserda, A., Hichame, A.: Pointwise estimate for the Bergman kernel of the weighted Bergman spaces with exponential weights. C. R. Math. Acad. Sci. Paris., 352(1), 13–16 (2014)

    Article  MATH  Google Scholar 

  6. Borichev, A., Dhuez, R., Kellay, K.: Sampling and interpolation in large Bergman and Fock spaces. J. Funct. Anal., 242(2), 563–606 (2007)

    Article  MATH  Google Scholar 

  7. Constantin, O., Peláez, J. Á.: Boundedness of the Bergman projection on Lp spaces with exponential weights. Bull. Sci. Math., 139(3), 245–268 (2015)

    Article  MATH  Google Scholar 

  8. Constantin, O., Peláez, J. Á.: Integral operators, embedding theorems and a Littlewood-Paley formula on weighted Fock spaces. J. Geom. Anal., 26(2), 1109–1154 (2016)

    Article  MATH  Google Scholar 

  9. Contreras, M. D., Hernández-Díaz, A. G.: Weighted composition operators between different Hardy spaces. Integral Equations Operator Theory, 46(2), 165–188 (2003)

    Article  MATH  Google Scholar 

  10. Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995

    Google Scholar 

  11. Čučković, Z., Zhao, R.: Weighted composition operators on the Bergman spaces. J. Lond. Math. Soc., 70(2), 499–511 (2004)

    Article  MATH  Google Scholar 

  12. Čučković, Z., Zhao, R.: Weighted composition operators between different weighted Bergman spaces and different Hardy spaces. Illinois J. Math., 51(2), 479–498 (2007)

    Article  MATH  Google Scholar 

  13. Forelli, F.: The isometries of Hp. Can. J. Math., 16, 721–728 (1964)

    Article  MATH  Google Scholar 

  14. Galanopoulos, P., Pau, J.: Hankel operators on large weighted Bergman spaces. Ann. Acad. Sci. Fenn. Math., 37(2), 635–648 (2012)

    Article  MATH  Google Scholar 

  15. He, Z., Wang, X., Liu, Y.: Toeplitz operators between Bergman spaces with exponential weights \(A_\varphi^p\) and \(A_\varphi^\infty\). Acta Math. Sinica, Chin. Ser., 64(3), 655–668 (2021)

    MATH  Google Scholar 

  16. Hu, Z., Lv, X., Schuster, A.: Bergman spaces with exponential weights. J. Funct. Anal., 276(5), 1402–1429 (2019)

    Article  MATH  Google Scholar 

  17. Hyvärinen, O., Lindström, M., Nieminen, I., et al., Spectra of weighted composition operators with automorphic symbols. J. Funct. Anal., 265(8), 1749–1777 (2013)

    Article  MATH  Google Scholar 

  18. Kolaski, C. J.: Isometries of weighted Bergman spaces. Can. J. Math., 34(4), 910–915 (1982)

    Article  MATH  Google Scholar 

  19. Lin, P., Rochberg, R.: Hankel operators on the weighted Bergman spaces with exponential weights. Integral Equations Operator Theory, 21(4), 460–483 (1995)

    Article  MATH  Google Scholar 

  20. Lin, P., Rochberg, R.: Trace ideal criteria for Toeplitz and Hankel operators on the weighted Bergman spaces with exponential type weights. Pacific J. Math., 173(1), 127–146 (1996)

    Article  MATH  Google Scholar 

  21. Luecking, D. H.: Embedding theorems for spaces of analytic functions via Khinchine’s inequality. Michigan Math. J., 40(2), 333–358 (1993)

    Article  MATH  Google Scholar 

  22. MacCluer, B. D., Zhao, R.: Essential norms of weighted composition operators between Bloch-type spaces. Rocky Mountain J. Math., 33(4), 1437–1458 (2003)

    Article  MATH  Google Scholar 

  23. Manhas, J. S.: Compact differences of weighted composition operators on weighted Banach spaces of analytic functions. Integral Equations Operator Theory, 62(3), 419–428 (2008)

    Article  MATH  Google Scholar 

  24. Mengestie, T.: Carleson type measures for Fock-Sobolev spaces. Complex Anal. Oper. Theory, 8(6), 1225–1256 (2014)

    Article  MATH  Google Scholar 

  25. Oleinik, V. L.: Embedding theorems for weighted classes of harmonic and analytic functions. J. Sov. Math., 9, 228–243 (1978)

    Article  MATH  Google Scholar 

  26. Oleinik, V. L., Perel’man, G. S.: Carleson’s embedding theorems for a weighted Bergman spaces. Math. Notes, 7(5–6), 577–581 (1990)

    Article  Google Scholar 

  27. Park, I.: The weighted composition operators on the large weighted Bergman spaces. Complex Anal. Oper. Theory, 13(1), 223–239 (2019)

    Article  MATH  Google Scholar 

  28. Park, I.: Compact differences of composition operators on large weighted Bergman spaces. J. Math. Anal. Appl., 479(2), 1715–1737 (2019)

    Article  MATH  Google Scholar 

  29. Pau, J., Peláez, J. Á.: Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights. J. Funct. Anal., 259(10), 2727–2756 (2010)

    Article  MATH  Google Scholar 

  30. Pau, J., Peláez, J. Á: Volterra type operators on Bergman spaces with exponential weights. Contemp. Math., 561(1), 239–252 (2012)

    Article  MATH  Google Scholar 

  31. Shapiro, J. H.: Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993

    Google Scholar 

  32. Singh, R. K., Manhas, J. S.: Composition Operators on Function Spaces, North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1993

    Google Scholar 

  33. Zhang, Y., Wang, X., Hu, Z.: Toeplitz operators on Bergman spaces with exponential weights. Complex Var. Elliptic Equ., (2022) https://www.webofscience.com/wos/alldb/full-record/WOS:000758715700001

    Google Scholar 

  34. Zhu, K.: Operator Theory in Function Spaces, Second edition, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007

    Book  MATH  Google Scholar 

Download references

Acknowledgements

We thank the referees for their time and comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, P. R. China

    Guang Fu Cao, Li He & Yi Yuan Zhang

Authors
  1. Guang Fu Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yi Yuan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li He.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 12071155, 11871170), the open project of Key Laboratory, School of Mathematical Sciences, Chongqing Normal University (Grant No. CSSXKFKTM202002) and the Innovation Research for the Postgraduates of Guangzhou University (Grant No. 2020GDJC-D08). We are co-first authors.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, G.F., He, L. & Zhang, Y.Y. Weighted Composition Operators between Bergman Spaces with Exponential Weights. Acta. Math. Sin.-English Ser. 38, 2231–2252 (2022). https://doi.org/10.1007/s10114-022-2048-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-022-2048-8

Keywords

MR(2010) Subject Classification

Search

Navigation