Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 591–607 | Cite as

Bergman–Toeplitz operators on weakly pseudoconvex domains

  • Tran Vu KhanhEmail author
  • Jiakun Liu
  • Phung Trong Thuc


We prove that for certain classes of pseudoconvex domains of finite type, the Bergman–Toeplitz operator \(T_{\psi }\) with symbol \(\psi =K^{-\alpha }\) maps from \(L^{p}\) to \(L^{q}\) continuously with \(1< p\le q<\infty \) if and only if \(\alpha \ge \frac{1}{p}-\frac{1}{q}\), where K is the Bergman kernel on diagonal. This work generalises the results on strongly pseudoconvex domains by Čučković and McNeal, and Abate, Raissy and Saracco.


Bergman kernel Berman projection Bergman–Toeplitz operator Schur’s test Pseudoconvex domain of finite type 

Mathematics Subject Classification

Primary 47B35 32T25 Secondary 32A25 32A36 


  1. 1.
    Abate, M., Raissy, J., Saracco, A.: Toeplitz operators and Carleson measures in strongly pseudoconvex domains. J. Funct. Anal. 263(11), 3449–3491 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Boas, H.: Extension of Kerzman’s theorem on differentiability of the Bergman kernel function. Indiana Univ. Math. J. 37(3), 495–499 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Böttcher, A., Silbermann, B.: Analysis of Toeplitz Operators. Springer Monographs in Mathematics, 2nd edn. Springer, Berlin (2006). (Prepared jointly with Alexei Karlovich) zbMATHGoogle Scholar
  4. 4.
    Catlin, D.W.: Estimates of invariant metrics on pseudoconvex domains of dimension two. Math. Z. 200(3), 429–466 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cho, S.: Boundary behavior of the Bergman kernel function on some pseudoconvex domains in \({ C}^n\). Trans. Am. Math. Soc. 345(2), 803–817 (1994)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cho, S.: Estimates of the Bergman kernel function on certain pseudoconvex domains in \({ C}^n\). Math. Z. 222(2), 329–339 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cho, S.: Estimates of the Bergman kernel function on pseudoconvex domains with comparable Levi form. J. Korean Math. Soc. 39(3), 425–437 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cuckovic, Z.: Estimates of the \(L^p\) norms of the Bergman projection on strongly pseudoconvex domains. Integr. Equ. Oper. Theory 88(3), 331–338 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cuckovic, Z., McNeal, J.D.: Special Toeplitz operators on strongly pseudoconvex domains. Rev. Mat. Iberoam. 22(3), 851–866 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fefferman, C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26, 1–65 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kerzman, N.: The Bergman kernel function. Differentiability at the boundary. Math. Ann. 195, 149–158 (1972)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Khanh, T.V., Raich, A.: Local regularity of the Bergman projection on a class of pseudoconvex domains of finite type. (submitted). arXiv:1406.6532
  13. 13.
    Lieb, E.H., Loss, M.: Analysis, Volume 14 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2001)Google Scholar
  14. 14.
    McNeal, J.D.: Local geometry of decoupled pseudoconvex domains. Complex Analysis (Wuppertal, 1991) Aspects Math., vol. E17, pp. 223–230. Friedr. Vieweg, Braunschweig (1991)Google Scholar
  15. 15.
    McNeal, J.D.: The Bergman projection as a singular integral operator. J. Geom. Anal. 4, 91–104 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    McNeal, J.D.: Estimates on the Bergman kernels of convex domains. Adv. Math. 109, 108–139 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    McNeal, J.D.: Subelliptic estimates and scaling in the \(\bar{\partial }\)-Neumann problem. Explorations in Complex and Riemannian Geometry, Volume of 332 of Contemp. Math., pp. 197–217. Amer. Math. Soc., Providence (2003)CrossRefGoogle Scholar
  18. 18.
    McNeal, J.D., Stein, E.M.: Mapping properties of the Bergman projection on convex domains of finite type. Duke Math. J. 73, 177–199 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nagel, A., Rosay, J.-P., Stein, E.M., Wainger, S.: Estimates for the Bergman and Szegö kernels in \({{\mathbb{C}}}^2\). Ann. Math. 129, 113–149 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Phong, D.H., Stein, E.M.: Estimates for the Bergman and Szegö projections on strongly pseudo-convex domains. Duke Math. J. 44(3), 695–704 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Salinas, N., Sheu, A., Upmeier, H.: Toeplitz operators on pseudoconvex domains and foliation \(C^*\)-algebras. Ann. Math. (2) 130(3), 531–565 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Upmeier, H.: Toeplitz operators and index theory in several complex variables. Operator Theory: Advances and Applications, vol. 81. Birkhäuse, Basel (1996)Google Scholar
  23. 23.
    Yu, J.Y.: Peak functions on weakly pseudoconvex domains. Indiana Univ. Math. J. 43(4), 1271–1295 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhao, R.: Generalization of Schur’s test and its application to a class of integral operators on the unit ball of \(\mathbb{C}^n\). Integral Equ. Oper. Theory 82(4), 519–532 (2015)CrossRefzbMATHGoogle Scholar
  25. 25.
    K, Zhu: A sharp norm estimate of the Bergman projection on \(L^p\) spaces. In Bergman Spaces and Related Topics in Complex Analysis, Volume 404 of Contemp. Math., pp. 199–205. Amer. Math. Soc., Providence (2006)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Mathematics and its Applications, School of Mathematics and Applied StatisticsUniversity of WollongongNSWAustralia

Personalised recommendations