Integral Equations and Operator Theory

, Volume 66, Issue 4, pp 495–528 | Cite as

On Weighted Toeplitz, Big Hankel Operators and Carleson Measures

  • Carme Cascante
  • Joan Fàbrega
  • Joaquín M. Ortega
Article
First Online:
Received:
Revised:

Abstract

We compute the norm of pointwise multiplication operators, Toeplitz and Big Hankel operators with antiholomorphic symbols, defined on Besov spaces. These norms will be given in terms of Carleson measures for Besov spaces related to the symbol.

Mathematics Subject Classification (2000)

Primary 47B35 Secondary 32A37 

Keywords

Toeplitz operators Hankel operators Carleson measures 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams D.R., Hedberg L.I.: Function Spaces and Potential Theory. Springer, Berlin (1996)Google Scholar
  2. 2.
    Ahern P., Cohn W.S.: Exceptional sets for Hardy–Sobolev spaces. Indiana Math. J. 39, 417–451 (1989)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Aleksandrov, A.B.: Function theory in the ball. Several complex variables. II. Function theory in classical domains. Complex potential theory. Encyclopaedia of Mathematical Sciences, vol. 8, pp. 107–178. Springer, Berlin (1994)Google Scholar
  4. 4.
    Arcozzi N., Rochberg R., Sawyer E.: Carleson measures for analytic Besov spaces. Rev. Mat. Iberoamericana 18, 443–510 (2002)MATHMathSciNetGoogle Scholar
  5. 5.
    Arcozzi, N., Rochberg, R., Sawyer, E.: Carleson measures and interpolation sequences for Besov spaces on complex balls. Mem. Am. Math. Soc. 182(859), vi+163 (2006)Google Scholar
  6. 6.
    Arcozzi N., Rochberg R., Sawyer E.: Carleson measures for Drury–Arvesson Hardy space and other Besov–Sobolev spaces on complex balls. Adv. Math. 218(4), 1107–1180 (2008)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Arcozzi, N., Rochberg, R., Sawyer, E., Wick, B.D.: Bilinear forms on the Dirichlet space (25 Nov 2008). arXiv:0811.4107v1Google Scholar
  8. 8.
    Beatrous F.: Estimates for derivatives of holmorphic functions in pseudoconvex domains. Math. Z. 191, 91–116 (1986)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Beatrous, F., Burbea, J.: Sobolev spaces of holomorphic functions in the ball. Dissertationes Math. 276 (1989)Google Scholar
  10. 10.
    Békollé D., Berger C.A., Coburn L.A., Zhu K.H.: BMO in the Bergman metric on bounded simmetric domains. J. Funct. Anal. 93, 310–350 (1990)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Cascante C., Ortega J.M.: Carleson measures on spaces of Hardy–Sobolev type. Can. J. Math. 47, 1177–1200 (1995)MATHMathSciNetGoogle Scholar
  12. 12.
    Charpentier Ph.: Formules explicites pour les solutions minimales de l’equation ∂u = f dans la boule et dans le polydisque de \({\mathbb{C}^n}\) . Ann. Inst. Fourier Grenoble 30, 121–154 (1980)MATHMathSciNetGoogle Scholar
  13. 13.
    Cohn W.S., Verbitsky I.E.: Trace inequalities for Hardy–Sobolev functions in the unit ball of \({\mathbb{C}^n}\) . Indiana Univ. Math. J. 43, 1079–1097 (1994)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Cohn W.S., Verbitsky I.E.: Non-linear potential theory on the ball, with applications to exceptional and boundary interpolation sets. Michigan Math. J. 42, 79–97 (1995)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Costea, S., Sawyer, E.T., Wick, B.D.: The corona theorem for the Drury- Arveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in \({\mathbb{C}^n}\) (8 Apr 2009). arXiv:0811.0627v7Google Scholar
  16. 16.
    Hytönen, T., Martikainen, H.: Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces (23 Nov 2009). arXiv:0911.4387v1Google Scholar
  17. 17.
    Maz’ya, V.G., Shaposhnikova, T.O.: Theory of multipliers in spaces of differentiable functions. Pitmann (1985)Google Scholar
  18. 18.
    Nikolski, N.K.: Operators, functions, and systems: an easy reading, vol. 1. Hardy, Hankel, and Toeplitz. Mathematical Surveys and Monographs, 92. American Mathematical Society, Providence (2002)Google Scholar
  19. 19.
    Ortega J.M., Fàbrega J.: Pointwise multipliers and decomposition theorems in analytic Besov spaces. Math. Z. 235, 53–81 (2000)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Peller V.V.: Hankel Operators and their Applications. Springer Monographs in Mathematics. Springer, New York (2003)Google Scholar
  21. 21.
    Stegenga D.A.: Multipliers of the Dirichlet space. Illinois J. Math. 24, 113–139 (1980)MATHMathSciNetGoogle Scholar
  22. 22.
    Tchoundja E.: Carleson measures for the generalized Bergman spaces via a T(1)-type theorem. Ark. Mat. 46, 377–406 (2008)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Triebel H.: Theory of function spaces. Monographs in Mathematics, vol. 78. Birkhäuser, Basel (1993)Google Scholar
  24. 24.
    Upmeier H.: Toeplitz operators and index theory in several complex variables. Operator Theory: Advances and Applications, vol. 81. Birkhäuser, Basel (1996)Google Scholar
  25. 25.
    Volberg, A., Wick, B.D.: Bergman-type singular operator and the characterization of Carleson measures for Besov–Sobolev spaces on the complex ball (8 Oct 2009). arXiv:0910.1142v2Google Scholar
  26. 26.
    Verbitsky I.E.: Multipliers in spaces of fractional norms, and inner functions. Sibirsk. Mat. Zh. 26, 51–72 (1985, Russian)MathSciNetGoogle Scholar
  27. 27.
    Xia J.: Boundedness and compactness of Hankel operators on the sphere. J. Funct. Anal. 255(1), 25–45 (2008)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Wu Z., Zhao R., Zorboska N.: Toeplitz operators on analytic Besov spaces. Int. Equ. Oper. Theory 60, 435–449 (2008)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Zhu K.: Spaces of holomorphic functions in the unit ball. Graduate Texts in Mathematics, vol. 226. Springer, New York (2005)Google Scholar

Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  • Carme Cascante
    • 1
  • Joan Fàbrega
    • 1
  • Joaquín M. Ortega
    • 1
  1. 1.Department Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

Personalised recommendations