Integral Equations and Operator Theory

, Volume 21, Issue 4, pp 460–483 | Cite as

Hankel operators on the weighted Bergman spaces with exponential type weights

  • Peng Lin
  • Richard Rochberg
Article

Abstract

Let\(AL_\varphi ^2 \left( \mathbb{D} \right)\) denote the closed subspace of\(L^2 \left( {\mathbb{D},e^{ - 2\varphi } dA} \right)\) consisting of analytic functions in the unit disk\(\mathbb{D}\). For certain class of subharmonic\(\varphi :\mathbb{D} \to \mathbb{Z}\), the Hankel operatorHb on\(AL_\varphi ^2 \left( \mathbb{D} \right)\) with symbol\(b \in L^2 \left( \mathbb{D} \right)\) is studied. Criteria for boundedness and compactness of such kind of Hankel operators are presented.

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References

  1. [AFP] J. Arazy, S. Fisher and J. Peetre,Hankel operators on weighted Bergman spaces, Amer. J. Math.110 (1988), 989–1054.Google Scholar
  2. [B] B. Berndtsson, Weighted estimates for\(\bar \partial \) in domains in C1, Duke Math. J.66 (1992), 239–255.Google Scholar
  3. [H1] L. Hörmander,An Introduction to Complex Analysis in Several Variables, 3rd, ed. rev., North Holland, Amsterdan, 1990.Google Scholar
  4. [H2] L. Hörmander,An Introduction to Complex Analysis in Several Variables, New York: Van Nortrand Reinhold, 1966.Google Scholar
  5. [JPR] S. Janson, J. Peetre and R. Rochberg,Hankel forms and the Fock space, Rev. Mat. Iberoamericana3 (1987), 61–138.Google Scholar
  6. [K] S. G. Krantz,Function Theory of Several Complex Variables, 2nd. ed. Wadsworth, Belmont, 1992.Google Scholar
  7. [KM] T. L. Kriete III and B. D. MacCluer,Composition operators on large weighted Bergman spaces, Indiana Univ. Math. J.41 (1992), 755–788.Google Scholar
  8. [LR] Peng Lin and R. Rochberg,The essential norm of Hankel operator on the Bergman space, Integral Equations and Operator Theory.17 (1993), 361–372.Google Scholar
  9. [Lu] D. Luecking,Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk. J. Functional Analysis110 (1992), 247–271.Google Scholar
  10. [O] V. L. Oleinik,Embedding theorems for weighted classes of harmonic and analytic functions, J. Soviet Math.9 (1978), 228–243.Google Scholar
  11. [OP] V. L. Oleinik and G. S. Perel'man,Carleson's imbedding theorem for a weighted Bergman space, Mathematical Notes57 (1990), 577–581.Google Scholar
  12. [S] E. M. Stein,Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press. 1970.Google Scholar
  13. [St] K. Stroethoff,Hankel and Toeplitz operators on the Fock space, Michigan Math. J.39 (1992), 3–16.Google Scholar
  14. [Tr] T. Trent,A measure inequality. preprint.Google Scholar

Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Peng Lin
    • 1
  • Richard Rochberg
    • 1
  1. 1.Department of MathematicsWashington UniversitySt. LouisUSA

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