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Hankel Operators on Pseudoconvex Domains of Finite Type in ℂ2

Published online by Cambridge University Press:  20 November 2018

Frédéric Symesak*
Affiliation:
UPESA 6086 Groupes de Lie et géométrie Département de Mathématiques Université de Poitiers 86022 Poitiers Cedex France e-mail: symesak@mathrs.univ-poitiers.fr
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Abstract

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The aim of this paper is to study small Hankel operators $h$ on the Hardy space or on weighted Bergman spaces,where $\Omega $ is a finite type domain in ${{\mathbb{C}}^{2}}$ or a strictly pseudoconvex domain in ${{\mathbb{C}}^{n}}$ . We give a sufficient condition on the symbol $f$ so that $h$ belongs to the Schatten class ${{S}_{p}}$ , $1\,\le \,p\,<\,+\infty $ .

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1998

References

[AFP] Arazy, A., Fisher, S. and Peetre, J., Hankel operators on weighted Bergman spaces. Amer. J. Math. 111(1988), 9891054.Google Scholar
[BL1] Beatrous, F. and Li, S-Y., On the boundedness and compactness of operators of Hankel type. J. Funct. Anal. 111(1993), 350379.Google Scholar
[BL2] Beatrous, F., Trace Ideals Criteria for Operators of Hankel type. Illinois J. Math. 39(1995), 723754.Google Scholar
[B] Boas, H., The Szegö projection: Sobolev estimates in regular domains. Trans. Amer. Math. Soc. 300(1987), 109132.Google Scholar
[B1] Bell, S., A duality theorem for harmonic functions. Michigan Math. J. 29(1982), 123128.Google Scholar
[B2] Bell, S., Biholomorphic mappings and the δ -problem. Ann. of Math. 114(1981), 103114.Google Scholar
[BCG] Bonami, A., Der-Chen Chang and Grellier, S., Commutation properties and Lipschitz estimates for the Bergman and Szegö projection. Math Z. 223(1996), 275302.Google Scholar
[BPS1] Bonami, A., Peloso, M. and Symesak, F., Powers of the Szegö kernel and Hankel operators on Hardy spaces. Preprint.Google Scholar
[BPS2] Bonami, A., Factorization theorems for Hardy spaces and Hankel operators. In preparation.Google Scholar
[Ca] Catlin, D., Estimates of invariant metrics on pseudoconvex domains of dimension two. Math.Z. 200(1989), 429466.Google Scholar
[CR] Coifman, R. and Rochberg, R., Representation theorem for holomorphic and harmonic functions in Lp. Astérisque 77(1980), 165.Google Scholar
[CRW] Coifman, R., Rochberg, R. and G.Weiss, Factorization theorems for Hardy spaces in several variables. Ann. of Math. 103(1976), 611635.Google Scholar
[CW] Coifman, R. et Weiss, G., Analyse harmonique non commutative sur certains espaces homog`enes. Lecture Notes in Math. 242, Springer-Verlag, 1971.CrossRefGoogle Scholar
[Co] Coupet, B., D´ecomposition atomique des espaces de Bergman. Indiana Math. J. 38(1989), 917941.Google Scholar
[FR] Feldman, M. and Rochberg, R., Singular value estimates for commutators and Hankel operators on the unit ball and the Heisenberg group. Lecture Notes in Pure and Appl. Math. 122 (Analysis and partial differential equations), Marcel Dekker, New York, 1990.Google Scholar
[GK] Gohberg, I. and Krein, M.G., Introduction to the theory of non-self adjoint operators. Trans. Math. Monograph. 18, Amer. Math. Soc., Providence, RI, 1969.Google Scholar
[J] Janson, S., On functions with conditions on the mean oscillation. ArkMat. 14(1976), 189190.Google Scholar
[KL1] Krantz, S. and Li, S-Y., On decomposition theorems for Hardy spaces on domains in Cn and applications. J. Fourier Anal. App. 2(1995), 65107.Google Scholar
[KL2] Krantz, S., Hardy classes, integral operators, and duality on spaces of homogeneous type. Preprint.Google Scholar
[KLR] Krantz, S., Li, S-Y. and Rochberg, R., The effect of boundary geometry on Hankel operators belonging to the trace ideals of Bergman spaces. Preprint.Google Scholar
[KLLR] Krantz, S., Li, S-Y., Lin, P. and Rochberg, R., The effect of regularity on the singular number of Friedrichs operators on Bergman spaces. Michigan Math. J. 43(1996), 337–348.Google Scholar
[L] Li, H., Schatten class Hankel operators on Bergman space of strongly pseudoconvex domain. Integral Equations Operator Theory 19(1994), 458–476.CrossRefGoogle Scholar
[NSW] Nagel, A., Stein, E. and Wainger, S., Balls and metrics defined by vector fields I: basic properties. Acta Math. 155(1985), 103–147.CrossRefGoogle Scholar
[NRSW] Nagel, A., Rosay, J-P., Stein, E. and Wainger, S., Estimates for the Bergman and the Szegö kernel in C2 . Ann. of Math. 129(1989), 113–149.CrossRefGoogle Scholar
[Pell] Peller, V. V., Hankel operators of class Cp and their application (rational approximation, Gaussian processes, the problem of majorizing operators). Math. of the USSR-Sbornik 41(1982), 443–479.CrossRefGoogle Scholar
[Pelo] Peloso, M., Hankel operators on weighted Bergman spaces on strictly pseudoconvex domains. Illinois J. Math. 38(1995), 223–249.Google Scholar
[RS1] Rochberg, R. and Semmes, S., A decomposition theorem for BMO and application. J. Funct. Anal. 67(1986), 228–263.CrossRefGoogle Scholar
[RS2] Rochberg, R. and Semmes, S., Nearly weakly orthonormal sequences, singular value estimates and Calderon-Zygmund operators. J. Funct. Anal. 86(1989), 237–306.CrossRefGoogle Scholar
[S1] Symesak, F., Décomposition atomique des espaces de Bergman. Publ. Mat. 39(1995), 285–299.CrossRefGoogle Scholar
[S2] Symesak, F., Hankel operators on complex ellipsoids. Illinois. J. Math. 40(1996), 632–647.Google Scholar
[Zha] Zhang, G., Hankel operators on Hardy spaces and Shatten classes. Chinese Ann. Math. Ser. B 12(1991), 282–294.Google Scholar
[Zhe] Zheng, D., Shatten class Hankel operators on Bergman spaces. Integral Equations Operator Theory 13(1990), 442–459.CrossRefGoogle Scholar
[Zhu] Zhu, K., Operator Theory in function spaces. Decker, New York, 1990.Google Scholar
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