Abstract
The characterization of thosef for which the Hankel operatorsH f belongs to various trace ideals over Bergman spaces on pseudoconvex domains of finite type in complex dimension two is given. In particular, we determine how the cutoff values are affected by the boundary geometry.
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[AFP1] J. Arazy, S. Fisher, and J. Peetre, Hankel operators on weighted Bergman spaces,Amer. J. Math. 111 (1988), 989–1054.
[AFP2] J. Arazy, S. Fisher, S. Janson and J. Peetre, Hankel operators on planar domains,Constructive Approximation 6(1990), 113–138.
[AFJP] J. Arazy, S. Fisher, S. Janson and J. Peetre, Membership of Hankel operators on the unit ball in unitary ideals.J. London Math. Soc. 43(1991), 485–508.
[BL1] F. Beatrous and Song-Ying Yi, On the boundedness and compactness of operators of Hankel type,Jour. of Funct. Anal. 111(1993), 350–379.
[BL2] Beatrous, F., S.-Y. Li, Trace ideal criteria for operators of Hankel type,Illinois J. Math. 39(1995), 723–754.
[BBCZ] D. Békollé, C. A. Berger, L. A. Coburn, and K. H. Zhu,BMO in the Bergman metric on bounded symmetric domains,J. Funct. Anal. 93(1990), 310–350.
[CAT] D. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two,Math. Z. 200(1989), 429–466.
[CS] M. Cotlar and C. Sadosky, Abstract, Weighted, and Multidimensional Adamjan-Aronov-Krein Theorems and Singular Number of Sarason Commutants,Int. Equations and Op. Thy. 17(1993), 171–200.
[FR] M. Feldman and R. Rochberg, Singular value estimates for commutators and Hankel operators on the unit ball and Heisenberg group,Analysis and P.D.E., Lecture Notes in Pure and Applied Math. 122, Decker, New York, 1990.
[J] S. Janson, Hankel operators between weighted Bergman spaces,Ark. Math. 26(1988), 205–219.
[K] S. G. Krantz,Function Theory of Several Complex Variables, 2nd Ed., Wadsworth, Belmont, 1992.
[KL1] S. G. Krantz and S-Y. Li, On the decomposition theorems for Hardy spaces and applications in domains in ℂn,J. of Fourier Analysis and Applications 2(1995), 65–107.
[KL2] S. G. Krantz and S-Y. Li, A Note on Hardy spaces and functions of bounded mean oscillation on domains in ℂn,Michigan J. of Math 41(1994), 51–72.
[KL3] S. G. Krantz and S-Y. Li, Duality theorems for Hardy and Bergman spaces on convex domains of finite type in ℂn,Ann. of Fourier Institute 45(1995), 1305–1327.
[L] H. Li, Schatten Class Hankel Operators on Bergman Space of Strongly Pseudoconvex Domain,IEOPT, 1995.
[MCN1] J. McNeal, Boundary behavior of the Bergman kernel function in ℂ2,Duke Math. J. 58(1989), 499–512.
[MCN2] J. McNeal, Estimates on the Bergman kernels of convex domains,Advances in Math. 109(1994), 108–139.
[MS] J. D. McNeal and E. M. Stein, Mapping properties of the Bergman projection on convex domains of finite type,Duke Math. J. 73(1994), 177–199.
[NRSW] A. Nagel, Rosay, E. M. Stein and Wainger, Estimates for the Bergman and Szegö kernels in ℂ2,Ann. of Math. 129(1989), 113–149.
[Pe] V. V. Peller, Hankel operators of classS p and applications,Mat. Sb. 113(1980), 538–581.
[P] M. Peloso, Hankel operators on weighted Bergman spaces on strongly pseudoconvex domains,Ill. J. of Math. 38(1994), 223–249.
[W] R. Wallstén, Hankel operators between Bergman spaces in the ball,Ark. Math. 28(1990), 183–192.
[Zh] D. Zheng, Schatten class Hankel operators on Bergman spaces,Int. Equations and Op. Thy. 13(1990), 442–459.
[Z] Kehe Zhu, Schatten class Hankel operators on the Bergman space of the unit ball,Amer. J. Math. 113(1991), 147–167.
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All three authors supported by grants from the National Science Foundation
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Krantz, S.G., Li, SY. & Rochberg, R. The effect of boundary geometry on Hankel operators belonging to the trace ideals of Bergman spaces. Integr equ oper theory 28, 196–213 (1997). https://doi.org/10.1007/BF01191818
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DOI: https://doi.org/10.1007/BF01191818