BMO on strongly pseudoconvex domains: Hankel operators, duality and $\overline \partial$-estimates
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- by Huiping Li and Daniel H. Luecking PDF
- Trans. Amer. Math. Soc. 346 (1994), 661-691 Request permission
Abstract:
We study the condition that characterizes the symbols of bounded Hankel operators on the Bergman space of a strongly pseudoconvex domain and show that it is equivalent to $BMO$ plus analytic. (Here we mean the Bergman metric $BMO$ of Berger, Coburn and Zhu.) In the course of the proof we obtain new $\overline \partial$-estimates that may be of independent interest. Some applications include a decomposition of $BMO$ similar to the classical ${L^\infty } + \widetilde {{L^\infty }}$, and two characterizations of the dual of $VMO$ (which is also a predual of $BMO$). In addition, we obtain some partial results on the boundedness of Hankel operators in ${L^1}$ norm.References
- Sheldon Axler, Bergman spaces and their operators, Surveys of some recent results in operator theory, Vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, pp. 1–50. MR 958569
- Sheldon Axler, The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53 (1986), no. 2, 315–332. MR 850538, DOI 10.1215/S0012-7094-86-05320-2
- Frank Beatrous and Song-Ying Li, On the boundedness and compactness of operators of Hankel type, J. Funct. Anal. 111 (1993), no. 2, 350–379. MR 1203458, DOI 10.1006/jfan.1993.1017
- 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), no. 2, 310–350. MR 1073289, DOI 10.1016/0022-1236(90)90131-4
- C. A. Berger, L. A. Coburn, and K. H. Zhu, BMO on the Bergman spaces of the classical domains, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 133–136. MR 888889, DOI 10.1090/S0273-0979-1987-15539-X
- B. Berndtsson and M. Andersson, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, v–vi, 91–110 (English, with French summary). MR 688022, DOI 10.5802/aif.881
- F. F. Bonsall, Decompositions of functions as sums of elementary functions, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 146, 129–136. MR 841422, DOI 10.1093/qmath/37.2.129
- Bernard Coupet, Décomposition atomique des espaces de Bergman, Indiana Univ. Math. J. 38 (1989), no. 4, 917–941 (French, with English summary). MR 1029682, DOI 10.1512/iumj.1989.38.38042
- Š. A. Dautov and G. M. Henkin, Zeros of holomorphic functions of finite order and weighted estimates for the solutions of the $\bar \partial$-equation, Mat. Sb. (N.S.) 107(149) (1978), no. 2, 163–174, 317 (Russian). MR 512005
- Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65. MR 350069, DOI 10.1007/BF01406845
- John Erik Fornaess, Embedding strictly pseudoconvex domains in convex domains, Amer. J. Math. 98 (1976), no. 2, 529–569. MR 422683, DOI 10.2307/2373900
- David Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), no. 1, 27–42. MR 523600
- Ian Graham, Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in $C^{n}$ with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219–240. MR 372252, DOI 10.1090/S0002-9947-1975-0372252-8
- Steven G. Krantz and Daowei Ma, Bloch functions on strongly pseudoconvex domains, Indiana Univ. Math. J. 37 (1988), no. 1, 145–163. MR 942099, DOI 10.1512/iumj.1988.37.37007
- Huiping Li, BMO, VMO and Hankel operators on the Bergman space of strongly pseudoconvex domains, J. Funct. Anal. 106 (1992), no. 2, 375–408. MR 1165861, DOI 10.1016/0022-1236(92)90054-M —, Hankel operators on the Bergman space of strongly pseudoconvex domains, Integral Equations Operator Theory (to appear).
- Huiping Li, Schatten class Hankel operators on the Bergman spaces of strongly pseudoconvex domains, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1211–1221. MR 1169879, DOI 10.1090/S0002-9939-1993-1169879-9
- Daniel Luecking, A technique for characterizing Carleson measures on Bergman spaces, Proc. Amer. Math. Soc. 87 (1983), no. 4, 656–660. MR 687635, DOI 10.1090/S0002-9939-1983-0687635-6
- Daniel H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 107 (1985), no. 1, 85–111. MR 778090, DOI 10.2307/2374458
- Daniel H. Luecking, Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk, J. Funct. Anal. 110 (1992), no. 2, 247–271. MR 1194989, DOI 10.1016/0022-1236(92)90034-G
- R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR 847923, DOI 10.1007/978-1-4757-1918-5
- N. Th. Varopoulos, BMO functions and the $\overline \partial$-equation, Pacific J. Math. 71 (1977), no. 1, 221–273. MR 508035, DOI 10.2140/pjm.1977.71.221
- Ke He Zhu, Multipliers of BMO in the Bergman metric with applications to Toeplitz operators, J. Funct. Anal. 87 (1989), no. 1, 31–50. MR 1025882, DOI 10.1016/0022-1236(89)90003-7
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 346 (1994), 661-691
- MSC: Primary 47B35; Secondary 32A37, 32F20, 32H10, 46E15
- DOI: https://doi.org/10.1090/S0002-9947-1994-1273537-5
- MathSciNet review: 1273537