zbMATH — the first resource for mathematics

Schatten class Hankel operators on the Bergman spaces of strongly pseudoconvex domains. (English) Zbl 0802.47022
Consider on a strongly pseudoconvex domain in \(\mathbb{C}^ n\) the Hankel operator \(H_{\overline{f}}\) given by \(H_{\overline {f}} (g)= (I-P) (\overline {f} g)\), where \(f\) is a square-integrable holomorphic function, and \(P\) is the Bergman projection. The author shows that if \(p>2n\), then \(H_{\overline {f}}\) belongs to the Schatten class \(S_ p\) if and only if \(f\) belongs to the holomorphic Besov space \(B_ p\), while if \(p\leq 2n\), then \(H_{\overline {f}}\in S_ p\) if and only if \(f\) is a constant function.
The result was obtained independently by Marco M. Peloso [Ill. J. Math. 38, No. 2, 223-249 (1994)].
The proof uses integral formulas for solving the \(\overline {\partial}\)- equation.
For the special case of the unit ball, the result was found previously by R. Wallsten [Ark. Mat. 28, No. 1, 183-192 (1990; Zbl 0705.47023)], J. Arazy, S. Fisher, S. Janson, and J. Peetre [J. Lond. Math. Soc., II. Ser. 43, No. 3, 485–508 (1991; Zbl 0747.47019)], and K. Zhu [Am. J. Math. 113, No. 1, 147-167 (1991; Zbl 0734.47017)].

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
Full Text: DOI
[1] L. A. Aĭzenberg and Sh. A. Dautov, Differential forms orthogonal to holomorphic functions or forms, and their properties, Translations of Mathematical Monographs, vol. 56, American Mathematical Society, Providence, R.I., 1983. Translated from the Russian by R. R. Simha. · Zbl 0511.32002
[2] Jonathan Arazy, Stephen D. Fisher, Svante Janson, and Jaak Peetre, Membership of Hankel operators on the ball in unitary ideals, J. London Math. Soc. (2) 43 (1991), no. 3, 485 – 508. · Zbl 0747.47019
[3] Frank Beatrous Jr., \?^{\?}-estimates for extensions of holomorphic functions, Michigan Math. J. 32 (1985), no. 3, 361 – 380. · Zbl 0584.32024
[4] 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. · Zbl 0765.32005
[5] 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). · Zbl 0466.32001
[6] E. M. Chirka and G. M. Khenkin, Boundary properties of holomorphic functions of several complex variables, J. Soviet Math. 5 (1976), 612-687. · Zbl 0375.32005
[7] Bernard Coupet, Décomposition atomique des espaces de Bergman, Indiana Univ. Math. J. 38 (1989), no. 4, 917 – 941 (French, with English summary). · Zbl 0702.32028
[8] Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1 – 65. · Zbl 0289.32012
[9] Ian Graham, Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in \?\(^{n}\) with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219 – 240. · Zbl 0305.32011
[10] Sandrine Grellier, Comportement des fonctions holomorphes dans les directions complexes tangentes, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 3, 267 – 270 (French, with English summary). · Zbl 0719.32008
[11] J. J. Kohn and Hugo Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of Math. (2) 81 (1965), 451 – 472. · Zbl 0166.33802
[12] Huiping Li, BMO, VMO and Hankel operators on the Bergman space of strongly pseudoconvex domains, J. Funct. Anal. 106 (1992), no. 2, 375 – 408. · Zbl 0793.47025
[13] R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. · Zbl 0591.32002
[14] Bernard Russo, On the Hausdorff-Young theorem for integral operators, Pacific J. Math. 68 (1977), no. 1, 241 – 253. · Zbl 0367.47028
[15] E. M. Stein, Boundary behavior of holomorphic functions of several complex variables, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Mathematical Notes, No. 11. · Zbl 0242.32005
[16] Robert Wallstén, Hankel operators between weighted Bergman spaces in the ball, Ark. Mat. 28 (1990), no. 1, 183 – 192. · Zbl 0705.47023
[17] Ke He Zhu, Schatten class Hankel operators on the Bergman space of the unit ball, Amer. J. Math. 113 (1991), no. 1, 147 – 167. · Zbl 0734.47017
[18] D. H. Phong and E. M. Stein, Estimates for the Bergman and Szegö projections on strongly pseudo-convex domains, Duke Math. J. 44 (1977), no. 3, 695 – 704. · Zbl 0392.32014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.