Hankel operators on the Bergman space of the unit polydisc

Li, Huiping

doi:10.1090/S0002-9939-1994-1176483-6

Hankel operators on the Bergman space of the unit polydisc
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Proc. Amer. Math. Soc. 120 (1994), 1113-1121 Request permission

Abstract:

Let $D$ be the unit polydisc in ${\mathbb {C}^n}$. Let ${H^2}(D)$ be the Bergman space of $D$. In this paper, by using the integral representations of solutions to the $\overline \partial$-equations, we give function theoretic characterizations of functions $f \in {L^2}(D)$ such that the Hankel operators ${H_f}$ from the Bergman space to ${L^2}(D)$ are bounded and compact, respectively.

References

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
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
Philippe Charpentier, Formules explicites pour les solutions minimales de l’équation $\bar \partial u=f$ dans la boule et dans le polydisque de $\textbf {C}^{n}$, Ann. Inst. Fourier (Grenoble) 30 (1980), no. 4, 121–154 (French). MR 599627
Paul Richard Halmos and Viakalathur Shankar Sunder, Bounded integral operators on $L^{2}$ spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 96, Springer-Verlag, Berlin-New York, 1978. MR 517709
Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
Huiping Li, Hankel operators on the Bergman spaces of strongly pseudoconvex domains, Integral Equations Operator Theory 19 (1994), no. 4, 458–476. MR 1285493, DOI 10.1007/BF01299844
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
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. MR 450623
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
Henri Skoda, Valeurs au bord pour les solutions de l’opérateur $d''$, et caractérisation des zéros des fonctions de la classe de Nevanlinna, Bull. Soc. Math. France 104 (1976), no. 3, 225–299 (French). MR 450620
Karel Stroethoff, Compact Hankel operators on the Bergman space, Illinois J. Math. 34 (1990), no. 1, 159–174. MR 1031892