BIG HANKEL OPERATORS ON HARDY SPACES OF STRONGLY PSEUDOCONVEX DOMAINS

doi:10.1007/s10473-024-0301-1

摘要/Abstract

摘要： In this article, we investigate the (big) Hankel operator $H_f$ on the Hardy spaces of bounded strongly pseudoconvex domains $\Omega$ in $\mathbb{C}^n$. We observe that $H_f$ is bounded on $H^p(\Omega)$ ($1< p<\infty$) if $f$ belongs to BMO and we obtain some characterizations for $H_f$ on $H^2(\Omega)$ of other pseudoconvex domains. In these arguments, Amar's $L^p$-estimations and Berndtsson's $L^2$-estimations for solutions of the $\bar{\partial}_b$-equation play a crucial role. In addition, we solve Gleason's problem for Hardy spaces $H^p(\Omega)$ ($1\le p\le\infty$) of bounded strongly pseudoconvex domains.

关键词: Hankel operator, Hardy space, Bergman space, pseudoconvex domain

Abstract: In this article, we investigate the (big) Hankel operator $H_f$ on the Hardy spaces of bounded strongly pseudoconvex domains $\Omega$ in $\mathbb{C}^n$. We observe that $H_f$ is bounded on $H^p(\Omega)$ ($1< p<\infty$) if $f$ belongs to BMO and we obtain some characterizations for $H_f$ on $H^2(\Omega)$ of other pseudoconvex domains. In these arguments, Amar's $L^p$-estimations and Berndtsson's $L^2$-estimations for solutions of the $\bar{\partial}_b$-equation play a crucial role. In addition, we solve Gleason's problem for Hardy spaces $H^p(\Omega)$ ($1\le p\le\infty$) of bounded strongly pseudoconvex domains.

Key words: Hankel operator, Hardy space, Bergman space, pseudoconvex domain

中图分类号:

47B35

Boyong Chen, Liangying Jiang. BIG HANKEL OPERATORS ON HARDY SPACES OF STRONGLY PSEUDOCONVEX DOMAINS[J]. 数学物理学报(英文版), 2024, 44(3): 789-809.

Boyong Chen, Liangying Jiang. BIG HANKEL OPERATORS ON HARDY SPACES OF STRONGLY PSEUDOCONVEX DOMAINS[J]. Acta mathematica scientia,Series B, 2024, 44(3): 789-809.

参考文献

[1] Ahern P, Schneider R. Holomorphic Lipschitz functions in pseudoconvex domains. Amer J Math, 1979, 101: 543-565
[2] Amar E. Big Hankel operator and $\overline{\partial}_b$-equation. J Oper Theory, 1995, 33: 223-233
[3] Amar E, Bonami A. Measures de Carleson d'ordre $\alpha$ et solutions au bord de l'équation $\overline{\partial}$. Bull Soc Math France, 1979, 107: 23-48
[4] Andersson M, Carlsson H. Estimates of solutions of the $H^p$ and BMOA corona problem. Math Ann, 2000, 316: 83-102
[5] Asserda S. The essential norm of Hankel operators on the Bergman spaces of strongly pseudoconvex domain. Integral Equ Oper Theory, 2000, 36: 379-395
[6] Beatrous F. $L^p$ estimates for extensions of holomorphic functions. Michigan Math J, 1985, 32: 361-380
[7] Beatrous F, Li S. Trace ideal criteria for operators of Hankel type. Illinois J Math, 1995, 39: 723-754
[8] Berndtsson B. $\overline{\partial}_b$ and Carleson inequalities//Berenstein C. Complex Analysis II: Proceedings of the Special Year held at the University of Maryland, College Park, 1985-86. Berlin: Springer, 1987: 42-54
[9] Berndtsson B. Weighted estimates for the $\bar{\partial}$-equation//McNeal J D. Complex Analysis and Complex Geometry. Berlin: De Gruyter, 2001: 43-57
[10] Bonami A, Peloso M, Symesak F. Factorization of Hardy spaces and Hankel operators on convex domains in $\mathbb{C}^n$. J Geometric Anal, 2001, 11: 363-397
[11] Charpentier P, Dupain Y. Weighted and boundary $L^p$ estimates for solutions of the $\partial$-equation on lineally convex domains of finite type and applications. Math Z, 2018, 290: 195-220
[12] Chen B, Fu S. Comparison of the Bergman and Szegö kernels. Adv Math, 2011, 228: 2366-2384
[13] Chen S, Shaw M.Partial Differential Equations in Several Complex Variables. Providence, RI: Amer Math Soc, 2001
[14] Coifman R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann Math, 1976, 103: 611-635
[15] Donnelly H, Fefferman C. $L^2$-cohomology and index theorem for the Bergman metric. Ann Math, 1983, 118: 593-618
[16] Fang Q, Xia J. Schatten class membership of Hankel operators on the unit sphere. J Funct Anal, 2009, 257: 3082-3134
[17] Feldman M, Rochberg R. Singular value estimates for commutators and Hankel operators on the unit ball and the Heisenberg group//Sadosky C. Analysis and Partial Differential Equations: A collection of Papers Dedicated to Mischa Cotlar. New York: Marcel Dekker, 1989: 121-159
[18] Folland G, Kohn J.The Neumann Problem for Cauchy-Riemann Complex. Princeton: Princeton Univ Press, 1972
[19] Fornaess J. Embedding strictly pseudoconvex domains in convex domains. Amer J Math, 1976, 98: 529-569
[20] Gao J, Hu Z. Approximation in weighted Bergman spaces and Hankel operators on strongly pseudoconvex domains. Math Z, 2020, 297: 1483-1505
[21] Garnett J. Bounded Analytic Functions.New York: Academic Press, 1981
[22] Gleason A. Finitely generated ideals in Banach algebras. J Math Mech, 1964, 13: 125-132
[23] Hartman P. On completely continuous Hankel metrices. Proc Amer Math Soc, 1958, 9: 862-866
[24] Henkin G. Integral representation of a function in a strictly pseudoconvex domain and applications to the $\overline{\partial}$-problem (Russian). Math Sb, 1970, 82: 300-308
[25] Hörmander L. $L^p$-estimates for plurisubharmonic functions. Math Scand, 1967, 20: 65-78
[26] Hörmander L.An Introduction to Complex Analysis in Several Variables. New York: North-Holland, 1973
[27] Krantz S, Li S. On decomposition theorems for Hardy spaces on domains in $\mathbb{C}^n$ and applications. J Fourier Anal Appl, 1995, 2: 65-107
[28] Krantz S, Li S.Hardy classes, integral operators on spaces of homogenous type. arXiv:math/9601210
[29] Li H. Schatten class Hankel operators on strongly pseudoconvex domains. Proc Amer Math Soc, 1993, 199: 1211-1221
[30] Li H. Hankel operators on the Bergman spaces of strongly pseudoconvex domains. Integral Equ Oper Theory, 1994, 19: 458-476
[31] Li H, Luecking D. BMO on strongly pseudoconvex domains: Hankel operators, duality and $\overline{\partial}$-estimates. Trans Amer Math Soc, 1994, 346: 661-691
[32] Li S, Luo W. Analysis on Besov spaces II: embedding and duality theorems. J Math Anal Appl, 2007, 333: 1189-1202
[33] Lin P, Rochberg R. Hankel operators on the weighted Bergman spaces with exponential type weights. Integral Equ Oper Theory, 1995, 21: 460-483
[34] Luecking D. Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disc. J Funct Anal, 1992, 110: 247-271
[35] Nehari Z. On bounded linear forms. Ann Math, 1957, 65: 153-162
[36] Øvrelid N. Integral representation formulas and $L^p$ estimates for a equation. Math Scand, 1971, 29: 137-160
[37] Partington J.An Introduction to Hankel Operators. Cambridge: Cambridge University Press, 1988
[38] Peller V.Hankel Operators and Their Applications. New York: Springer-Verlag, 2003
[39] Power S.Hankel Operators on Hilbert Spaces. London: Pitman, 1982
[40] Range R.Holomorphic Functions and Integral Representations in Several Complex Variables. New York: Springer-Verlag, 1986
[41] Rudin W.Function Theory in the Unit Ball of $\mathbb{C}^n$. New York: Springer-Verlag, 1980
[42] Sato H, Yabuta K. Toeplitz operators on strongly pseudoconvex domains in stein spaces. Tohoku Math J, 1978, 30: 153-162
[43] Skoda H. Valeurs au bord pour les solutions de opérateur $d''$ et caracérisation de zéros des fonctions de la classe de Nevanlinna. Bull Soc Math France, 1976, 104: 225-299
[44] Stein E.Boundary Behavior of Holomorphic Functions of Several Complex Variables. Princeton: Princeton University Press, 1972
[45] Stout E. $H^p$-Functions on strictly pseudoconvex domains. Amer J Math, 1976, 98: 821-852
[46] Varopoulos N. BMO functions and the $\overline{\partial}$-equation. Pacific J Math, 1977, 71: 221-273
[47] Xia J. Bounded functions of vanishing mean oscillation on compact metric spaces. J Funct Anal, 2004, 209: 444-467
[48] Xia J. Boundedness and compactness of Hankel operators on the sphere. J Funct Anal, 2008, 255: 25-45
[49] Zheng D. Toeplitz operators and Hankel operators on the Hardy space of the unit sphere. J Funct Anal, 1997, 149: 1-24
[50] Zhu K.Operator Theory in Function Spaces. Providence, RI: American Mathematical Society, 2007