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On Hankel Forms of Higher Weights: The Case of Hardy Spaces

Published online by Cambridge University Press:  20 November 2018

Marcus Sundhäll  and
Edgar Tchoundja
Marcus Sundhäll
Affiliation:
Mathematical Institute, Silesian University in Opava, Na Rybnicku 1, CZ-746 01, Czech Republic, e-mail: Marcus.Sundhall@oru.se
Edgar Tchoundja
Affiliation:
Department of Mathematical Sciences, Division of Mathematics, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden, e-mail: etchoundja@crm.cat
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Abstract

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In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by Sundhäll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.

Keywords

32A2532A3532A3747B35Hankel formsSchatten–von Neumann classesBergman spacesHardy spacesBesov spacestransvectantsunitary representationsMöbius group
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 2 , 01 April 2010 , pp. 439 - 455
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Aleman, A. and Constantin, O., Hankel operators on Bergman spaces and similarity to contractions. Int. Math. Res. Not. 2004, no. 35, 1785–1801.Google Scholar
[2] Beatrous, F. and Burbea, J., Holomorphic Sobolev spaces on the ball. Dissertationes Math. (Rozprawy Mat.) 276(1989).Google Scholar
[3] Burbea, J., Boundary behavior of holomorphic functions in the ball. Pacific J. Math. 127(1987), no. 1, 1–17.Google Scholar
[4] Coifman, R., Rochberg, R., and G.Weiss, Factorization theorems for Hardy spaces in several variables. Ann. of Math. 103(1976), no. 3, 611–635. doi:10.2307/1970954Google Scholar
[5] Feldman, M. and Rochberg, R., Singular value estimates for commutators and Hankel operators on the unit ball and the Heisenberg group. In: Analysis and Partial Differential Equations. Lecture Notes in Pure and Appl. Math. 122. Dekker, New York, 1990, pp. 121–159.Google Scholar
[6] Janson, S. and Peetre, J., A new generalization of Hankel operators (the case of higher weights). Math. Nachr. 132(1987), 313–328. doi:10.1002/mana.19871320121Google Scholar
[7] Janson, S., Peetre, J., and Rochberg, R., Hankel forms and the Fock space. Revista Mat. Iberoamericana 3(1987), no. 1, 61-138.Google Scholar
[8] Peller, V. V., Vectorial Hankel operators, commutators and related operators of the Schatten–von Neumann class °p. Integral Equations Operator Theory 5(1982), no. 2, 244–272. doi:10.1007/BF01694041Google Scholar
[9] Peller, V. V., Hankel operators and their applications. Springer-Verlag, New York, 2003.Google Scholar
[10] Peng, L. and Zhang, G., Tensor products of holomorphic representations and bilinear differential operators. J. Funct. Anal. 210(2004), no. 1, 171–192. doi:10.1016/j.jfa.2003.09.006Google Scholar
[11] Rochberg, R., Trace ideal criteria for Hankel operators and commutators. Indiana Univ. Math. J. 31(1982), no. 6, 913–925. doi:10.1512/iumj.1982.31.31062Google Scholar
[12] Rudin, W., The radial variation of analytic functions. Duke Math. J. 22(1955), 235–242. doi:10.1215/S0012-7094-55-02224-9Google Scholar
[13] Rudin, W., Function Theory in the Unit Ball of Cn. Grundlehren der Mathematischen Wissenschaften 241. Springer-Verlag, New York, 1980.Google Scholar
[14] Sundhäll, M., Schatten-von Neumann properties of bilinear Hankel forms of higher weights. Math. Scand. 98(2006), no. 2, 283–319.Google Scholar
[15] Sundhäll, M., Trace class criteria for bilinear Hankel forms of higher weights. Proc. Amer. Math. Soc. 135(2007), no. 5, 1377–1388. doi:10.1090/S0002-9939-06-08583-2Google Scholar
[16] Zhang, G., Hankel operators on Hardy spaces and Schatten classes. Chinese Ann. Math. Ser. B, 12(1991), no. 3, 282–294.Google Scholar
[17] Zhao, R. and Zhu, K., Theory of Bergman spaces in the unit ball of Cn. To appear in Mem. Soc. Math. France.Google Scholar
[18] Zhu, K., Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics 226. Springer-Verlag, New York, 2005.Google Scholar
[19] Zhu, K., A class of Möbius invariant function spaces. Illinois J. Math. 51(2007), no. 3, 977–1002.Google Scholar