Boundedness of Hankel operators on $ {\mathcal{H}}^{1}({\mathbb{B}}^{n})$

Bonami, Aline; Grellier, Sandrine; Sehba, Benoît F.

doi:10.1016/j.crma.2007.05.004

Complex Analysis

Boundedness of Hankel operators on

H^{1} (B^{n})

Presented by: Yves Meyer

Bonami, Aline ¹; Grellier, Sandrine ¹; Sehba, Benoît F.²

¹ Fédération Denis-Poisson, MAPMO-UMR 6628, département de mathématiques, université d'Orléans, 45067 Orléans cedex 2, France
² Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK

Comptes Rendus. Mathématique, Volume 344 (2007) no. 12, pp. 749-752.

Abstract
Résumé

We prove that the Hankel operator $h_{b}$ associated to the Szegö projection on the unit ball $B^{n}$ is bounded on the Hardy space $H^{1} (B^{n})$ if and only if its symbol b has logarithmic mean oscillation on the unit sphere.

On démontre que l'opérateur de Hankel $h_{b}$ associé au projecteur de Szegö sur la boule unité s'étend continûment à l'espace de Hardy $H^{1} (B^{n})$ si et seulement si b est à oscillation moyenne logarithmique sur la sphère unité.

Received: 2007-02-09
Accepted: 2007-05-15
Published online: 2007-06-15

DOI: 10.1016/j.crma.2007.05.004

Author's affiliations:

Bonami, Aline ¹; Grellier, Sandrine ¹; Sehba, Benoît F. ²

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     title = {Boundedness of {Hankel} operators on $ {\mathcal{H}}^{1}({\mathbb{B}}^{n})$},
     journal = {Comptes Rendus. Math\'ematique},
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Bonami, Aline; Grellier, Sandrine; Sehba, Benoît F. Boundedness of Hankel operators on $ {\mathcal{H}}^{1}({\mathbb{B}}^{n})$. Comptes Rendus. Mathématique, Volume 344 (2007) no. 12, pp. 749-752. doi : 10.1016/j.crma.2007.05.004. https://www.numdam.org/articles/10.1016/j.crma.2007.05.004/

References
Cited by

[1] A. Bonami, S. Grellier, Decomposition theorems for Hardy–Orlicz spaces and weak factorization, Preprint, 2007

[2] Bonami, A.; Peloso, M.; Symesak, F. Factorization of Hardy spaces and Hankel operators on convex domains in $C^{n}$ , J. Geom. Anal., Volume 11 (2001) no. 3, pp. 363-397

[3] Janson, S.; Peetre, J.; Semmes, S. On the action of Hankel and Toeplitz operators on some function spaces, Duke Math. J., Volume 51 (1984) no. 4, pp. 937-958

[4] Rudin, W. Function Theory in the Unit Ball of $C^{n}$ , Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Science, vol. 241, Springer-Verlag, New York–Berlin, 1980 (xiii+436 pp) (ISBN: 0-387-90514-6)

[5] Smith, W.S. $BMO (ρ)$ and Carleson measures, Trans. Amer. Math. Soc., Volume 287 (1985) no. 1, pp. 107-126

[6] Tolokonnikov, V.A. Hankel and Toeplitz operators in Hardy spaces, Soviet Math., Volume 37 (1987), pp. 1359-1364

[7] Zhao, R. On logarithmic Carleson measures, Acta Sci. Math. (Szeged), Volume 69 (2003), pp. 605-618

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