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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a characterization of bilinear forms on the Dirichlet space


Authors: Carme Cascante and Joaquin M. Ortega
Journal: Proc. Amer. Math. Soc. 140 (2012), 2429-2440
MSC (2010): Primary 31C25, 31C15, 47B35
Published electronically: November 15, 2011
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Abstract: Arcozzi, Rochberg, Sawyer and Wick obtained a characterization of the holomorphic functions $ b$ such that the Hankel type bilinear form $ T_{b}(f,g)= \int _{\D }(I+R)(fg)(z) \overline {(I+R)b(z)}dv(z)$ is bounded on $ {\mathcal D}\times {\mathcal D}$, where $ {\mathcal D}$ is the Dirichlet space. In this paper we give an alternative proof of this characterization which tries to understand the similarity with the results of Maz$ '$ya and Verbitsky relative to the Schrödinger forms on the Sobolev spaces $ L_2^1(\mathbb{R}^n)$.


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Additional Information

Carme Cascante
Affiliation: Departmento Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain
Email: cascante@ub.edu

Joaquin M. Ortega
Affiliation: Departmento Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain
Email: ortega@ub.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11409-6
PII: S 0002-9939(2011)11409-6
Keywords: Dirichlet spaces, Carleson measures
Received by editor(s): February 23, 2011
Published electronically: November 15, 2011
Additional Notes: The authors were partially supported by DGICYT Grant MTM2011-27932-C02-01, DURSI Grant 2009SGR 1303 and Grant MTM2008-02928-E
Communicated by: Richard Rochberg
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.