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Transactions of the American Mathematical Society

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A Toeplitz-type operator on Hardy spaces in the unit ball

Authors: Jordi Pau and Antti Perälä
Journal: Trans. Amer. Math. Soc. 373 (2020), 3031-3062
MSC (2010): Primary 47B35, 30H10
Published electronically: February 19, 2020
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Abstract: We study a Toeplitz-type operator $ Q_\mu $ between the holomorphic Hardy spaces $ H^p$ and $ H^q$ of the unit ball. Here the generating symbol $ \mu $ is assumed to be a positive Borel measure. This kind of operator is related to many classical mappings acting on Hardy spaces, such as composition operators, the Volterra-type integration operators, and Carleson embeddings. We completely characterize the boundedness and compactness of $ Q_\mu :H^p\to H^q$ for the full range $ 1<p,q<\infty $; and also describe the membership in the Schatten classes of $ H^2$. In the last section of the paper, we demonstrate the usefulness of $ Q_\mu $ through applications.

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Additional Information

Jordi Pau
Affiliation: Departament de Matemàtiques i Informàtica, Universitat de Barcelona, 08007 Barcelona, Catalonia, Spain

Antti Perälä
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg SE-412 96, Sweden

Keywords: Hardy spaces, Toeplitz operators, tent spaces, Schatten classes
Received by editor(s): May 11, 2018
Received by editor(s) in revised form: February 15, 2019
Published electronically: February 19, 2020
Additional Notes: The first author was partially supported by DGICYT grant MTM2014-51834-P (MCyT/MEC) and the grant 2017SGR358 (Generalitat de Catalunya). The second author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445). Both authors were also supported by the grant MTM2017-83499-P (Ministerio de Educación y Ciencia).
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