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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Contractive inequalities for Bergman spaces and multiplicative Hankel forms


Authors: Frédéric Bayart, Ole Fredrik Brevig, Antti Haimi, Joaquim Ortega-Cerdà and Karl-Mikael Perfekt
Journal: Trans. Amer. Math. Soc. 371 (2019), 681-707
MSC (2010): Primary 30H20; Secondary 47B35, 30B50
DOI: https://doi.org/10.1090/tran/7290
Published electronically: August 21, 2018
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Abstract: We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield corresponding inequalities for the Bergman spaces of Dirichlet series. We use these results to study weighted multiplicative Hankel forms associated with the Bergman spaces of Dirichlet series, reproducing most of the known results on multiplicative Hankel forms associated with the Hardy spaces of Dirichlet series. In addition, we find a direct relationship between the two types of forms which does not exist in lower dimensions. Finally, we produce some counterexamples concerning Carleson measures on the infinite polydisc.


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Additional Information

Frédéric Bayart
Affiliation: Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France
Email: bayart@math.univ-bpclermont.fr

Ole Fredrik Brevig
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway
Email: ole.brevig@math.ntnu.no

Antti Haimi
Affiliation: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
Email: antti.haimi@univie.ac.at

Joaquim Ortega-Cerdà
Affiliation: Department de Matemàtiques i Informàtica, Universitat de Barcelona & Barcelona Graduate school in mathematics, Gran Via 585, 08007 Barcelona, Spain
Email: jortega@ub.edu

Karl-Mikael Perfekt
Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996
Email: karlmikael.perfekt@gmail.com

DOI: https://doi.org/10.1090/tran/7290
Received by editor(s): January 31, 2017
Received by editor(s) in revised form: May 9, 2017, and May 10, 2017
Published electronically: August 21, 2018
Additional Notes: The second-named author was supported by Grant 227768 of the Research Council of Norway. The third-named author was supported by Lise Meitner grant of Austrian Science Fund (FWF). The fourth-named author was supported by the MTM2014-51834-P grant by the Ministerio de Economía y Competitividad, and by the Generalitat de Catalunya (project 2014 SGR 289).
Article copyright: © Copyright 2018 American Mathematical Society

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