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On the $ H^1$-$ L^1$ boundedness of operators

Authors: Stefano Meda, Peter Sjögren and Maria Vallarino
Journal: Proc. Amer. Math. Soc. 136 (2008), 2921-2931
MSC (2000): Primary 42B30, 46A22
Published electronically: April 3, 2008
MathSciNet review: 2399059
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Abstract: We prove that if $ q$ is in $ (1,\infty)$, $ Y$ is a Banach space, and $ T$ is a linear operator defined on the space of finite linear combinations of $ (1,q)$-atoms in $ \mathbb{R}^n$ with the property that

$\displaystyle \sup\{\Vert{Ta}\Vert {Y}: \hbox{$a$ is a $(1,q)$-atom} \} < \infty, $

then $ T$ admits a (unique) continuous extension to a bounded linear operator from $ H^1({\mathbb{R}^n})$ to $ Y$. We show that the same is true if we replace $ (1,q)$-atoms by continuous $ (1,\infty)$-atoms. This is known to be false for $ (1,\infty)$-atoms.

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Additional Information

Stefano Meda
Affiliation: Dipartimento di Matematica e Applicazioni, Universitá degli Studi di Milano–Bicocca, Via Cozzi, 53, 20125 Milano, Italy

Peter Sjögren
Affiliation: Department of Mathematical Sciences, University of Gothenburg, SE-412 96 Göteborg, Sweden; and Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg, Sweden

Maria Vallarino
Affiliation: Laboratoire MAPMO UMR 6628, Fédération Denis Poisson, Université d’Orléans, UFR Sciences, Bâtiment de mathématiques – Route de Chartres, B.P. 6759 – 45067 Orléans cedex 2, France

Keywords: BMO, atomic Hardy space, extension of operators.
Received by editor(s): June 18, 2007
Published electronically: April 3, 2008
Additional Notes: This work was partially supported by the Progetto Cofinanziat “Analisi Armonica”.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2008 American Mathematical Society

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