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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Duality in spaces of finite linear combinations of atoms


Authors: Fulvio Ricci and Joan Verdera
Journal: Trans. Amer. Math. Soc. 363 (2011), 1311-1323
MSC (2010): Primary 42B30; Secondary 46J99
Published electronically: October 15, 2010
MathSciNet review: 2737267
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Abstract: In this paper we describe the dual and the completion of the space of finite linear combinations of $ (p,\infty)$-atoms, $ 0<p\leq 1 $. As an application, we show an extension result for operators uniformly bounded on $ (p,\infty)$-atoms, $ 0<p < 1$, whose analogue for $ p=1$ is known to be false. Let $ 0 < p <1$ and let $ T$ be a linear operator defined on the space of finite linear combinations of $ (p,\infty)$-atoms, $ 0<p < 1 $, which takes values in a Banach space $ B$. If $ T$ is uniformly bounded on $ (p,\infty)$-atoms, then $ T$ extends to a bounded operator from $ H^p(\mathbb{R}^n)$ into $ B$.


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Additional Information

Fulvio Ricci
Affiliation: Reparto di Matematica, Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italia
Email: fricci@sns.it

Joan Verdera
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra, Barcelona, Catalonia
Email: jvm@mat.uab.cat

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05036-6
PII: S 0002-9947(2010)05036-6
Received by editor(s): October 23, 2008
Published electronically: October 15, 2010
Article copyright: © Copyright 2010 American Mathematical Society