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Boundedness of operators on Hardy spaces via atomic decompositions


Author: Marcin Bownik
Journal: Proc. Amer. Math. Soc. 133 (2005), 3535-3542
MSC (2000): Primary 42B30
DOI: https://doi.org/10.1090/S0002-9939-05-07892-5
Published electronically: June 6, 2005
MathSciNet review: 2163588
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Abstract: An example of a linear functional defined on a dense subspace of the Hardy space $H^1(\mathbb{R}^n)$ is constructed. It is shown that despite the fact that this functional is uniformly bounded on all atoms, it does not extend to a bounded functional on the whole $H^1$. Therefore, this shows that in general it is not enough to verify that an operator or a functional is bounded on atoms to conclude that it extends boundedly to the whole space. The construction is based on the fact due to Y. Meyer which states that quasi-norms corresponding to finite and infinite atomic decompositions in $H^p$, $0<p \le 1$, are not equivalent.


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Additional Information

Marcin Bownik
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403–1222
Email: mbownik@uoregon.edu

DOI: https://doi.org/10.1090/S0002-9939-05-07892-5
Received by editor(s): July 8, 2004
Published electronically: June 6, 2005
Additional Notes: The author was partially supported by NSF grant DMS-0441817
Communicated by: Andreas Seeger
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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