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ISSN 1088-6826(e) ISSN 0002-9939(p)

Boundedness of operators on Hardy spaces via atomic decompositions

Author(s): Marcin Bownik
Journal: Proc. Amer. Math. Soc. 133 (2005), 3535-3542.
MSC (2000): Primary 42B30
Posted: 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 . 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 , $0<p \le 1$ , are not equivalent.

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