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

 

Bilinear operators on Herz-type Hardy spaces


Authors: Loukas Grafakos, Xinwei Li and Dachun Yang
Journal: Trans. Amer. Math. Soc. 350 (1998), 1249-1275
MSC (1991): Primary 47H19, 42B20, 42B30
MathSciNet review: 1407489
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Abstract: The authors prove that bilinear operators given by finite sums of products of Calderón-Zygmund operators on $\mathbb{R}^{n}$ are bounded from $ H\dot K_{q_{1}}^{\alpha _{1},p_{1}}\times H\dot K_{q_{2}}^{\alpha _{2},p_{2}}$ into $ H\dot K_{q}^{\alpha ,p}$ if and only if they have vanishing moments up to a certain order dictated by the target space. Here $H\dot K_{q}^{\alpha ,p}$ are homogeneous Herz-type Hardy spaces with $1/p=1/p_{1}+1/p_{2},$ $0<p_{i}\le \infty ,$ $1/q=1/q_{1}+1/q_{2},$ $1<q_{1},q_{2}<\infty ,$ $1\le q<\infty ,$ $\alpha =\alpha _{1}+\alpha _{2}$ and $-n/q_{i}<\alpha _{i}<\infty $. As an application they obtain that the commutator of a Calderón-Zygmund operator with a BMO function maps a Herz space into itself.


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Additional Information

Loukas Grafakos
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211-0001
Email: loukas@math.missouri.edu

Xinwei Li
Affiliation: Department of Mathematics, Washington University, Campus Box 1146, St. Louis, Missouri 63130-4899
Email: li@math.wustl.edu

Dachun Yang
Affiliation: Department of Mathematics, Beijing Normal University, 100875 Beijing, The People’s Republic of China
Email: dcyang@bnu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01878-9
PII: S 0002-9947(98)01878-9
Keywords: Herz spaces, Beurling algebras, Hardy spaces, atoms, bilinear operators, Calder\'{o}n-Zygmund operators
Received by editor(s): January 15, 1996
Received by editor(s) in revised form: July 15, 1996
Additional Notes: The first author’s research was supported by the University of Missouri Research Board
Article copyright: © Copyright 1998 American Mathematical Society