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

 

Classes of Hardy spaces associated with operators, duality theorem and applications


Author: Lixin Yan
Journal: Trans. Amer. Math. Soc. 360 (2008), 4383-4408
MSC (2000): Primary 42B30, 42B35, 47B38
Published electronically: March 20, 2008
MathSciNet review: 2395177
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Abstract: Let $ L$ be the infinitesimal generator of an analytic semigroup on $ L^2({\mathbb{R}}^n)$ with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space $ H^1_L({\mathbb{R}}^n)$ and a $ {\rm BMO}_L({\mathbb{R}}^n)$ space associated with the operator $ L$ were introduced and studied. In this paper we define a class of $ H^p_L({\mathbb{R}}^n)$ spaces associated with the operator $ L$ for a range of $ p<1$ acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical $ H^p({\mathbb{R}}^n)$ spaces. We then establish a duality theorem between the $ H^p_L({\mathbb{R}}^n)$ spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on $ H^p_L({\mathbb{R}}^n)$ and give the inclusion between the classical $ H^p({\mathbb{R}}^n)$ spaces and the $ H^p_L({\mathbb{R}}^n)$ spaces associated with operators.


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Additional Information

Lixin Yan
Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
Email: mcsylx@mail.sysu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04476-0
PII: S 0002-9947(08)04476-0
Received by editor(s): July 15, 2005
Received by editor(s) in revised form: September 5, 2006
Published electronically: March 20, 2008
Additional Notes: The author was supported by NNSF of China (Grant No. 10571182/10771221) and by a grant from the Australia Research Council.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.