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

 

Multiparameter Hardy space theory on Carnot-Carathéodory spaces and product spaces of homogeneous type


Authors: Yongsheng Han, Ji Li and Guozhen Lu
Journal: Trans. Amer. Math. Soc. 365 (2013), 319-360
MSC (2010): Primary 42B35; Secondary 32T25, 32W30
Published electronically: July 24, 2012
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Abstract: This paper is inspired by the work of Nagel and Stein in which the $ L^p$ $ (1<p<\infty )$ theory has been developed in the setting of the product Carnot-Carathéodory spaces $ \widetilde {M}=M_1\times \cdots \times M_n$ formed by vector fields satisfying Hörmander's finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparameter Hardy space theory on product spaces of homogeneous type. This theory includes the product Hardy space, its dual, the product $ BMO$ space, the boundedness of singular integral operators and Calderón-Zygmund decomposition and interpolation of operators. As a consequence, we obtain the endpoint estimates for those singular integral operators considered by Nagel and Stein (2004). In fact, we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric. All of our results are established by introducing certain Banach spaces of test functions and distributions, developing discrete Calderón identity and discrete Littlewood-Paley-Stein theory. Our methods do not rely on the Journé-type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product Hardy spaces.


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Additional Information

Yongsheng Han
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849
Email: hanyong@auburn.edu

Ji Li
Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
Email: liji6@mail.sysu.edu.cn

Guozhen Lu
Affiliation: School of Mathematical Science, Beijing Normal University, Beijing, 100875, People’s Republic of China – and – Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: gzlu@math.wayne.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05638-8
PII: S 0002-9947(2012)05638-8
Keywords: Kohn-Laplacian, heat equation, Shilov boundary, finite type domains, multiparameter Hardy space, Carleson measure space, Calderón’s reproducing formula, almost orthogonality estimate, sup-inf-type inequality, sequence spaces, duality, singular integral operators.
Received by editor(s): May 13, 2010
Received by editor(s) in revised form: February 21, 2011
Published electronically: July 24, 2012
Additional Notes: The first two authors were supported by NNSF of China (Grant No. 11001275).
The third author is the corresponding author and was partly supported by US National Science Foundation grants DMS0500853 and DMS0901761.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.