Multiparameter Hardy space theory on Carnot-Carathéodory spaces and product spaces of homogeneous type
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- by Yongsheng Han, Ji Li and Guozhen Lu PDF
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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.References
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Additional Information
- Yongsheng Han
- Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849
- MR Author ID: 209888
- 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
- MR Author ID: 322112
- Email: gzlu@math.wayne.edu
- 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. - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 319-360
- MSC (2010): Primary 42B35; Secondary 32T25, 32W30
- DOI: https://doi.org/10.1090/S0002-9947-2012-05638-8
- MathSciNet review: 2984061