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Duality of multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular integral operators


Authors: Wei Ding and Guozhen Lu
Journal: Trans. Amer. Math. Soc. 368 (2016), 7119-7152
MSC (2010): Primary 42B35, 42B30, 42B25, 42B20
DOI: https://doi.org/10.1090/tran/6576
Published electronically: January 21, 2016
MathSciNet review: 3471087
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Abstract: In this paper, we study the duality theory of the multi-parameter Triebel-Lizorkin spaces $ \dot F^{\alpha ,q}_{p}(\mathbb{R}^{m})$ associated with the composition of two singular integral operators on $ \mathbb{R}^m$ of different homogeneities. Such composition of two singular operators was considered by Phong and Stein in 1982. For
$ 1<p<\infty $, we establish the dual spaces of such spaces as $ (\dot F^{\alpha ,q}_{p}(\mathbb{R}^{m}))^*=\dot F^{-\alpha ,q'}_{p'}(\mathbb{R}^{m})$, and for $ 0<p\leq 1$ we prove $ (\dot F^{\alpha ,q}_{p}(\mathbb{R}^{m}))^*=CMO^{-\alpha ,q'}_{p}(\mathbb{R}^m)$. We then prove the boundedness of the composition of two Calderón-Zygmund singular integral operators with different homogeneities on the spaces $ CMO^{-\alpha ,q'}_{p}$. Surprisingly, such dual spaces are substantially different from those for the classical one-parameter Triebel-Lizorkin spaces $ \dot {\mathcal {F}}^{\alpha ,q}_p(\mathbb{R}^{m})$. Our work requires more complicated analysis associated with the underlying geometry generated by the multi-parameter structures of the composition of two singular integral operators with different homogeneities. Therefore, it is more difficult to deal with than the duality result of the Triebel-Lizorkin spaces in the one-paramter settings. We note that for $ 0<p\leq 1$, $ q=2$ and $ \alpha =0$, $ \dot F^{\alpha ,q}_{p}(\mathbb{R}^{m})$ is the Hardy space associated with the composition of two singular operators considered in Rev. Mat. Iberoam. 29 (2013), 1127-1157. Our work appears to be the first effort on duality for Triebel-Lizorkin spaces in the multi-parameter setting.


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Additional Information

Wei Ding
Affiliation: School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, Beijing Normal University, Beijing, 100875, People’s Republic of China – and – School of Sciences, Nantong University, Nantong 226007, People’s Republic of China
Email: dingwei@ntu.edu.cn

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

DOI: https://doi.org/10.1090/tran/6576
Keywords: Multi-parameter Triebel-Lizorkin spaces, Calder\'on-Zygmund operators, non-isotropic dilations, composition of operators, discrete Calder\'on's identity, duality
Received by editor(s): February 14, 2014
Received by editor(s) in revised form: September 5, 2014
Published electronically: January 21, 2016
Additional Notes: The research of the first author was partly supported by NNSF of China grants (No. 11371056 and No. 11271209) and the second author was partly supported by US NSF grant DMS#1301595. The second author is the corresponding author.
Article copyright: © Copyright 2016 American Mathematical Society