Dual spaces of anisotropic mixed-norm Hardy spaces
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- by Long Huang, Jun Liu, Dachun Yang and Wen Yuan PDF
- Proc. Amer. Math. Soc. 147 (2019), 1201-1215 Request permission
Abstract:
Let $\vec {a}:=(a_1,\ldots ,a_n)\in [1,\infty )^n$, $\vec {p}:=(p_1,\ldots ,p_n)\in (0,\infty )^n$ and $H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$ be the anisotropic mixed-norm Hardy space associated with $\vec {a}$ defined via the non-tangential grand maximal function. In this article, the authors give the dual space of $H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$, which was asked by Cleanthous et al. in [J. Geom. Anal. 27 (2017), pp. 2758-2787]. More precisely, applying the known atomic and finite atomic characterizations of $H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$, the authors prove that the dual space of $H_{\vec {a}}^{\vec {p}}(\mathbb {R}^n)$, with $\vec {p}\in (0,1]^n$, is the anisotropic mixed-norm Campanato space $\mathcal {L}_{\vec {p}, r, s}^{\vec {a}}(\mathbb {R}^n)$ for every $r\in [1,\infty )$ and $s\in [\lfloor \frac {\nu }{a_-}(\frac {1}{p_-}-1) \rfloor ,\infty )\cap \mathbb {Z}_+$, where $\nu :=a_1+\cdots +a_n$, $a_-:=\min \{a_1,\ldots ,a_n\}$, $p_-:=\min \{p_1,\ldots ,p_n\}$ and, for any $t\in \mathbb {R}$, $\lfloor t\rfloor$ denotes the largest integer not greater than $t$. This duality result is new even for the isotropic mixed-norm Hardy spaces on $\mathbb {R}^n$.References
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Additional Information
- Long Huang
- Affiliation: Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
- Email: longhuang@mail.bnu.edu.cn
- Jun Liu
- Affiliation: Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
- Email: junliu@mail.bnu.edu.cn
- Dachun Yang
- Affiliation: Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
- MR Author ID: 317762
- Email: dcyang@bnu.edu.cn
- Wen Yuan
- Affiliation: Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
- MR Author ID: 743517
- Email: wenyuan@bnu.edu.cn
- Received by editor(s): March 26, 2018
- Received by editor(s) in revised form: July 1, 2018
- Published electronically: November 8, 2018
- Additional Notes: The third author was the corresponding author.
This project was supported by the National Natural Science Foundation of China (Grant Nos. 11761131002, 11571039, 11726621, and 11471042) and also by the Joint Research Project Between China Scholarship Council and German Academic Exchange Service (PPP) (Grant No. LiuJinOu [2016]6052). - Communicated by: Svitlana Mayboroda
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1201-1215
- MSC (2010): Primary 42B35; Secondary 42B30, 46E30
- DOI: https://doi.org/10.1090/proc/14348
- MathSciNet review: 3896067