Dual spaces of anisotropic mixed-norm Hardy spaces

By Long Huang, Jun Liu, Dachun Yang, Wen Yuan. Proceedings of the American Mathematical Society, 2018

95 citations recorded by the source corpus.

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), 2758-2787]. More precisely, via first introducing the anisotropic mixed-norm Campanato space $\mathcal{L}_{\vec{p},\,q,\,s}^{\vec{a}}(\mathbb{R}^n)$ with $q\in[1,\infty]$ and $s\in\mathbb{Z}_+:=\{0,1,\ldots\}$, and 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 $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ is the space $\mathcal{L}_{\vec{p},\,r',\,s}^{\vec{a}}(\mathbb{R}^n)$ with $\vec{p}\in(0,1]^n$, $r\in(1,\infty]$, $1/r+1/r'=1$ 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 $\mathbb{R}^n$.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/45bf576aba97c80fff5e396584735076791cec89

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Dachun Yang
2. Wen Yuan
3. Yangyang Zhang
4. Songbai Wang
5. T. Nogayama
6. Long Huang
7. Yoshihiro Sawano
8. Jun Liu

In Mathematical Physics

1. Dachun Yang
2. Wen Yuan
3. Yangyang Zhang
4. Songbai Wang
5. T. Nogayama
6. Yoshihiro Sawano
7. Ciqiang Zhuo
8. Jun Liu

Citations in the corpus

• Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation (2019, background): “some other fields, the real-variable theory of mixed-norm function spaces, including mixed-norm Morrey spaces, mixed-norm Hardy spaces, mixed-norm Besov spaces and mixed-norm Triebel– Lizorkin spaces, has rapidly been developed in recent years (see, for instance, [20, 32, 63, 42, 43, 44]).”
• Boundedness of commutators of fractional integral operators on mixed Morrey spaces (2019, background): “Moreover, other spaces with mixed norm, mixed Lorentz spaces [10–12], mixed Besov spaces and Triebel–Lizorkin spaces [13,14], andmixedHardy spaces [15,16] have been studied in recent years.”
• On Function Spaces with Mixed Norms — A Survey (2019, unknown): “To present this, we first introduce the notion of the anisotropic mixed-norm Campanato space L~a ~p,q,s(R n) given in [43].”