Dual Spaces of Anisotropic Mixed-Norm Hardy Spaces

Abstract

Let $\vec{a}:=(a_1,\dots ,a_n)\in [1,\infty {)}^n$, $\vec{p}:=(p_1,\dots ,p_n)\in (0,\infty {)}^n$ 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 $s\in {\mathbb{Z}}_{+}:=\{0,1,\dots \}$, 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 ${\mathcal{L}}_{\vec{p},r^{\prime },s}^{\vec{a}}({\mathbb{R}}^n)$ with $\vec{p}\in (0,1{]}^n$, $r\in (1,\infty ]$, $1/r+1/r^{\prime }=1$ and $s\in [\lfloor \frac{\nu }{a_{-}}(\frac{1}{p_{-}}-1)\rfloor ,\infty )\cap {\mathbb{Z}}_{+}$, where $\nu :=a_1+\cdots +a_n$, $a_{-}:=\min \{a_1,\dots ,a_n\}$, $p_{-}:=\min \{p_1,\dots ,p_n\}$ 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 ${\mathbb{R}}^n$.

15 pages; Submitted