Dual spaces of anisotropic mixed-norm Hardy spaces

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 $\mathbb {R}^n$.