  Let $X$ be a ball quasi-Banach function space satisfying some mild
assumptions and $H_X(\mathbb{R}^n)$ the Hardy space associated with $X$. In
this article, the authors introduce both the Hardy space
$H_X(\mathbb{R}^{n+1}_+)$ of harmonic functions and the Hardy space
$\mathbb{H}_X(\mathbb{R}^{n+1}_+)$ of harmonic vectors, associated with $X$,
and then establish the isomorphisms among $H_X(\mathbb{R}^n)$,
$H_{X,2}(\mathbb{R}^{n+1}_+)$, and $\mathbb{H}_{X,2}(\mathbb{R}^{n+1}_+)$,
where $H_{X,2}(\mathbb{R}^{n+1}_+)$ and $\mathbb{H}_{X,2}(\mathbb{R}^{n+1}_+)$
are, respectively, certain subspaces of $H_X(\mathbb{R}^{n+1}_+)$ and
$\mathbb{H}_X(\mathbb{R}^{n+1}_+)$. Using these isomorphisms, the authors
establish the first order Riesz transform characterization of
$H_X(\mathbb{R}^n)$. The higher order Riesz transform characterization of
$H_X(\mathbb{R}^n)$ is also obtained. The results obtained in this article have
a wide range of generality and can be applied to the classical Hardy space, the
weighted Hardy space, the Herz-Hardy space, the Lorentz-Hardy space, the
variable Hardy space, the mixed-norm Hardy space, the local generalized
Herz-Hardy space, and the mixed-norm Herz-Hardy space.
