  In this paper we study Hardy spaces $\mathcal{H}^{p,q}(\mathbb{R}^d)$,
$0<p,q<\infty$, modeled over amalgam spaces $(L^p,\ell^q)(\mathbb{R}^d)$. We
characterize $\mathcal{H}^{p,q}(\mathbb{R}^d)$ by using first order classical
Riesz transforms and compositions of first order Riesz transforms depending on
the values of the exponents $p$ and $q$. Also, we describe the distributions in
$\mathcal{H}^{p,q}(\mathbb{R}^d)$ as the boundary values of solutions of
harmonic and caloric Cauchy-Riemann systems. We remark that caloric
Cauchy-Riemann systems involve fractional derivative in the time variable.
Finally we characterize the functions in $L^2(\mathbb{R}^d) \cap
\mathcal{H}^{p,q}(\mathbb{R}^d)$ by means of Fourier multipliers $m_\theta$
with symbol $\theta(\cdot/|\cdot|)$, where $\theta \in
C^\infty(\mathbb{S}^{d-1})$ and $\mathbb{S}^{d-1}$ denotes the unit sphere in
$\mathbb{R}^d$.
