Riesz transforms, Cauchy-Riemann systems and amalgam Hardy spaces

In this paper we study Hardy spaces ${\mathcal{H}}^{p,q}({\mathbb{R}}^d)$, $0<p,q<\mathrm{\infty }$, modeled over amalgam spaces $(L^p,{\mathrm{\ell }}^q)({\mathbb{R}}^d)$. 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)$bymeansofFouriermultipliers$m_\theta$withsymbol$\theta(\cdot/|\cdot|)$,where$\theta \in C^\infty($\mathbb{S}$^{d-1})$and$$\mathbb{S}$^{d-1}$denotestheunitspherein$$\mathbb{R}$^d$.