Riesz transforms, Cauchy-Riemann systems and amalgam Hardy spaces

Abstract

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 ${\mathbb{R}}^d$.

24 pages