Riesz transforms, Cauchy-Riemann systems and amalgam Hardy spaces

TL;DR

This paper explores Hardy spaces over amalgam spaces, characterizes them using Riesz transforms and Cauchy-Riemann systems, and describes their boundary values and Fourier multiplier characterizations.

Contribution

It provides new characterizations of amalgam Hardy spaces via Riesz transforms, boundary value descriptions through harmonic and caloric Cauchy-Riemann systems, and Fourier multiplier criteria.

Findings

Characterization of Hardy spaces using Riesz transforms.

Boundary value description via harmonic and caloric Cauchy-Riemann systems.

Fourier multiplier characterization of functions in the intersection with L^2.

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 $\mathbb{R}^d$.