Hörmander's multiplier theorem on $H^p$-spaces in the rational Dunkl setting

Abstract

On ${\mathbb{R}}^N$ equipped with a normalized root system $\mathcal{R}$ and a multiplicity function $k\ge 0$, let $dw(\mathbf{x})={\Pi }_{\alpha \in \mathcal{R}}|⟨\mathbf{x},\alpha ⟩{|}^{k(\alpha )}d\mathbf{x}$, $\mathbf{N}=N+{\sum }_{\alpha \in \mathcal{R}}k(\alpha )$ denote the associated measure and the homogeneous dimension of the system $(\mathcal{R},k)$ respectively. Let $\mathcal{F}$ stand for the Dunkl transform. For $0<p\le 1$, let $m$ be a bounded function on ${\mathbb{R}}^N$, which satisfies the classical Hörmander's condition with smoothness $s>\mathbf{N}/p$. We show that the multiplier operator ${\mathcal{T}}_{m}f={\mathcal{F}}^{-1}(m\mathcal{F}f)$, initially defined on $H_{\mathrm{Dunkl}}^p\cap L^2(dw)$, has a unique extension to a bounded operator in $H_{\mathrm{Dunkl}}^p$, where the space $H_{\mathrm{Dunkl}}^p$ is defined by means of a Littlewood-Paley square function. To prove the theorem, we use special atomic and molecule characterizations of $H_{\mathrm{Dunkl}}^p$.

27 pages