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 $k\geq 0$, let $dw(\mathbf x)=Π_{α\in \mathcal R}|\langle \mathbf x,α\rangle|^{k(α)}\, d\mathbf x$, $\mathbf{N}=N+\sum_{α\in \mathcal R}k(α)$ denote the associated measure and the homogeneous dimension of the system $(\mathcal R,k)$ respectively. Let $\mathcal F$ stand for the Dunkl transform. For $0<p\leq 1$, 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 $\mathcal T_mf=\mathcal F^{-1}(m\mathcal Ff)$, initially defined on $H^p_{\mathrm{Dunkl}}\cap L^2(dw)$, has a unique extension to a bounded operator in $H^p_{\mathrm{Dunkl}}$, where the space $H^p_{\mathrm{Dunkl}}$ is defined by means of a Littlewood-Paley square function. To prove the theorem, we use special atomic and molecule characterizations of $H^p_{\mathrm{Dunkl}}$.

Theorems & Definitions (48)

• Definition 1.1
• Theorem 1.2
• Lemma 2.1
• Theorem 2.2
• Theorem 2.3: Theorem 4.1, [DH]
• Theorem 2.4
• Lemma 2.5
• proof
• Lemma 3.1
• proof