Multipliers and convolution spaces for the Hankel space and its dual on the half space $[0,+\infty [ \times\mathbb{R}^n$

Abstract

We define the Hankel space $\mathbb{H}_\mu(]0,+\infty[\times\mathbb{R}^n)$; $\mu\geqslant -\frac{1}{2}$, and its dual $\mathbb{H'}_\mu(]0,+\infty[\times\mathbb{R}^n)$. First, we characterize the space $\mathscr{M}_\mu([0,+\infty[\times\mathbb{R}^n)$ of multipliers of the space $\mathbb{H}_\mu(]0,+\infty[\times\mathbb{R}^n)$. Next, we define a subspace $\mathbb{O}'_\mu([0,+\infty[\times \mathbb{R}^n)$ of the dual $\mathbb{H'}_\mu(]0,+\infty[\times\mathbb{R}^n)$ which permits to define and study a convolution product $\ast$ on $\mathbb{H'}_\mu(]0,+\infty[\times\mathbb{R}^n)$ and we give nice properties.