Products and Convolutions in Hermite-Sobolev spaces

In this paper, we show that the product, or equivalently the convolutions of two functions in the Hermite-Sobolev spaces ${\mathcal{S}}_p({\mathbb{R}}^d)$ is again in the same space, for $p$ depending on the dimension $d$. As a consequence, for such $p$ we show that the product, or equivalently the convolutions of $\varphi \in {\mathcal{S}}_p({\mathbb{R}}^d)$ and $\psi \in {\mathcal{S}}_{-p}({\mathbb{R}}^d)$ is in ${\mathcal{S}}_{-p}({\mathbb{R}}^d)$. As a further consequence, we show that the operators of translation by $x$ on ${\mathcal{S}}_p({\mathbb{R}}^d)$ are bounded uniformly in $x\in {\mathbb{R}}^d$.