I have a question concerning the subspace $\mathcal{S}'_h$ of tempered distributions defined by
$u\in\mathcal{S}'_h\Leftrightarrow\lim_{\lambda\rightarrow\infty}\Vert\theta(\lambda D)u\Vert_{\infty}=0\quad\forall \theta\in\mathcal{D}(\mathbb{R}^n)$,
where $\theta(\lambda D)u=\mathcal{F}^{-1}(\theta(\lambda\cdot)\mathcal{F}u)$. I shall proof that if $\theta(D)u$ belongs to $L^p$ for some $p\in\lbrack 1,\infty)$ then $u\in\mathcal{S}'_h$. Unfortunately I don't have any idea how to do this. Besides I don't even know how the $L^p$-Norm of a tempered distribution is defined. So I'm hoping for your help :)
greets Puki94