# Subclasses of distributions

I am wondering if there exists some known useful distribution spaces which are larger than tempered distributions, but that are defined from Banach test function spaces, as Schwartz space. For instance, one may think of the order $n$ norm $$\sup_{\vert \alpha \vert \leq n , x \in \mathbb{R}^d} \vert \partial^\alpha \varphi (x) \vert \chi (\vert x \vert )$$ where $\chi$ is a weighting function (say, exponential).

The advantage is obviously that order $n$ distributions are dual Banach spaces.

Have you any basic reference on this ?

Thanks.