I am studying a certain model kinetic equation. To study that system, I have to find a function space, which is a subspace of $L^1 (\mathbb{R}^d)$ and the operator $f \rightarrow x \cdot \nabla\_x f$, which corresponds to a drift, is continuous in the space. Of course Schwartz space is such a space, but are there any normed spaces (say, we only need to look at finitely many derivatives and finite order polynomials multiplied) satisfying the property?


Thank you in advance for any comments or suggestions.



A nice function space closed with the operation $x \cdot \nabla $
This [article](http://popups.ulg.ac.be/0037-9565/index.php?id=1768) of Langenbruch and Voigt might be relevant for you. It shows that every Banach space $E$ continuously included in $\mathscr D'(\mathbb R^d)$ which is closed under differentiation is already contained in a weighted space of entire functions $\mathscr H\_A=\lbrace f\in H(\mathbb C^d): \int |f(z)|^2 \exp(-2A|z|)dz$ for some $A>0$ (more precisely, every $h\in E$ extends uniquely to some $f\in \mathscr H\_A$). 


