On Sobolev norms for Lie group representations

We define Sobolev norms of arbitrary real order for a Banach representation $(\pi, E)$ Lie group, with regard to single differential operator $D=d\pi(R^2+\Delta)$. Here, $\Delta$ is Laplace element in the universal enveloping algebra, and $R>0$ depends explicitly on growth rate representation. In particular, we obtain spectral gap $D$ space smooth vectors $E$. The main tool novel factorization delta distribution group.