On Sobolev norms for Lie group representations

Abstract

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

10 pages, to appear in Journal of Functional Analysis