Sobolev spaces on Lie groups: embedding theorems and algebra properties

Let $G$ be a noncompact connected Lie group, denote with $\rho$ a right Haar measure and choose a family of linearly independent left-invariant vector fields $\mathbf{X}$ on $G$ satisfying H"ormander’s condition. Let $\chi$ be a positive character of $G$ and consider the measure ${\mu }_{\chi }$ whose density with respect to $\rho$ is $\chi$. In this paper, we introduce Sobolev spaces $L_{\alpha }^p({\mu }_{\chi })$ adapted to $\mathbf{X}$ and ${\mu }_{\chi }$ ($1<p<\mathrm{\infty }$, $\alpha \ge 0$) and study embedding theorems and algebra properties of these spaces. As an application, we prove local well-posedness and regularity results of solutions of some nonlinear heat and Schr"odinger equations on the group.