Sobolev spaces on Lie groups: embedding theorems and algebra properties

Abstract

Let $G$ be a noncompact connected Lie group, denote with $ρ$ a right Haar measure and choose a family of linearly independent left-invariant vector fields $\mathbf{X}$ on $G$ satisfying Hörmander's condition. Let $χ$ be a positive character of $G$ and consider the measure $μ_χ$ whose density with respect to $ρ$ is $χ$. In this paper, we introduce Sobolev spaces $L^p_α(μ_χ)$ adapted to $\mathbf{X}$ and $μ_χ$ ($1<p<\infty$, $α\geq 0$) 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 Schrödinger equations on the group.