Sobolev spaces on Lie groups: Embedding theorems and algebra properties

Let G be a noncompact connected Lie group, denote with ρ a right Haar measure and choose a family of linearly independent leftinvariant vector fields 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 Lpα(μχ) adapted to X and μ_χ (1<p<∞,α≥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.