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 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 αp (μ χ ) adapted to X and μ χ (1<∞ 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.