Sobolev spaces on Lie groups: embedding theorems and algebra properties

TL;DR

This paper develops Sobolev spaces on noncompact Lie groups with specific measures and vector fields, establishing embedding theorems, algebra properties, and applying these to analyze nonlinear heat and Schrödinger equations.

Contribution

It introduces new Sobolev spaces on Lie groups with measures influenced by characters, and proves embedding, algebra, and well-posedness results for PDEs on these groups.

Findings

Established embedding theorems for Sobolev spaces on Lie groups.

Proved algebra properties of these Sobolev spaces.

Demonstrated local well-posedness for nonlinear heat and Schrödinger equations.

Abstract

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 $L^p_\alpha(\mu_\chi)$ adapted to $\mathbf{X}$ and $\mu_\chi$ ($1<p<\infty$, $\alpha\geq 0$) 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 Schr\"odinger equations on the group.