# Potential spaces on Lie groups

## TL;DR

This paper studies advanced function spaces on noncompact Lie groups, providing characterizations and properties of Triebel–Lizorkin and Besov spaces defined via sub-Laplacians with drift.

## Contribution

It establishes norm characterizations, density of test functions, and isomorphism properties for these function spaces on Lie groups, extending previous definitions.

## Key findings

- Norm characterization via finite differences
- Density of test functions proven
- Isomorphism properties established

## Abstract

In this paper we discuss function spaces on a general noncompact Lie group, namely the scales of Triebel--Lizorkin and Besov spaces, defined in terms of a sub-Laplacian with drift. The sub-Laplacian is written as negative the sum of squares of a collection of left-invariant vector fields satisfying H\"ormander's condition. These spaces were recently introduced by the authors. In this paper we prove a norm characterization in terms of finite differences, the density of test functions, and related isomorphism properties.