# Boundedness of pseudo-differential operators in subelliptic Sobolev and Besov spaces on compact Lie groups

## TL;DR

This paper studies the properties of Besov and Sobolev spaces on compact Lie groups in a subelliptic setting, establishing embeddings, estimates, and interpolation results for pseudo-differential operators associated with the H"ormander condition.

## Contribution

It introduces new embedding and boundedness results for pseudo-differential operators in subelliptic Besov and Sobolev spaces on compact Lie groups, linking them with matrix-valued quantisation.

## Key findings

- Embedding properties between subelliptic and classical Besov spaces.
- Boundedness of pseudo-differential operators in subelliptic Sobolev and Besov spaces.
- Interpolation results between Besov and Triebel-Lizorkin spaces.

## Abstract

In this paper we investigate the Besov spaces on compact Lie groups in a subelliptic setting, that is, associated with a family of vector fields, satisfying the H\"ormander condition, and their corresponding sub-Laplacian. Embedding properties between subelliptic Besov spaces and Besov spaces associated to the Laplacian on the group are proved. We link the description of subelliptic Sobolev spaces with the matrix-valued quantisation procedure of pseudo-differential operators in order to provide subelliptic Sobolev and Besov estimates for operators in the H\"ormander classes. Interpolation properties between Besov spaces and Triebel-Lizorkin spaces are also investigated.