Hormander class of pseudo-differential operators on compact Lie groups and global hypoellipticity

TL;DR

This paper characterizes the Hormander class of pseudo-differential operators on compact Lie groups, providing criteria for ellipticity and hypoellipticity via matrix-valued symbols, with examples and explicit constructions.

Contribution

It offers new global characterizations and criteria for ellipticity and hypoellipticity of pseudo-differential operators on compact Lie groups, including explicit examples.

Findings

Criteria for ellipticity and hypoellipticity based on symbols

Examples of globally hypoelliptic differential operators

Explicit constructions where hypoellipticity fails

Abstract

In this paper we give several global characterisations of the Hormander class of pseudo-differential operators on compact Lie groups. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.