Subelliptic sharp G\r{a}rding inequality on compact Lie groups

TL;DR

This paper proves a sharp G {a}rding inequality for subelliptic pseudo-differential operators on compact Lie groups, extending previous results and including the torus case, with conditions on the symbols' nonnegativity.

Contribution

It establishes a new sharp G {a}rding inequality for subelliptic pseudo-differential operators on compact Lie groups, broadening the classes of symbols and settings covered.

Findings

Validates the inequality for symbols in global subelliptic H"ormander classes.

Extends previous results to broader symbol classes $ ext{S}^m_{ ho, ho}(G)$.

Proves the inequality is sharp on the torus.

Abstract

In this work we establish a subelliptic sharp G\r{a}rding inequality on compact Lie groups for pseudo-differential operators with symbols belonging to global subelliptic H\"ormander classes. In order for the inequality to hold we require the global matrix-valued symbol to satisfy the suitable classical nonnegativity condition in our setting. Our result extends to $\mathscr{S}^m_{\rho,\delta}(G)$-classes, $0\leq \delta<\rho$, the one in [26] about the validity of the sharp G\r{a}rding inequality for the class $\mathscr{S}^m_{1,0}(G)$. We remark that the result we prove here is already new and sharp in the case of the torus.