Subelliptic sharp Gårding inequality on compact Lie groups

In this work we establish a subelliptic sharp Gårding inequality on compact Lie groups for pseudo-differential operators with symbols belonging to global subelliptic Hörmander 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 [29] about the validity of the sharp Gårding 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.