Vector Valued G rding Inequality for pseudo-differential operators on compact homogeneous manifolds

We prove sufficient conditions in order to obtain a sharp G rding inequality for pseudo-differential operators acting on vector-valued functions on compact Lie groups. As a consequence, we obtain a sharp G rding inequality for compact homogeneous vector bundles and compact homogeneous manifolds. The sharp G rding inequality is the strongest lower bound estimate known to hold for systems on Rⁿ, and the aim of this paper is to extend this property to systems on compact Lie groups and compact homogeneous manifolds. Our results extend previous works by Lax and Nirenberg [Comm. Pure Appl. Math., Vol. 8, 129-209, (1966)], and by Ruzhansky and Turunen [J. Funct. Anal., Vol. 267, 144-172, (2011)]. As an application, we establish existence and uniqueness of solution to a class of systems of initial value problems of pseudo-differential equations on compact Lie groups and compact homogeneous manifolds.