Global hypoellipticity of systems of Fourier multipliers on compact Lie groups

Abstract

We apply the characterization of global hypoellipticity for $G$-invariant operators on homogeneous vector bundles obtained by Cardona and Kowacs [J. Pseudo-Differ. Oper. Appl. 16, 23 (2025)] to obtain a necessary and sufficient condition for an arbitrary system of left-invariant operators on a compact Lie group to be globally hypoelliptic, providing a full proof independent of the bundle structure of that paper. We then prove alternative sufficient conditions for globally hypoellipticity for a large class of particular cases of systems making use of lower bounds for the smallest singular value of complex matrices.

v2: Added in page 7 a relevant citation related to the characterization of global hypoellipticity in the scalar-valued case. v3: Fixed some minor typos, such as the denominator 2 in the statement of Corollary 3.6, and made this statement and the one of Theorem 3.5 more clear replacing d by dim(G)