Global hypoellipticity of systems of Fourier multipliers on compact Lie groups

TL;DR

This paper establishes necessary and sufficient conditions for the global hypoellipticity of systems of Fourier multipliers on compact Lie groups, extending previous characterizations and providing new criteria for specific cases.

Contribution

It generalizes the characterization of global hypoellipticity to arbitrary systems of left-invariant operators on compact Lie groups, with new sufficient conditions for particular cases.

Findings

Derived necessary and sufficient conditions for global hypoellipticity.

Provided alternative sufficient conditions using lower bounds on singular values.

Extended previous results to a broader class of operators 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.