Global solutions for systems of strongly invariant operators on closed manifolds

TL;DR

This paper investigates the global hypoellipticity and solvability of strongly invariant operators on closed manifolds using spectral analysis, providing characterizations and explicit solutions based on eigenvalue estimates.

Contribution

It introduces a spectral decomposition approach to analyze global properties of invariant operators and derives explicit solutions for systems of normal operators.

Findings

Characterization of global hypoellipticity via asymptotic matrix symbol estimates

Explicit solution formulas for systems of normal strongly invariant operators

Sufficient conditions for solvability based on eigenvalue analysis

Abstract

We study the global hypoellipticity and solvability of strongly invariant operators and systems of strongly invariant operators on closed manifolds. Our approach is based on the Fourier analysis induced by an elliptic pseudo-differential operator, which provides a spectral decomposition of $L^2(M)$ into finite-dimensional eigenspaces. This framework allows us to characterize these global properties through asymptotic estimates on the matrix symbols of the operators. Additionally, for systems of normal strongly invariant operators, we derive an explicit solution formula and establish sufficient conditions for global hypoellipticity and solvability in terms of their eigenvalues.