Global solutions for systems of strongly invariant operators on closed manifolds
Alexandre Kirilov, Wagner Augusto Almeida de Moraes, Pedro Meyer Tokoro
TL;DR
The paper addresses global regularity and solvability for systems of strongly invariant operators on closed manifolds by translating PDE properties into spectral data from an elliptic operator $E$ of order $\nu$. Using the Fourier decomposition from $E$, the authors derive necessary and sufficient spectral conditions on matrix symbols $\sigma_P(k)$ and their systems to characterize global hypoellipticity and global solvability, including equivalences with almost global hypoellipticity. In the special case where operators are normal, explicit solution formulas are obtained via eigenvalues $\mu_j(k)_\ell$, with $m(\sigma_P(k))$ governing the global properties. When the system is commuting, the results specialize to classical torus and compact Lie-group settings, connecting to established spectral criteria for left-invariant vector fields.
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.