2024/12/21 by Paulo da Silva, da Silva, Paulo L. Dattori, Alexandre Kirilov +2 · 1 citation
Mathematics · #35H10 #35R03 #43A80 #58D25 #Analysis of PDEs (math.AP) #FOS: Mathematics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2412.16623
openalex publication_date 2024/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the global hypoellipticity and global solvability of systems of left-invariant differential operators on compact Lie groups. Focusing on diagonal systems, we establish necessary and sufficient conditions for these global properties. Specifically, we show that global solvability is characterized by a Diophantine condition on the symbol of the system, while global hypoellipticity further requires that a set depending on the symbol to be finite. As an application, we provide a complete characterization of these properties for systems of vector fields defined on products of tori and spheres. Additionally, we present illustrative examples, including systems involving higher-order differential operators. Finally, we extend our analysis to triangular systems on compact Lie groups, introducing an additional condition related to the boundedness of the dimensions of the representations in these Lie groups.