Global properties of vector fields on compact Lie groups in Komatsu classes. II. Normal forms

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ G_1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ G_2 $\end{document}</tex-math></inline-formula> be compact Lie groups, <inline-formula><tex-math id="M3">\begin{document}$ X_1 \in \mathfrak{g}_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ X_2 \in \mathfrak{g}_2 $\end{document}</tex-math></inline-formula> and consider the operator <inline-formula><tex-math id="M5">\begin{document}$ L_{aq} = X_1 + a(x_1)X_2 + q(x_1,x_2), $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M6">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q $\end{document}</tex-math></inline-formula> are ultradifferentiable functions in the sense of Komatsu, and <inline-formula><tex-math id="M8">\begin{document}$ a $\end{document}</tex-math></inline-formula> is real-valued. Assuming certain condition on <inline-formula><tex-math id="M9">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ q $\end{document}</tex-math></inline-formula> we characterize completely the global hypoellipticity and the global solvability of <inline-formula><tex-math id="M11">\begin{document}$ L_{aq} $\end{document}</tex-math></inline-formula> in the sense of Komatsu. For this, we present a conjugation between <inline-formula><tex-math id="M12">\begin{document}$ L_{aq} $\end{document}</tex-math></inline-formula> and a constant-coefficient operator that preserves these global properties in Komatsu classes. We also present examples of globally hypoelliptic and globally solvable operators on <inline-formula><tex-math id="M13">\begin{document}$ \mathbb{T}^1\times \mathbb{S}^3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ \mathbb{S}^3\times \mathbb{S}^3 $\end{document}</tex-math></inline-formula> in the sense of Komatsu. In particular, we give examples of differential operators which are not globally <inline-formula><tex-math id="M15">\begin{document}$ C^\infty $\end{document}</tex-math></inline-formula>–solvable, but are globally solvable in Gevrey spaces.</p>