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

Abstract

Let G 1 and G 2 be compact Lie groups, X 1 ∈ g 1 , X 2 ∈ g 2 and consider the operatorwhere a and q are ultradifferentiable functions in the sense of Komatsu, and a is real-valued. We characterize completely the global hypoellipticity and the global solvability of Laq in the sense of Komatsu. For this, we present a conjugation between Laq 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 T 1 × S 3 and S 3 × S 3 in the sense of Komatsu. In particular, we give examples of differential operators which are not globally C ∞ -solvable, but are globally solvable in Gevrey spaces.

Global analytic hypoellipticity for a class of evolution operators on $\mathbb{T}^1\times\mathbb{S}^3$

“…We have the following characterizations of the spaces C ∞ (T 1 × S 3 ), D ′ (T 1 × S 3 ) and C ω (T 1 × S 3 ) (see [19] and [20]).…”