Global Properties of Vector Fields on Compact Lie Groups in Komatsu Classes

Abstract

In this paper we characterize completely the global hypoellipticity and global solvability in the sense of Komatsu (of Roumieu and Beurling types) of constant-coefficients vector fields on compact Lie groups. We also analyze the influence of perturbations by lower order terms in the preservation of these properties.ators. Theory and Applications. Birkhäuser Verlag, Basel, 2010. Background analysis and advanced topics.

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

“…Introduction. The current article is a continuation of [17], in which we have characterized the global hypoellipticity and global solvability in the sense of Komatsu (of Roumieu and Beurling types) of constant-coefficients vector fields defined on compact Lie groups, as well as the influence of lower-order perturbations in the preservation of these properties.…”

“…In this section, we will present the characterization of ultradifferentiable functions and ultradistributions in Komatsu classes of both Roumieu and Beurling types through the analysis of the behavior of their partial Fourier series. This will allow us to study global properties of a variable coefficient operator on a product of compact Lie groups analyzing its normal form, which was completely characterized in [17].…”

“…The case where a is a constant was studied in [17] and we have the following characterization of the global properties of L a : Theorem 5.1 (Thms 3.2, 3.4, 3.6 and 3.8 of [17]). The operator L a = X 1 + aX 2 , with a ∈ C, is globally Γ {M k } -hypoelliptic (respectively, globally Γ (M k ) -hypoelliptic) if and only if the following conditions hold: 1.…”

“…Moreover, the operator L a is globally Γ {M k } -solvable (respectively, globally Γ (M k )solvable) if and only if the condition 2. above is satisfied. [17] for more details). In particular, f belongs to the following set…”

“…From the automorphism Ψ a we recover for the operator L a the connection between the different notions of global hypoellipticity and global solvability, obtained in [17] for constant-coefficients vector fields, summarized in the following diagram:…”