Global Hypoellipticity for a Class of Pseudo-differential Operators on the Torus

Global hypoellipticity for a class of periodic Cauchy operators

“…Remark 3.5. We emphasize that the phenomena of hypoellipticity with changes of sign in the imaginary part ℑM j (t, ξ), for operators with symbols satisfying the logarithm growth |p j (ξ)| = O(log(|ξ|)), when |ξ| → ∞, is discussed in [10], as the reader can see in Sections 4 and 5 of that paper.…”

“…In the next example, we exhibit an operator satisfying (7) which cannot be captured by the approach presented in [10].…”

Global hypoellipticity for a class of overdetermined systems of pseudo-differential operators on the torus

“…Proposition 3.3 in [7] proves that set Z = {ξ ∈ Z N ; p(ξ) ∈ Z} being finite is a necessary condition to the global hypoellipticity of a single operator L = D t + P (D x ) on T 1+N . However, the following example not only shows that the reciprocal of Corollary 2.3 is not true, but also exhibits a globally hypoelliptic system, where each Z j in Corollary 2.4 is infinite.…”

“…By taking inspiration from work [7], the main goal of this section is to investigate the global hypoellipticity of a special class of constant coefficient systems on T n t × T 1 x , where each operator P j (D x ), defined on T 1 , has a homogeneous symbol of positive and rational order κ, namely,…”

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

“…Such a reduction ensures that the original operator and the conjugated constant-coefficient operator have the same type of global properties in Komatsu sense. This equivalence was inspired in reduction to normal forms, which is a technique widely used in this context, see for example [5,[8][9][10]12,17]. It should be emphasized here that, as far as we know, this is the first time that this technique is extended to ultradifferentiable functions and ultradistributions in Komatsu classes, especially in the Beurling setting.…”