Gevrey semiglobal solvability for a class of elliptic vector fields with degeneracies

Abstract

We deal with Gevrey solvability of a class of complex vector fields defined on , given by , , near the characteristic set . Diophantine conditions appear in a natural way in our results.

A note on solvability of vector fields in Denjoy-Carleman classes

“…By using properties of strongly regular sequences and arguing as in [1], in this present paper we give an alternative proof of the theorem 1.3. For this purpose we extend to ultradifferentiable classes of functions, given by strongly regular sequences, some results previously obtained to Gevrey classes in [1]. We present examples and some useful properties of the class of functions (for more examples and properties see, for instance, [5]).…”

“…s−1 } in Ω, usually denoted by G s (Ω). Hence, G 1 (Ω) is the space of real-analytic functions on Ω. For more about Gevrey functions see [15].…”

“…Motived by [1], we will consider the following notion of solvability: given a sequence of real numbers M = {m j } j∈Z + satisfying (M.1)-(M.3), and assuming that coefficients of L are in E M (Ω), we say that L is E M −solvable at Σ if for any f belonging to a subspace of finite codimension of E M (Ω) there exists a solution u ∈ E M to the equation Lu = f defined in a neighborhood of Σ.…”