Ultradifferential operators in the study of Gevrey solvability and regularity

Abstract

In this work we present a new representation formula for ultradistributions using the so‐called ultradifferential operators. The main difference between our representation result and other works is that here we do not break the duality of Gevrey functions of other s and their ultradistributions, i.e., we locally represent an element of Ds′ by an infinite order operator acting on a function of class Gs. Our main application was in the local solvability of the differential complex associated to a locally integrable structure in a Gevrey environment.

The Baouendi-Treves approximation Theorem for Gevrey classes and applications

“…Analogously, we denote by G s (Ω; L, M) the space of Gevrey functions with respect to the vector fields considered in (2.5), associated with a locally integrable structure. Since L is a G s -locally integrable structure, it was proved in [Rag19] that…”

A Fourier-Type Characterisation for Gevrey Vectors on Hypo-Analytic Structures and Propagation of Gevrey Singularities

“…. ,M m } are pairwise commuting and M j Z k (x,t) = δ j,k , we can use formula (11) to calculate M α e iζ•(Z(x,t)−z )− ζ Z(x,t)−z 2 , obtaining…”

“…We point out that since the hypo-analytic manifold is in principle only smooth, we do not have the usual Gevrey classes defined on it. Now, if the manifold has Gevrey- s regularity, then both notions coincide; see, for instance, [11]. Then we use our characterisation to obtain a propagation of Gevrey singularities (Theorem 4.3).…”

An FBI characterization for Gevrey vectors on hypo-analytic structures and propagation of Gevrey singularities

“…Now for Gevrey regularity little is known concerning propagation of singularities on hypo-analytic structures. In 2000 P. Caetano started the study of Gevrey vectors on hypo-analytic structures of maximum codimension in his Ph.D dissertation ( [6]), and his work was continued in [7] and [11], but their aim was solvability questions for the associate differential complex. Our goal here is to initiate the study of regularity problems on these structures, for instance, propagation of Gevrey singularities.…”