Gevrey local solvability in locally integrable structures

TL;DR

This paper investigates the local solvability of complex differential operators in Gevrey and ultradistribution spaces within real-analytic structures, establishing the necessity of condition Y(q) for solvability.

Contribution

It extends the necessity of the Y(q) condition for local solvability to Gevrey and ultradistribution contexts in real-analytic structures.

Findings

Condition Y(q) is necessary for local solvability in Gevrey spaces.

The results apply to ultradistributions, broadening the scope of solvability conditions.

The study links geometric properties of Levi form to functional analytic solvability.

Abstract

We consider a locally integrable real-analytic structure, and we investigate the local solvability in the category of Gevrey functions and ultradistributions of the complex d' naturally induced by the de Rham complex. We prove that the so-called condition Y(q) on the signature of the Levi form, for local solvability of d' u=f, is still necessary even if we take f in the classes of Gevrey functions and look for solutions u in the corresponding spaces of ultradistributions.