Regularity and solvability of linear differential operators in Gevrey spaces

Abstract

We are interested in the following question: when regularity properties of a linear differential operator imply solvability of its transpose in the sense of Gevrey ultradistributions? This question is studied for a class of abstract operators that contains the usual differential operators with real‐analytic coefficients. We obtain a new proof of a global result on compact manifolds (global Gevrey hypoellipticity implying global solvability of the transpose), as well as some results in the non‐compact case by means of the so‐called property of non‐confinement of singularities. We provide applications to Hörmander operators, to operators of constant strength and to locally integrable systems of vector fields. We also analyze a conjecture stated in a recent paper of Malaspina and Nicola, which asserts that, in differential complexes naturally arising from locally integrable structures, local solvability in the sense of ultradistributions implies local solvability in the sense of distributions. We establish the validity of the conjecture when the cotangent structure bundle is spanned by the differential of a single first integral.

Solvability in Gevrey Classes of Some Nonlinear Fractional Functional Differential Equations

“…e question then arises of the choice of a suitable framework for the study of the solvability of these equations. But, since the functional Gevrey spaces play an important role in various branches of partial and ordinary differential equations [37][38][39][40], we think that these functional spaces can play the role of such convenient framework. However, let us point out that in order to make these spaces adequate to our specific setting, it is necessary to make a modification to their definition.…”

Solvability in Gevrey classes of some nonlinear fractional functional differential equations

“…The question then arises of the choice of a suitable framework for the study of the solvability of these equations. But since the functional Gevrey spaces play an important role in various branches of partial and ordinary differential equations ( [2], [11], [22], [43]), we think that these functional spaces can play the role of such convenient framework. However let us pointwise that in order to make these spaces adequate to our specific setting, it is necessary to make a modification to their definition.…”