On Gevrey solvability and regularity

Abstract

In this paper we study global C∞ and Gevrey solvability for a class of sublaplacian defined on the torus T3. We also prove Gevrey regularity for a class of solutions of certain operators that are globally C∞ hypoelliptic in the N ‐dimensional torus (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Global regularity in ultradifferentiable classes

“…Actually, we prove the ω-regularity of solutions of operators of type P = P (t, D t , Dx) defined on the torus T m+n with real valued coefficients in E * (T m ) and which are globally hypoelliptic in T m+n . Therefore, we extend the previous work for Gevrey classes of Himonas and Petronilho [32,43] (see, Theorem 3.1). As a consequence, we obtain some applications to sublaplacians that may satisfy the finite type condition or may be of infinite type at most points, see §4.…”

“…Now, we study the global * -hypoellipticity for some classes of sublaplacians, whose global G s -hypoellipticity for s ≥ 1 was treated in [15,16,32,33,43]. For the notation see, for example, [15,16].…”

“…[16,32,33,43]. In particular, Himonas and Petronilho [32,43] showed the global analytic hypoellipticity (and also the global Gevrey hypoellipticity) of certain operators of the type P = P (t, D t , Dx) defined on the torus T m+n with real valued coefficients in A(T m ) and globally hypoelliptic in T m+n (see [15,16,33]). …”

“…Motivated by the recent work developed in [15,16,32,33,43] and in [2,3], we investigate the global hypoellipticity of linear partial differential operators defined on the torus T N in a bigger scale of spaces, namely, in the setting of ultradifferentiable classes as introduced in [10]. Actually, we prove the ω-regularity of solutions of operators of type P = P (t, D t , Dx) defined on the torus T m+n with real valued coefficients in E * (T m ) and which are globally hypoelliptic in T m+n .…”