Gevrey hypoellipticity and solvability on the multidimensional torus of some classes of linear partial differential operators (Q867744)

The authors consider partial differential operators of the form \[ P= D^2_t+ \Biggl(D_t+ \sum^m_{j=1} a_j(t)\,D_{x_j}\Biggr)^2 \] on the torus \(\mathbb{T}^{1+n}\), with variables \(t\), \(x= (x_1,\dots, x_n)\). The coefficients \(a_j(t)\) are assumed to be real-valued, belonging to the Gevrey space \(G^s(\mathbb{T})\), \(s\geq 1\). A complete characterization of \(s\)-global hypoellipticity and \(s\)-global solvability of \(P\) on \(\mathbb{T}^{1+n}\) is given, in terms of Diophantine approximation properties of the coefficients \(a_j(t)\). The same operator \(P\) was studied in the frame of the class \(C^\infty(\mathbb{T}^{1+n})\) by \textit{A. A. Himonas} and \textit{G. Petronilho} [Mich. Math. J. 50, No. 3, 471--481 (2002; Zbl 1028.35049)].