Some examples of global Gevrey hypoellipticity and solvability (Q1327774)

The authors carry on their general plan of research [Rend. Semin. Mat. Univ. Politec. Torino 51, No. 2, 145-172 (1993)] concerning global properties of linear partial differential equations on the two dimensional torus \(\mathbb{T}^ 2\). Here they consider an operator \(P\) of the form \(D_ x + a(x) D_ y\). Suitable analytic \(a(x)\) are found such that global hypoellipticity and solvability of \(P\) in the Gevrey class \(G^ s (\mathbb{T}^ 2)\) depend on the Gevrey index \(s\). This makes an interesting contrast to local results for operators of real principal type, which are non-hypoelliptic and locally solvable independently of the index \(s\), see the book of the reviewer [\textit{L. Rodino}, Linear partial differential operators in Gevrey spaces (1993)].