Global hypoellipticity and global solvability in Gevrey classes on the \(n\)-dimensional torus
The authors prove global hypoellipticity and global solvability for linear differential operators with variable coefficients in \(G^s(\mathbb T^n)\), which is called \(s\)-globally hypoellipticity and \(s\)-globally solvability, where \(G^s(\mathbb T^n)\) is the Gevrey class of order \(s(\geq 1)\) on the \(n\)-dimensional torus \(\mathbb T^n\); this result is an extension of a well-know result by Hörmander in the \(C^\infty\)-framework. The proof of their main theorem is based on some properties on \(G^2(\mathbb T^n)\), and they are given as key lemmas in this paper. These lemmas can be generalized to other classes of functions \(F(K)\) of \(C^\infty(K)\), which have the same structure that the key lemmas require, where \(K\) is a compact subset of \(\mathbb R^n\). Consequently, they also obtain more general results of the first theorem, so-called \(F\)-globally hypoellipticity and \(F\)-globally solvability in \(K\).
scientific article; zbMATH DE number 2104511
scientific article