A class of non \(s\)-solvable operators (Q2704049)

The author of this note considers non-solvability of some inhomogeneous equation \(Lu= f\), \(f\) being a function of Gevrey class, concerning a linear partial differential operator \(L\) which is called a generalization of the Mizohata operator by the author.NEWLINENEWLINENEWLINEThe main result is as follows:NEWLINENEWLINENEWLINETheorem. Let us consider the operator NEWLINE\[NEWLINEL={\partial\over \partial x_1}+ ib(x_1){\partial\over\partial x_2}.NEWLINE\]NEWLINE Let us suppose that the function \(b(x_1)\) satisfies the following hypotheses:NEWLINENEWLINENEWLINE1. \(b(x_1)\in{\mathcal G}^{s'}(\mathbb{R})\), \(s'\geq 1\); \({\mathcal G}^{s'}(\mathbb{R})\) is Gevrey class functions on \(\mathbb{R}\);NEWLINENEWLINENEWLINE2. \(b(0)= 0\);NEWLINENEWLINENEWLINE3. \(b(x_1)> 0\) for \(x_1< 0\), \(b(x_1)< 0\) for \(x_1> 0\).NEWLINENEWLINENEWLINEThen the operator \(L\) is not \(s\)-solvable at the origin for every \(s> s'\).NEWLINENEWLINENEWLINEThe operator \(L\) is considered in \(\mathbb{R}^2\) (two-dimensional Euclidean space).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00030].