Nonsolvability for differential operators with multiple complex characteristics (Q1326734)

Let \(A\) be a classical analytic pseudodifferential operator of principal type modelled microlocally near a point by the Mizohata operator \(D_ n+ iy^ k_ n D_ 1\), \(k\in \mathbb{N}\). The main result is the local non- solvability of the operator \(P=A^ r+\) lower order terms, in the Gevrey classes \(G^ \sigma\) with \(1< \sigma\leq \infty\), where \(r\geq 2\), \(k\) is odd. Here \(G^ \infty= C^ \infty\). So the results of Cardoso, Treves, Popivanov and of Goldman are recovered. On the other hand, if \(k\) is even, the author finds the sufficient conditions for the operators to be solvable in \(G^ \sigma\) with \(1< \sigma< \sigma_ 0\), but not solvable if \(\sigma> \sigma_ 1\).