Solvability of subprincipal type operators (Q2417806)

The author studies the solvability of pseudodifferential operators \(P\) defined on a \(C^\infty\) manifold \(X\) of dimension \(n\). It is assumed that the principal symbol of \(P\) vanishes of at least second order at a nonradial involutive manifold \(\Sigma_2\subset T^* X\setminus 0\). Moreover, it is assumed that the operator is of subprincipal type. The last assumption means that the \(k\)th inhomogeneous blowup at \(\Sigma_2\) of the refined principal symbol is of principal type with Hamiltonian vector field parallel to the base \(\Sigma_2\), but transversal to the symplectic leaves of \(\Sigma_2\) at the characteristics. It is also assumed that the blowup is essentially constant on the leaves of \(\Sigma_2\) and does not satisfy the Nirenberg-Treves condition \(\Psi\). It is proved that under these conditions the operator \(P\) is not solvable. For the entire collection see [Zbl 1411.46003].