In this paper we consider the solvability of pseudodifferential operators in the case when the principal symbol vanishes of at least second order at a non-radial involutive manifold $\Sigma_2$. We shall assume that the Taylor expansion at $\Sigma_2$ of the refined principal symbol is of principal type with Hamilton vector field parallel to the base $\Sigma_2$ at the characteristics, but transversal to the symplectic leaves of $\Sigma_2$. We shall also assume that this Taylor expansion is essentially constant on the leaves of $\Sigma_2$, and does not satisfying the Nirenberg-Treves condition (${\Psi}$). In the case when the sign change of this condition is of infinite order, we also need some conditions on the vanishing of the gradient and the Hessian of the principal symbol at $\Sigma_2$. Under these conditions, we show that $P$ is not solvable. 查看全文>>