Sufficient Conditions for Solvability of Operators of Subprincipal Type

In this paper we show that condition $\operatorname{Sub_r}(\Psi)$ on the subprincipal symbol is sufficient for local solvability of linear pseudodifferential operators of real subprincipal type. These are the operators having real principal symbol, which is of principal type and vanishes of second order on an involutive manifold where the subprincipal symbol is of principal type. Condition $\operatorname{Sub_r}(\Psi)$ is a condition on the sign changes of the imaginary part of the subprincipal symbol, which has previously been shown by the author to be necessary for local solvability of linear pseudodifferential operators of real subprincipal type. In the appendix, we study the local solvability of quasilinear second order partial differential operators of real principal type.