Local Solvability for a Class of Evolution Equations

Motivated by the celebrated example of Y. Kannai of a linear partial differential operator which is hypoelliptic but not locally solvable, we consider a class
of evolution operators with real-analytic coefficients and study their local solvability both in $L^2$ as well as in the weak sense. In order to do so we are led to propose a generalization of the Nirenberg--Treves condition $(\psi)$ which is suitable to our
study.