On the Solvability of Certain Degenerate Partial Differential Operators

In this paper I will give a survey of the problem of solvability of degenerate operators that are not of principal type. In the first place, I will describe some recent results obtained with C. Parenti about semi-global solvability of degenerate operators with symplectic multiple characteristics. I will then describe some other recent results obtained with S. Federico about local solvability in the Sobolev spaces of a class of degenerate operators which are an elaboration of the class considered by Colombini-Cordaro-Pernazza (in turn, an elaboration of the Kannai operator).