Local solvability of a class of degenerate second order operators

In this paper we will first present some results about the local solvability property of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration (which in turn is a generalization of the Kannai operator) exhibits a degeneracy due to the interplay between the singularity associated with the characteristic set of a system of vector fields and the vanishing of a function. Afterward we will also discuss some local solvability results for two classes of degenerate second order linear partial differential operators with non-smooth coefficients which are a variation of the main class presented above.