Local solvability of a class of degenerate second order operators

We study the local solvability of a class of operators with multiple characteristics. The class introduced in this paper 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 smooth function. It generalizes in part the class introduced by Colombini, Cordaro and Pernazza, for which they proved L^2 local solvability. It is shown here, among other results, that local solvability with right-hand side in H^(-1/2)_(loc) or H^(-1)_(loc) and solution in L^2_(loc) is also possible. In addition, we extend the solvability result to certain systems of complex vector fields, and show that also in this case one has local solvability with right-hand side in H^s_(loc), with s = 0 or s = −1/r, r ≥ 1 an integer, with solution in L^2_(loc).