Local Solvability for Partial Differential Equationswith Multiple Characteristics in Mixed Gevrey-$C^\infty$ Spaces

In this paper we consider a class of linear partial differential equations with multiple characteristics, whose principal part is elliptic in a set of variables. We assume that the subprincipal symbol has real part different from zero and that its imaginary part does not change sign. We then prove the local solvability of such a class of operators in mixed Gevrey-$C^\infty$ spaces, in the sense that the linear equation admits a local solution when the datum is Gevrey in some variables and smooth in the other ones.