Let WY D)=C,,,., c,(x)D” be a linear partial differential operator with analytic coefficients c,, D = -ia, and L2 an open subset of R”; we denote by G’“)(Q) the class of Gevrey functions of order S, 1 es < 03. The operator L is said to be @)-solvable in 52 if for every f in G:)(Q), the space of all the Gevrey functions of order s with compact support, there exists a solution U, in the space of ultradistributions Gtr(sZ), of the equation Lu =f: We shall speak similarly of (co )-solvable operators when, in the above sentence, the class CF of all the indefinitely differentiable functions with compact support takes the place of Gt’ (and 9’ of Gr”). Owing to these definitions of solvability, when an operator L is not (s)-solvable then it is also not (t)-solvable for any t, s < t < co; we can therefore obtain non-solvability results in the class of indefinitely differentiable functions from non-solvability results in some Gevrey classes. Vice versa, dealing with operators whose unsolvability in the C” category is known, we could study up to which orders s they remain (s)-unsolvable. This way of considering the problem of the solvability of differential operators was introduced by Rodino in [13] and then carried on by the author in [3]; this last work, in particular, is the starting point of the present paper (see also [4]). We shall study here necessary conditions for the local solvability (that is, in small neighborhoods of points in R”) in Gevrey classes of differential operators with analytic coefficients of the form