Abstract: | Let P be a linear partial differential operator with coefficients in the Gevrey class Gs. We prove first that if P is s‐hypoelliptic then its transposed operator tP is s‐locally solvable, thus extending to the Gevrey classes the well‐known analogous result in the C∞class. We prove also that if P is s‐hypoelliptic then its null space is finite dimensional and its range is closed; this implies an index theorem for s‐hypoelliptic operators. Generalizations of these results to other classes of functions are also considered. |