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.
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