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We consider p-evolution equations in (t,x) with real characteristics. We give sufficient conditions for the well-posedness of the Cauchy problem in Sobolev spaces, in terms of decay estimates of the coefficients as the space variable x goes to infinity

Let P be a linear partial differential operator with coefficients in the Gevrey class G s ðT n Þ where T n is the n-dimensional torus and sX1: We prove that if P is s-globally hypoelliptic in T n then its transposed operator t P is s-globally solvable in T n ; thus extending to the Gevrey classes the well-known analogous result in the corresponding C N class. r 2004 Elsevier Inc. All rights reserved.

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