JOU.RNAL OF DIFFERENTIAL EQUATIONS 13, 319-328 (3973) A t-Blmgren Type Theorem for Pseudo-Differential Operators in Gevrey Classes PAUL DUCHATEAU Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506 Received August 10, 1972 1, INTRODUCTION Consider the first order partial differential operator where the coefficients q(x, t) j = 0, l,..., n are assumed to be analytic on an open set U in Rn+l. Let S denote the hypersurface t = 0 in R@+l and let U+ denote the set {(x, t) E U : t > 01. Then a classical theorem, originally due to Halmgren, states that if u is any distribution in U satisfying Lu = 0 in U and having its support in U+, then u must vanish on some open neighborhood of s. It is easy to see (cf. [l]) that it represents no loss of generality to suppose 77 is a neighborhood of the origin in RR”” and that, in fact, U is of the form U = Co x B,, where 0 = neighborhood of the origin in Rn B,.=(t~R~:]tI-cr) Y > 0. Moreover, we may also suppose (again without loss of generality) that ZA(X, t) is a distribution in t with values in the space of distributions in x having compact support in 0. We shall denote this by writing u EB’(~,‘(S)). To further indicate that u has support in the half space t 2 0 we shall write u E ‘9+‘(&3E’( 0)). We seek then to prove a theorem analogous to the IElmgren theorem in which the operator L is replaced by a pseudo-differential operator. More specifically, let 6 > 0 be chosen so that 0 C Cd = (x E Rn: 1 xi 1 < (&r/Z) j = l,..., n> WI 319 Copyright 0 1973 by AcademicPress, Inc. AU tights of reproduction in any form reserved.