On certain partial differential operators of finite odd type

Abstract

Let P be a linear partial differential operator of order m > 1 with real-analytic coefficients defined in Q, an open set of Rn , and let y be in the cotangent space of Q minus the zero section. If P is of odd finite type k and if the Hormander numbers are 1 = ki < k2, k2 odd, then P is analytic hypoelliptic at y . These operators are not semirigid. INTRODUCTION In this paper we present a microlocal analytic hypoellipticity result for a class of linear partial differential operators of finite type with real-analytic coefficients. The assumption is that the type of the operator is an odd positive integer k and that there are two Hormander numbers k1, k2 with 1 = k, < k2 and k2 is odd. The main difference between this result and previous results obtained by the author is that the operators here may not be semirigid. Therefore we view it as a partial result of the more general problem of giving necessary and sufficient conditions for microlocal analytic hypoellipticity of operators of finite type. For CR structures a version of this problem has been recently solved by Tumanov [10] (sufficient conditions) and Baouendi-Rothschild [1] (necessary conditions). In the analytic case the necessity was proved in [2]. The condition used in [10] is a condition at a point in the base space called the minimality condition. An analytic CR structure is minimal at a point if it is of finite type at this point . For C? CR structures finite type implies minimality but not vice versa. The finite type condition used in this paper is microlocal. The new difficulty that arises in the proof of this result is that the phase function may contain terms of degree less than k which is the degree of the terms that result from the type of the operator. To make these lower degree terms negligible, we choose a special good contour and we use a more general sufficient condition than the one used in [6]. Received by the editors December 18, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 35H05; Secondary 58G15. The author was partially supported by NSF. ? 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page