Concatenations Applied to Analytic Hypoellipticity of Operators with Double Characteristics

Abstract

We use the method of concatenations to get a sufficient condition for a class of analytic pseudodifferential operators with double characteristics to be analytic hypoelliptic. Introduction. The present paper is concerned with analytic hypoellipticity for operators on an N-dimensional real-analytic manifold i2, of the form (1) P(x, D) = Idp(x, D) + Q(x, D), where Id is the identity d x d matrix, p(x, () a scalar analytic symbol, homogeneous of degree m in (, and Q(x, D) a d x d matrix of classical analytic pseudodifferential operators of order m 1 (classical means that its total formal symbol is a series of homogeneous terms whose homogeneous degrees drop by integers). We assume that its principal symbol, p(x, (), is nonnegative everywhere, that it vanishes exactly of order two on its characteristic set L, and that L is a symplectic real-analytic submanifold of T*2 0. For such an operator, analytic hypoellipticity was already obtained in F. Treves [13] under the additional hypothesis that (2) P ( x, D) is hypoelliptic with loss of one derivative. Recently, several other studies have been made for the analytic hypoellipticity of similar operators. For example, in [7], G. Metivier extended the result of [13] to the operators with multiple characteristics assuming suitable hypoellipticity in case that the characteristic manifold is symplectic. Whereas, in [6], A. Grigis and L. P. Rothschild gave a necessary and sufficient condition for a class of operators with polynomial coefficients to be analytic hypoelliptic (see also [10] and [11] for a different approach to similar problems). On the other hand, in [3], L. Boutet de Monvel and F. Treves obtained a necessary and sufficient condition for P(x, D) to be hypoelliptic with loss of one derivative by means of the method of concatenations (introduced first by F. Treves in [12]). According to their work, we can associate with P(x, D) a sequence of operators p(v), P > 0, with the same principal as P(x, D), which satisfy certain relations connecting them, called concatenations. In terms of concatenations, the condition (2) can be restated by saying that the lower order part of every p(v) (i.e., the one Received by the editors March 25, 1983 and, in revised form, September 15, 1983. 1980 Mathematics Subject Classification. Primary 35B65.