Concatenations applied to analytic hypoellipticity of operators with double characteristics

Kil Hyun Kwon
1984 Transactions of the American Mathematical Society  
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 /V-dimensional real-analytic manifold £2, of the form 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
more » ... dodifferential 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 2, and that 2 is a symplectic real-analytic submanifold of T*£l \ 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 PiP), v > 0, with the same principal part 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
doi:10.1090/s0002-9947-1984-0737898-9 fatcat:wbktotxbvbbvfiko3br7qjroz4