A necessary condition for local solvability for a class of operators with double characteristics

Abstract

In this paper we prove that condition sub(Y) (see below), stated as a conjecture in the introduction of [5], is necessary for local solvability at a point for certain pseudodifferential operators with double involutive characteristics. The main novelty of this result is not in the method of proof, which follows the pattern originally presented by Hormander in [2] (we shall rely heavily on Hormander [3]), but rather the fact that the operator is not of principal type. That the condition is on the subprincipal symbol should not come as a surprise since many results in the literature on solvability and well posedness of the Cauchy problem for operators with characteristics of multiplicity higher than 1 involve conditions on lower order terms of the full symbol of the operator. We shall work with a classical, properly supported operator P on an open set X in R” whose principal symbol p is real and factorizes microlocally, i.e., near any point in T*x\O, p =p,p2 with pi real valued, Cm and homogeneous. We assume the doubly characteristic set C = (V E r*x\O] p(v) = dp(v) = 0) to be an involutive submanifold of codimension 2 and that at points in C, H,,,, H,, and the cone direction are independent.