On the reduction of certain pseudodifferential operators with noninvolutive characteristics

Abstract

where D = -;a/& and K is a positive integer. All notations used in this note are those of Hiirmander [4]. In the hyperfunction category these operators were characterized by Sato et al. [6, 71. Moreover they proved the following: For a pseudodifferential operator P(x, D) with the principal symbol f,,% + ix,“&,, , one can choose elliptic pseudodifferential operators -4 and B such that APB is equal to the differential operator (M). In the proof they used the Cauchy-Kovalevskaja theorem. Thus the situation is quite different in the G” case. However, when K = 1, even in the Cm category, Duistermaat-Sjbtrand proved the same fact in [2]. We attempt in this paper to obtain the same result for general odd integers K. When k is an arbitrary integer, we do not know whether this result is true or not. However, we are still able to show hypoellipticity and a local solvability of our operator P(x, D). In fact, using an argument of a vector valued pseudodifferential operator, which was developed in [8], we can construct parametricies for our pseudodifferential operators. In this way we obtain a generalization of a theorem in [Z]. In [9] using our argument, we also construct parametricies for some class of pseudodifferential operators with double characteristics. We formulate our problem. Let X be a paracompact P-manifold of dimension II and P EL,*~(X) be a properly supported classical pseudodifferential operator on X of type 1, 0 and order nt, which in every local coordinate system UC X has a symbolic asymptotic expansion