Microlocal hypo-analyticity and hypo-analytic pseudodifferential operators

Let A be a hypo-analytic pseudodifferential operator on a hypoanalytic manifold Q of maximal dimension. It is proved here that the operator A decreases the hypo-analytic wave front set of a distribution. 1. HYPO-ANALYTIC MANIFOLD In [1] Baouendi, Chang, and Treves introduced generalizations of real analytic structures which they called hypo-analytic structures. Let Q2 be a C?? manifold of dimension m + n. A hypo-analytic structure on Q2 is the data of an open covering (U) of Q2 and for each index a, of m C?? functions zl ... Zm satisfying the following two conditions: (i) dZ ..dZm are linearly independent at each point of U; (ii) if Uan Afl U1 0, there are open neighborhoods o0l of Z. (LUr n Uq) and Oa of Za ( (Ur n Uq) and a holomorphic map F of Oa onto 0. , such that Z? = F ao Z on U Zfi=FOZa on uaflufi. We use the notation Za = (Z . Zm): Ua Cm. In what follows, we assume that n = 0 and we refer to Q2 as a hypo-analytic manifold. Definition 1.1. A distribution h defined on an open neighborhood of a point p0 of Q2 is called hypo-analytic at p0 if there is a local chart (U(k, Za) of the above type whose domain contains p0 and a holomorphic function h defined of an open neighborhood of Z(,(po) in Cm such that h = h o Z, in a neighborhood of pO. By a hypo-analytic local chart we mean an m + 1 -tuple (U, Z.1 Zm) [abbreviated (U, Z)] consisting of an. open subset U of Q2 and of m hypoanalytic functions Z', .Z. , zm whose differentials are linearly independent at every point of U. We observe that since dim Q2 = m the mapping Z = (Zl, . , Zm): U -Ctm is a local diffeomorphism onto Z(U). Received by the editors December 1, 1987 and, in revised form, August 17, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 35A20; Secondary 35S99. (D 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page