proceedings of the american mathematical society Volume 105, Number 3, March 1989 HYPO-ANALYTICPSEUDODIFFERENTIAL OPERATORS S. BERHANU (Communicated by Walter D. Littman) Abstract. Let ß be a hypo-analytic manifold of dimension m equipped with a hypo-analytic structure whose structure bundle T' has dimension m . This paper introduces hypo-analytic pseudodifferential operators and it is shown that such operators preserve the hypo-analyticity of a distribution. 1. INTRODUCTION The main concepts relating to hypo-analyticity were introduced by Baouendi, Chang, and Trêves in [1], Chapter 1. We shall summarize some of these in this section. Suppose Q isa C°° manifold of dimension m+n . A hypo-analytic structure on Q is the data of an open covering (Ua) of Q and for index a, of m C°° functions Za, ... , Z™ satisfying the following two conditions: (i) dZ'n , ... ,dZ™ are linearly independent at each point of Ua ; (ii) if Uan Ug ■£ 0, there are open neighborhoods <fa of Za(Ua U U~) and (f„ of ZJU(1¡n U„) and a holomorphic map F'l of 6ta onto (f„ , such that Zß = F'ß°Za 0n UanUß- When the ZJ are real-valued and n = 0, such a structure specializes to a real analytic structure. A distribution h defined in an open neighborhood of a point p0 of Q is hypo-analytic at p0 if there is a hypo-analytic local chart (Un , Zn) whose domain contains p0 and a holomorphic function hn defined on an open neighborhood of Zn(p0) in Cm such that h — hi% o Z in a neighborhood of Po- By a hypo-analytic local chart we mean an (m + l)-tuple (U ,Z , ... ,Zm) [abbreviated (U ,Z)] consisting of an open subset U of Q and of m hypo- analytic functions Zl , ... ,Zm whose differentials are linearly independent at every point of U . Received by the editors November 2, 1987 and, in revised form, August 17, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 35A20; Secondary 35S99. ©1989 American Mathematical Society 0002-9939/89 $1.00+ $.25 per page 582 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use