No abstract

No abstract

No abstract

No abstract

. Introduction Kashiwara and Kawai [16] defined microhyperbolicity and proved that the microlocal Cauchy problem for microhyperbolic pseudodifferential operators is well-posed in the framework of microfunctions, which is a microlocalization of the results obtained by Bony and Schapira [3].In the microlocal studies of pseudo-differential operators, the concept of microhyperbolicity is very useful.From their results one can obtain results on propagation of analytic singularities (propagation of micro-analyticities) of solutions for microhyperbolic operators (see [28]). On the other hand, Bronshtein [5] proved that the hyperbolic Cauchy problem is well-posed in some Gevrey classes which are intermediate spaces between the space of real analytic functions and C°° (see, also, [14], [15]). So we can generalize the definition of microhyperbolicity in the framework of some Gevrey classes, to say the least of it. In doing so, we expect to get a clue to a generalization of microhyperbolicity and microlocal studies of microhyperbolic operators in the framework of C°°.In this paper we shall consider microhyperbolic operators in Gevrey classes and prove microlocal well-posedness of the microlocal Cauchy problem and theorems on propagation of singularities for microhyperbolic operators. Our aims are to show how one can obtain microlocal results (microlocal well-posedness and, therefore, a microlocal version of Holmgren's uniqueness theorem) from methods to prove well-posedness of the Cauchy problem and to show that theorems on propagation of singularities are immediate consequences of a microlocal version of Holmgren's uniqueness theorem, using