WKB Analysis to Global Solvability and Hypoellipticity

This paper studies the global regularity and solvability of operators which could change their type at every point of the domain. Our object is to understand such operators from the viewpoints of a WKB analysis. To be more precise, let X be a compact manifold or an open domain in R. We denote by C°°(X) and C£°(X) the set of smooth functions on X and the set of smooth functions with compact supports respectively. We also denote the set of distributions on X by 9"'(X). We say that a differential operator P is globally solvable (resp. globally hypoelliptic) in X if for every /eC£°(X) there exists ue3i'(X) satisfying Pu = f. (resp. ueC°°(X) when PueC°°(X) and we£^ ' (X)) . The operator P is said to be locally solvable (resp. locally hypoelliptic) at a point p e X if there exists a neighborhood U of p such that for every /£C~(£7) , there exists ue9"'(U) satisfying Pu = f in U (resp. p £singsupp(Pu) implies p £singsupp(u)). By definition local hypoellipticity at each point p e X implies the global hypoellipticity in X, and the global solvability implies the local solvability at each point p e. X, while the corresponding inverse implications are not true. ([7]). Because the operators which we want to study are in general of mixed type the structures of local solutions may change drastically in every part of the domain. Therefore most of the methods such as those for degenerate elliptic operators, and for weakly hyperbolic operators are not applicable to such operators, (cf. [9], [17]). Moreover, because the structure of the characteristics is so complicated that the usual characteristic geometry does not seem adequate to