Some results on hypoelliptic pseudo-differential operators

In this paper, by extending the results of Yoshikawa (8), we obtain local a priori inequalities for hypoelliptic pseudo-differential operators. Using these inequalities we then show how the results of Hormander ((3), Theorem 8·7·2), on the solvability of the adjoint operator of a principally normal operator can be extended to the adjoint operator of a hypoelliptic pseudo-differential operator. Finally, we consider a class of operators which satisfy more particular a priori inequalities and we show that these operators are hypoelliptic. This class of operators was studied by Egorov (1), and he shows them to be ‘of principal type’. They include elliptic operators and also the subelliptic operators of Hormander (4).