Regularity Theorems for Elliptic and Hypoelliptic Operators Via the Global Relation

In this paperwe give a new proof regarding the regularity of solutions to hypoelliptic partial differential equations with constant coefficients. On the assumption of existence, we provide a spectral representation for the solution and use this spectral representation to deduce regularity results. By exploiting analyticity properties of the terms within the spectral representation, we are able to give simple estimates for the size of the derivatives of the solutions and interpret them in terms of Gevrey classes. AMS Subject Classifications: 46Fxx, 32Axx, 35H10, 35C15 Chinese Library Classifications: O175.27, O177.4