Solutions of Partial Differential Equations with Support on Leaves of Associated Foliations

Abstract

Suppose that the linear partial differential operator $P(x,D)$ has analytic coefficients and that it can be written in the form $P(x,D) = R(x,D)S(x,D)$ where $S(x,D)$ is a polynomial in the homogeneous first order operators ${A_1}(x,D), \cdots ,{A_r}(x,D)$. Then in a neighborhood of any point ${x^0}$ at which the principal part of $S(x,D)$ does not vanish identically, there is a solution of $P(x,D)u = 0$ with support the leaf through ${x^0}$ of the foliation induced by the Lie algebra generated by ${A_1}(x,D), \cdots ,{A_r}(x,D)$. This result yields necessary conditions for hypoellipticity and uniqueness in the Cauchy problem. An application to second order degenerate elliptic operators is also given.