Global Analytic Hypoellipticity of a Class of Degenerate Elliptic Operators on the Torus

Abstract

In this paper we prove global analytic hypoellipticity on the torus for certain classes of operators in the form of a sum of squares of vector fields with real-valued and real analytic coefficients. The main conditions are that every point of the torus is of finite type for the vector fields and that the coefficients are independent of some of the variables. We show, for instance, that the well known not locally analytic hypoelliptic operator of Baouendi and Goulaouic [1] is globally analytic hypoelliptic on the torus. Also, we show that the operators that were proved in Hanges-Himonas [11] and in Christ [6], [7] to be not locally analytic hypoelliptic, are globally analytic hypoelliptic. Global analytic hypoellipticity of sum of squares operators on a compact analytic manifold M is an open problem except for certain cases. From the complex analysis point of view, the most important situation is when M is the boundary of a domain in C n and the operator is Kohn’s Laplacian, b, for M. In the symplectic case the global analytic hypoellipticity of b follows from the fundamental result of Tartakoff [24], and Treves [26]. In