A Criterion for Analytic Hypoellipticity of a Class of Differential Operators with Polynomial Coefficients

Abstract

where SSa denotes the analytic singular support. Microlocal analytic hypoellipticity is defined in the same way, by use of the analytic wave front set WFa. (See e.g. [1] or [27, Definition 6.1] for the definition.) Analytic hypoellipticity for differential operators with variable coefficients and multiple characteristics has been studied by Treves [30], Tartakoff [29], Metivier [18] and others, in the case where the characteristic variety is symplectic. A sufficient condition for microlocal analytic hypoellipticity is the injectivity in some Schwartz space Y(R') of a differential operator with polynomial coefficients canonically associated to the operator under consideration at every characteristic point. For example their results apply to operators like