Note on Hypoellipticity of a First Order Linear Partial Differential Operator

Let Ω be a domain in the (n + 1)-dimensional euclidian space Rn+1. A linear partial differential operator P with coefficients in C∞(Ω) (resp. in Cω(Ω)) will be termed hypoelliptic (resp. analytic-hypoelliptic) in Ω if a distribution u on Ω (i.e. u ∈ D′(Ω)) is an infinitely differentiable function (resp. an analytic function) in every open set of Ω where Pu is an infinitely differentiable function (resp. an analytic function).