Local real analyticity of solutions for sums of squares of non-linear vector fields

Abstract

We consider quasilinear partial differential equations whose linearizations have a symplectic characteristic variety of codimension 2. We consider in detail a model case of a sum of squares of (non-linear) vector fields: P u = ( D x 2 + x 2 r h ( u , x , t ) D t 2 ) u = f with a positive definite, real analytic function h ( .,.,. ) and prove that moderately smooth solutions u must be real analytic locally where the right-hand side is. The techniques even in this case are new and we consider only this model in this first paper in order to avoid detailed consideration of the first author's complicated localization of high powers of ∂ / ∂ t introduced in Proc. Nat. Acad. Sci. USA 75 (1980) 3027; Acta Mathematica 145, 177.