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In this paper we prove the smoothness of solutions of a class of elliptic-parabolic nonlinear Levi type equations, represented as a sum of squares plus a vector field. By means of a freezing method the study of the operator is reduced to the analysis of a family Lξ 0 of left invariant operators on a free nilpotent Lie group. The fundamental solution Γξ 0 of the operator Lξ 0 is used as a parametrix of the fundamental solution of the Levi operator, and provides an explicit representation formula for the solution of the given equation. Differentiating this formula and applying a bootstrap method, we prove that the solution is C∞.

Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations