We establish the hypoellipticity of a large class of highly degenerate second order differential operators of Hörmander type. The hypotheses of our theorem allow Hörmander's general Lie algebra condition to fail on a collection of hypersurfaces. The proof of the theorem is probabilistic in nature. It is based on the Malliavin calculus and requires new sharp estimates for diffusion processes in Euclidean space.
Bell, Denis R. and Mohammed, Salah-Eldin A. "An Extension of Hörmander’s Theorem for Infinitely Degenerate Second-Order Operators." (Jun 1995).