An extension of Hörmander's theorem for infinitely degenerate second-order operators (Q1902021)

The hypoellipticity of a class of second-order operators \(L\) which do not satisfy Hörmander's conditions is proved. More precisely, it is assumed that the set of points where these conditions do not hold is contained in a submanifold of codimension 1; then the result is proved under another assumption concerning the behaviour near this submanifold. The parabolic equation associated to the operator \(L + \partial/ \partial t\) is also studied. The proofs of these results are probabilistic, and are based on Itô's and Malliavin's calculus.