The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, II

Abstract

Let L = ∑ j = 1 m X j 2 be sum of squares of vector fields in R n satisfying a Hörmander condition of order 2: span{ X j , [ X i , X j ]} is the full tangent space at each point. A point x ϵ ∂D of a smooth domain D is characteristic if X 1,…, X m are all tangent to ∂D at x. We prove sharp estimates in non-isotropic Lipschitz classes for the Dirichlet problem near (generic) isolated characteristic points in two special cases: (a) The Grushin operator ∂ 2 ∂x 2 + x 2 ∂ 2 ∂t 2 in R 2. (b) The real part of the Kohn Laplacian on the Heisenberg group ∑ j − 1 n ( ∂ ∂x j + 2y j ∂ ∂t ) 2 + ( ∂ ∂y j − 2x j ∂ ∂t ) 2 in R 2 n + 1 . In contrast to non-characteristic points, C ∞ regularity may fail at a characteristic point. The precise order of regularity depends on the shape of ∂D at x.