Let G be a connected Lie group with Lie algebra and an algebraic basis of . Further let denote the generators of left translations, acting on the -spaces formed with left Haar measure dg, in the directions . We consider second-order operators corresponding to a quadratic form with complex coefficients , , , . The principal coefficients are assumed to be Hölder continuous and the matrix is assumed to satisfy the (sub)ellipticity condition uniformly over G. We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel. Moreover, we establish Gaussian type bounds for the kernel and its derivatives. Similar theorems are proved for operators in nondivergence form for which the principal coefficients are at least once differentiable.
|Journal||Calculus of Variations and Partial Differential Equations|
|Publication status||Published - 1999|