Second-order subelliptic operators on Lie groups III: Holder continuous coefficients

Let G be a connected Lie group with Lie algebra g and alpha(1),..., alpha(d ') an algebraic basis of g. Further let A(i) denote the generators of left translations, acting on the L-p-spaces L-p(G; dg) formed with left Haar mea sure dg, in the directions a(i). We consider second-order operators [GRAPHICS] corresponding to a quadratic form with complex coefficients c(ij), c(i), c' (i), c(0) is an element of L-infinity. The principal coefficients c(ij) are assumed to be Holder continuous and the matrix C = (c(ij)) is assumed to s atisfy the (sub)ellipticity condition RC = 2(-1) (C + C*) greater than or equal to mu I > 0 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 b ounds for the kernel and its derivatives. Similar theorems are proved for operators [GRAPHICS] in nondivergence form for which the principal coefficients are at least onc e differentiable.