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

Abstract

Let G be a connected Lie group with Lie algebra \(\mathfrak{g}\) and \(a_1,\ldots,a_{d'}\) an algebraic basis of \(\mathfrak{g}\). Further let \(A_i\) denote the generators of left translations, acting on the \(L_p\)-spaces \(L_p(G\,;dg)\) formed with left Haar measure dg, in the directions \(a_i\). We consider second-order operators \(\) corresponding to a quadratic form with complex coefficients \(c_{ij}\), \(c_{i}\), \(c'_{i}\), \(c_{0}\in L_{\infty}\). The principal coefficients \(c_{ij}\) are assumed to be Holder continuous and the matrix \(C=(c_{ij})\) 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.