Second-order strongly elliptic operators on Lie groups with Hölder continuous coefficients

Abstract

AbstractLetGbe a connected Lie group with Lie algebra g anda1, …,adan algebraic basis of g. Further letAidenote the generators of left translations, acting on theLp-spacesLp(G; dg)formed with left Haar measuredg, in the directions ai. We consider second-order operatorsin divergence form corresponding to a quadratic form with complex coefficients, bounded Hölder continuous principal coefficientscijand lower order coefficientsci, c′ii, c0∈L∞such that the matrixC= (cij)of principal coefficients satisfies the subellipticity conditionuniformly overG.We discuss the hierarchy relating smoothness properties of the coefficients ofHwith smoothness of the kernel and smoothness of the domain of powers ofHon theLρ-spaces. Moreover, we present Gaussian type bounds for the kernel and its derivatives.Similar theorems are proved for strongly elliptic operatorsin non-divergence form for which the principal coefficients are at least once differentiable.