Second-order strongly elliptic operators on lie groups with hölder continuous coefficients

Abstract

Let G be a connected Lie group with Lie algebra g and a1 , . . . , ad′ an algebraic basis of g. Further let Ai denote the generators of left translations, acting on the Lp-spaces Lp(G; dg) formed with left Haar measure dg, in the directions ai. We consider second-order operators - ∑d′ i,j=l AiCijAj + ∑d′i=l(ciAi + Aic′i) + coI in divergence form corresponding to a quadratic form with complex coefficients, bounded Hölder continuous principal coefficients cij and lower order coefficients ci, c′i, c0 ∈ L∞ such that the matrix C = (cij) of principal coefficients satisfies the subellipticity condition ℛC = 2-1 (C + C*) ≥ μI > 0 uniformly over G. We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel and smoothness of the domain of powers of H on the Lp-spaces. Moreover, we present Gaussian type bounds for the kernel and its derivatives. Similar theorems are proved for strongly elliptic operators - ∑di,j=l cijAiAj + ∑di-lciAi + c0I in non-divergence form for which the principal coefficients are at least once differentiable. Show more