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

Let G be a connected Lie group with Lie algebra g and a₁, . . . , a_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 measure dg, in the directions a_i. We consider second-order operators H = - ∑_i,j=1^d′ A_iC_ijA_j + ∑_i=1^d′ (c_iA_i + A_ic′_i + c₀I corresponding to a quadratic form with complex coefficients c_ij, c_i, c′_i, c₀ ε L_∞. The principal coefficients C_ij are assumed to be Hölder continuous and the matrix C = (c_ij) is assumed to satisfy the (sub)ellipticity 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. Moreover, we establish Gaussian type bounds for the kernel and its derivatives. Similar theorems are proved for operators H′ = - ∑_i,j=1^d′ c_ijA_iA_j + ∑_i=1^d′ c_iA_i + c₀I in nondivergence form for which the principal coefficients are at least once differentiable.