Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds

Consider a Riemannian manifold M which is a Galois covering of a compact manifold, with nilpotent deck transformation group G. For the Laplace operator on M, we prove a precise estimate for the gradient of the heat kernel, and show that the Riesz transforms are bounded in L^p(M), 1 < p < . We also obtain estimates for discrete oscillations of the heat kernel, and boundedness of discrete Riesz transform operators, which are defined using the action of G on M.Mathematics Subject Classification (2000): 58J35, 35B65, 42B20in final form: 8 August 2003