Riesz transform via heat kernel and harmonic functions on non-compact manifolds

Let M be a complete non-compact manifold satisfying the volume doubling condition, with index N and reverse n, n≤N, both for large balls. Assume Gaussian upper bound heat kernel, an L2-Poincaré inequality outside compact set. If 22 on manifolds having at least two Euclidean ends dimension n. For p∈(max⁡{N,2},∞), fact (Rp), (RHp) essentially follows [22]; moreover, if non-parabolic, these conditions only one end. proof, develop new criterion boundedness which was nontrivially adapted [4], make essential application results [22]. result allows extensions non-smooth settings.