Harnack inequality and Liouville properties for L-harmonic operator ✩

For any complete manifold with nonnegative Bakry–Emery’s Ricci curvature, we prove the gradient estimate of L-harmonic function. As application, we use this gradient estimate to deduce the localized version of the Harnack inequality for L-harmonic operator and some Liouville properties of positive or bounded L-harmonic function. © 2006 Elsevier Inc. All rights reserved