Harmonic analysis for resistance forms

In this paper, we define the Green functions for a resistance form by using effective resistance and harmonic functions. Then the Green functions and harmonic functions are shown to be uniformly Lipschitz continuous with respect to the resistance metric. Making use of this fact, we construct the Green operator and the (measure valued) Laplacian. The domain of the Laplacian is shown to be a subset of uniformly Lipschitz continuous functions while the domain of the resistance form in general consists of uniformly 1/2-Holder continuous functions. (C) 2003 Elsevier Inc. All rights reserved.