Kantorovich-Rubinstein norm and its application in the theory of Lipschitz spaces

We obtain necessary and sufficient conditions on a compact metric space (K, p) that provide a natural isometric isomorphism between completion of the space of Borel measures on K with the Kantorovich-Rubinstein norm and the space (lip(K, p))* or equivalently between the spaces Lip(K, p) and (lip(K, p))** . Such metric spaces are studied and related properties of Lipschitz spaces are established. 1. NOTATION Let (K, p) be a metric space and M(K) be the set of all finite Borel measures on K. For a measure ,I E M(K), denote by ,u+, u_ its positive and negative variations, respectively, and set I uI = #U+ + # -u, Var,u = I1uI(K) . The Lipschitz space Lip(K, p) is defined as the set of all functions f on K with the finite norm If IIK, p = max{ IlfIIK, If IK, p}, where IlfIIK = Sup{lf(x)I: X E K}