Duality and distance formulas in Lipschitz-Holder spaces

For a compact metric space (K, rho), the predual of Lip(K, rho) can be identified with the normed space M(K) of finite (signed) Borel measures on K equipped with the Kantorovich-Rubinstein norm, this is due to Kantorovich [20]. Here we deduce atomic decomposition of M(K) by mean of some results from [10]. It is also known, under suitable assumption, that there is a natural isometric isomorphism between Lip(K, rho) and (lip(K, rho))** [15]. In this work we also show that the pair (lip(K, rho), Lip(K, rho)) can be framed in the theory of o-O type structures introduced by K. M. Perfekt.