Duality and distance formulas in Lipschitz–Hölder 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.