Duality and distance formulas in Lipschitz–Hölder spaces

Abstract

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 $\mathcal M(K)$ 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.