Kantorovich-Rubinstein Norm and Its Application in the Theory of Lipschitz Spaces

1992/06/01 by Leonid G. Hanin, Leonid Hanin · 2 citations
Mathematics · #Advanced Banach Space Theory #advanced mathematical theories #Approximation Theory and Sequence Spaces

paper · doi:10.2307/2159251

Abstract

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 [\ix)(K, />))* or equivalently between the spaces Lip(K, p) and (lip(ZY, /»))** .Such metric spaces are studied and related properties of Lipschitz spaces are established. NotationLet ÍK, p) be a metric space and ¿Vf(¿V) be the set of all finite Borel measures on K.For a measure p G MÍK), denote by p+ , /¿_ its positive and negative variations, respectively, and set \p = p+ + /z_ , Var/¿ = |/¿|(¿").The Lipschitz space Lip(¿V, p) is defined as the set of all functions f on K with the finite norm where and ||/||r,, = max||/||x,|/|jc,,, |l/|k-:sup|/(x)|:x€^ Functions / G LipiK, p) with the property lim /(*)-/Cr>=o p(x,y)-o pix,y) constitute the closed subspace lipiK, p) in LipiK, p).The notation B and

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