de Leeuw representations of functionals on Lipschitz spaces

Let Lip₀(M) be the space of Lipschitz functions on a complete metric space $(M,d)$ that vanish at a point 0∈ M. We investigate its dual Lip₀(M)^* using the de Leeuw transform, which allows representing each functional on Lip₀(M) as a (non-unique) measure on βM, where M is the space of pairs (x,y)∈ M× M, x≠ y. We distinguish a set of points of βM that are "away from infinity", which can be assigned coordinates belonging to the Lipschitz realcompactification MR of M. We define a natural metric d̄ on MR extending d and we show that optimal (i.e. positive and norm-minimal) de Leeuw representations of well-behaved functionals are characterised by d̄-cyclical monotonicity of their support, extending known results for functionals in F(M), the predual of Lip₀(M). We also extend the Kantorovich-Rubinstein theorem to normal Hausdorff spaces, in particular to MR, and use this to characterise measure-induced and majorisable functionals in Lip₀(M)^* as those admitting optimal representations with additional finiteness properties. Finally, we use de Leeuw representations to define a natural L-projection of Lip₀(M)^* onto F(M) under some conditions on M.