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, where M is the space of pairs (x, y) M M, x y. We distinguish a set of points of that are "away from infinity", which can be assigned coordinates belonging to the Lipschitz realcompactification M^R of M. We define a natural metric d on M^R extending d and we show that optimal (i. e. positive and norm-minimal) de Leeuw representations of well-behaved functionals are characterised by d-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 M^R, 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.