Abstract
Let Lipo(M) be the space of Lipschitz functions on a complete metric space (M.d) that vanish at a point 0 E M. We investigate its dual Lip (M) using the De Leeuw transform, which allows representing each functional on Lipo (M) as a (non-unique) measure on BM, where M is the space of pairs (x, y) is an element of MXM, xy. We distinguish a set of points of M that are "away from infinity", which can be assigned coordinates belonging to the Lipschitz realcompactification MR of M. We define a natural metric d on M 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 Lipo (M). We also extend the Kantorovich-Rubinstein theorem to normal Hausdorff spaces, in particular to M", and use this to characterise measure-induced and majorisable functionals in Lipo(M) as those admitting optimal representations with additional finiteness properties. Finally, we use De Leeuw representations to define a natural L-projection of Lipo (M)" onto F(M) under some conditions on M.
