De Leeuw representations of functionals on Lipschitz spaces

Let ${\mathrm{Lip}}_0\left(M\right)$ be the space of Lipschitz functions on a complete metric space $(M,d)$ that vanish at a point $0\in M$. We investigate its dual ${\mathrm{Lip}}_0{\left(M\right)}^{\ast }$ using the De Leeuw transform, which allows representing each functional on ${\mathrm{Lip}}_0\left(M\right)$ as a (non-unique) measure on $\beta \tilde{M}$, where $\tilde{M}$ is the space of pairs $(x,y)\in M\times M$, $x\ne y$. We distinguish a set of points of $\beta \tilde{M}$ that are “away from infinity”, which can be assigned coordinates belonging to the Lipschitz realcompactification $M^R$ of $M$. We define a natural metric $\bar{d}$ 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 $\bar{d}$-cyclical monotonicity of their support, extending known results for functionals in $F\left(M\right)$, the predual of ${\mathrm{Lip}}_0\left(M\right)$. 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 ${\mathrm{Lip}}_0{\left(M\right)}^{\ast }$ as those admitting optimal representations with additional finiteness properties. Finally, we use De Leeuw representations to define a natural L-projection of ${\mathrm{Lip}}_0{\left(M\right)}^{\ast }$ onto $F\left(M\right)$ under some conditions on $M$.