De Leeuw representations of functionals on Lipschitz spaces

Let $\mathrm{Lip}_0(M)$ 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 $\mathrm{Lip}_0(M)^*$ using the de Leeuw transform, which allows representing each functional on $\mathrm{Lip}_0(M)$ as a (non-unique) measure on $\beta\widetilde{M}$, where $\widetilde{M}$ is the space of pairs $(x,y)\in M\times M$, $x\neq y$. We distinguish a set of points of $\beta\widetilde{M}$ that are "away from infinity", which can be assigned coordinates belonging to the Lipschitz realcompactification $M^{\mathcal{R}}$ of $M$. We define a natural metric $\bar{d}$ on $M^{\mathcal{R}}$ 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 $\mathcal{F}(M)$, the predual of $\mathrm{Lip}_0(M)$. We also extend the Kantorovich-Rubinstein theorem to normal Hausdorff spaces, in particular to $M^{\mathcal{R}}$, and use this to characterise measure-induced and majorisable functionals in $\mathrm{Lip}_0(M)^*$ as those admitting optimal representations with additional finiteness properties. Finally, we use de Leeuw representations to define a natural L-projection of $\mathrm{Lip}_0(M)^*$ onto $\mathcal{F}(M)$ under some conditions on $M$.