De Leeuw representations of functionals on Lipschitz spaces

TL;DR

This work develops a De Leeuw-based framework to study the bidual Lip_0(M)^* by representing functionals as measures on the Stone–Čech compactification $\beta\widetilde{M}$ and by exploiting the Lipschitz realcompactification $M^{\mathcal{R}}$ with an extended metric $\bar{d}$. It extends optimal transport duality to $M^{\mathcal{R}}$, showing that optimal De Leeuw representations are governed by $\bar{d}$-cyclical monotonicity, and provides a detailed analysis of functionals that avoid infinity versus those concentrated at infinity. The paper also identifies structural links between measure-induced functionals and optimal representations, and furnishes an explicit L-projection from Lip_0(M)^*$ onto the Lipschitz-free space $\mathcal{F}(M)$ under suitable hypotheses. Together, these results illuminate the geometry of Lip_0(M)^*$, yield extendable Kantorovich–Rubinstein type dualities, and offer constructive tools for decomposing functionals via De Leeuw representations.

Abstract

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 $β\widetilde{M}$, where $\widetilde{M}$ is the space of pairs $(x,y)\in M\times M$, $x\neq y$. We distinguish a set of points of $β\widetilde{M}$ 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 $M$.

Theorems & Definitions (109)

• Proposition 1.1: APS23
• Proposition 1.2: APS23
• Lemma 1.3: APS23
• Proposition 2.1: Woods
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• Proposition 2.5