De Leeuw representations of functionals on Lipschitz spaces

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 ${\mathrm{Lip}}_0(M{)}^{\ast }$ using the de Leeuw transform, which allows representing each functional on ${\mathrm{Lip}}_0(M)$ 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 $M^{\mathcal{R}}$ 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 ${\mathrm{Lip}}_0(M{)}^{\ast }$ as those admitting optimal representations with additional finiteness properties. Finally, we use de Leeuw representations to define a natural L-projection of ${\mathrm{Lip}}_0(M{)}^{\ast }$ onto $\mathcal{F}(M)$ under some conditions on $M$.

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Cited by in corpus (1)

• Pełczyński's property (V${}^{\ast }$) in Lipschitz-free spaces