De Leeuw representations of functionals on Lipschitz spaces

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F25%3A00384195" target="_blank" >RIV/68407700:21240/25:00384195 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1016/j.na.2025.113851" target="_blank" >https://doi.org/10.1016/j.na.2025.113851</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.na.2025.113851" target="_blank" >10.1016/j.na.2025.113851</a>

Result language

angličtina

Original language name

De Leeuw representations of functionals on Lipschitz spaces

Original language description

Let $mathrm{Lip}_0(M)$ be the space of Lipschitz functions on a complete metric space $(M,d)$ that vanish at a point $0in M$. 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 $betawidetilde{M}$, where $widetilde{M}$ is the space of pairs $(x,y)in Mtimes M$, $xneq y$. We distinguish a set of points of $betawidetilde{M}$ 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 $M$.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GA22-32829S" target="_blank" >GA22-32829S: The structure of free Banach spaces and of their second duals</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2025

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Nonlinear Analysis: Theory, Methods & Applications

ISSN

0362-546X

e-ISSN

1873-5215

Volume of the periodical

260

Issue of the periodical within the volume

November

Country of publishing house

NL - THE KINGDOM OF THE NETHERLANDS

Number of pages

30

Pages from-to

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UT code for WoS article

001506245300001

EID of the result in the Scopus database

2-s2.0-105007061752