De Leeuw representations of functionals on Lipschitz spaces

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

De Leeuw representations of functionals on Lipschitz spaces

Popis výsledku v původním jazyce

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 $M$.

Název v anglickém jazyce

De Leeuw representations of functionals on Lipschitz spaces

Popis výsledku anglicky

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 $M$.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA22-32829S" target="_blank" >GA22-32829S: Struktura volných Banachových prostorů a jejich druhých duálů</a><br>

Návaznosti

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

Rok uplatnění

2025

Kód důvěrnosti údajů

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

Název periodika

Nonlinear Analysis: Theory, Methods & Applications

ISSN

0362-546X

e-ISSN

1873-5215

Svazek periodika

260

Číslo periodika v rámci svazku

November

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

30

Strana od-do

—

Kód UT WoS článku

001506245300001

EID výsledku v databázi Scopus

2-s2.0-105007061752