De Leeuw representations of functionals on Lipschitz spaces
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912966437240832 |
|---|---|
| author | Aliaga, Ramón J. Pernecká, E. Smith, Richard J. |
| author_facet | Aliaga, Ramón J. Pernecká, E. Smith, Richard J. |
| contents | 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 $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09546 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | De Leeuw representations of functionals on Lipschitz spaces Aliaga, Ramón J. Pernecká, E. Smith, Richard J. Functional Analysis Primary 46B20, 49Q22, Secondary 46E15, 54D35, 54E50 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 $M$. |
| title | De Leeuw representations of functionals on Lipschitz spaces |
| topic | Functional Analysis Primary 46B20, 49Q22, Secondary 46E15, 54D35, 54E50 |
| url | https://arxiv.org/abs/2403.09546 |