On the strongly subdifferentiable points in Lipschitz-free spaces
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912044811288576 |
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| author | Cobollo, Christian Dantas, Sheldon Hájek, Petr Jung, Mingu |
| author_facet | Cobollo, Christian Dantas, Sheldon Hájek, Petr Jung, Mingu |
| contents | In this paper, we present some sufficient conditions on a metric space $M$ for which every molecule is a strongly subdifferentiable (SSD, for short) point in the Lipschitz-free space $\mathcal{F}(M)$ over $M$. Our main result reads as follows: if $(M,d)$ is a metric space and $γ> 0$, then there exists a (not necessarily equivalent) metric $d_γ$ in $M$ such that every finitely supported element in $\mathcal{F}(M, d_γ)$ is an SSD point. As an application of the main result, it follows that if $M$ is uniformly discrete and $\varepsilon > 0$ is given, there exists a metric space $N$ and a $(1+\varepsilon)$-bi-Lipschitz map $ϕ: M \rightarrow N$ such that the set of all SSD points in $\mathcal{F}(N)$ is dense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01269 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the strongly subdifferentiable points in Lipschitz-free spaces Cobollo, Christian Dantas, Sheldon Hájek, Petr Jung, Mingu Functional Analysis In this paper, we present some sufficient conditions on a metric space $M$ for which every molecule is a strongly subdifferentiable (SSD, for short) point in the Lipschitz-free space $\mathcal{F}(M)$ over $M$. Our main result reads as follows: if $(M,d)$ is a metric space and $γ> 0$, then there exists a (not necessarily equivalent) metric $d_γ$ in $M$ such that every finitely supported element in $\mathcal{F}(M, d_γ)$ is an SSD point. As an application of the main result, it follows that if $M$ is uniformly discrete and $\varepsilon > 0$ is given, there exists a metric space $N$ and a $(1+\varepsilon)$-bi-Lipschitz map $ϕ: M \rightarrow N$ such that the set of all SSD points in $\mathcal{F}(N)$ is dense. |
| title | On the strongly subdifferentiable points in Lipschitz-free spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2406.01269 |