The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson’s inequalities

We show that the algebra of cylinder functions in the Wasserstein Sobolev space [Formula: see text] generated by a finite and positive Borel measure [Formula: see text] on the [Formula: see text] -Wasserstein space [Formula: see text] on a complete and separable metric space [Formula: see text] is d...

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Autor principal: Sodini, Giacomo Enrico
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer Berlin Heidelberg 2023
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10423185/
https://www.ncbi.nlm.nih.gov/pubmed/37581195
http://dx.doi.org/10.1007/s00526-023-02543-1
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author Sodini, Giacomo Enrico
author_facet Sodini, Giacomo Enrico
author_sort Sodini, Giacomo Enrico
collection PubMed
description We show that the algebra of cylinder functions in the Wasserstein Sobolev space [Formula: see text] generated by a finite and positive Borel measure [Formula: see text] on the [Formula: see text] -Wasserstein space [Formula: see text] on a complete and separable metric space [Formula: see text] is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space [Formula: see text] , then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if [Formula: see text] is reflexive (resp. if the dual of [Formula: see text] is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson’s type inequalities in the Wasserstein Sobolev space.
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spelling pubmed-104231852023-08-14 The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson’s inequalities Sodini, Giacomo Enrico Calc Var Partial Differ Equ Article We show that the algebra of cylinder functions in the Wasserstein Sobolev space [Formula: see text] generated by a finite and positive Borel measure [Formula: see text] on the [Formula: see text] -Wasserstein space [Formula: see text] on a complete and separable metric space [Formula: see text] is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space [Formula: see text] , then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if [Formula: see text] is reflexive (resp. if the dual of [Formula: see text] is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson’s type inequalities in the Wasserstein Sobolev space. Springer Berlin Heidelberg 2023-08-12 2023 /pmc/articles/PMC10423185/ /pubmed/37581195 http://dx.doi.org/10.1007/s00526-023-02543-1 Text en © The Author(s) 2023 https://creativecommons.org/licenses/by/4.0/Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Article
Sodini, Giacomo Enrico
The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson’s inequalities
title The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson’s inequalities
title_full The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson’s inequalities
title_fullStr The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson’s inequalities
title_full_unstemmed The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson’s inequalities
title_short The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson’s inequalities
title_sort general class of wasserstein sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and clarkson’s inequalities
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10423185/
https://www.ncbi.nlm.nih.gov/pubmed/37581195
http://dx.doi.org/10.1007/s00526-023-02543-1
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