A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space
Abstract
We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti–Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.
Main Authors
Format
Articles
Research article
Published
2020
Series
Subjects
Publication in research information system
Publisher
Institut de France
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-202012187289Käytä tätä linkitykseen.
Review status
Peer reviewed
ISSN
1631-073X
DOI
https://doi.org/10.5802/crmath.88
Language
English
Published in
Comptes Rendus Mathematique
Citation
- Di Marino, S., Lučić, D., & Pasqualetto, E. (2020). A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space. Comptes Rendus Mathematique, 358(7), 817-825. https://doi.org/10.5802/crmath.88
License
Funder(s)
Research Council of Finland
Research Council of Finland
Research Council of Finland
Research Council of Finland
Funding program(s)
Academy Research Fellow, AoF
Centre of Excellence, AoF
Research costs of Academy Research Fellow, AoF
Academy Project, AoF
Akatemiatutkija, SA
Huippuyksikkörahoitus, SA
Akatemiatutkijan tutkimuskulut, SA
Akatemiahanke, SA
Additional information about funding
The second and third named authors acknowledge the support by the Academy of Finland, projects 274372, 307333, 312488, and 314789. The first named author is a member of GNAMPA, INdAM.
Copyright© Académie des sciences, Paris and the authors, 2020