Universal Infinitesimal Hilbertianity of Sub-Riemannian Manifolds
Abstract
We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space of square-integrable sections of the horizontal bundle, which we obtain on all weighted sub-Finsler manifolds. As an intermediate tool, of independent interest, we show that any sub-Finsler distance can be monotonically approximated from below by Finsler ones. All the results are obtained in the general setting of possibly rank-varying structures.
Main Authors
Format
Articles
Research article
Published
2023
Series
Subjects
Publication in research information system
Publisher
Springer
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-202204132262Use this for linking
Review status
Peer reviewed
ISSN
0926-2601
DOI
https://doi.org/10.1007/s11118-021-09971-8
Language
English
Published in
Potential Analysis
Citation
- Le Donne, E., Lučić, D., & Pasqualetto, E. (2023). Universal Infinitesimal Hilbertianity of Sub-Riemannian Manifolds. Potential Analysis, 59(1), 349-374. https://doi.org/10.1007/s11118-021-09971-8
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Funder(s)
Research Council of Finland
Research Council of Finland
Research Council of Finland
Research Council of Finland
Research Council of Finland
Research Council of Finland
European Commission
Funding program(s)
Centre of Excellence, AoF
Academy Research Fellow, AoF
Academy Research Fellow, AoF
Academy Project, AoF
Research costs of Academy Research Fellow, AoF
Academy Project, AoF
ERC Starting Grant
Huippuyksikkörahoitus, SA
Akatemiatutkija, SA
Akatemiatutkija, SA
Akatemiahanke, SA
Akatemiatutkijan tutkimuskulut, SA
Akatemiahanke, SA
ERC Starting Grant
Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Education and Culture Executive Agency (EACEA). Neither the European Union nor EACEA can be held responsible for them.
Additional information about funding
Open access funding provided by University of Fribourg.
E.L.D. was partially supported by the Academy of Finland (grant 288501 ‘Geometry of subRiemannian groups’ and by grant 322898 ‘Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’). D.L. and E.P. were partially supported by the Academy of Finland, projects 274372, 307333, 312488, and 314789.
Copyright© 2022 the Authors