Universal Infinitesimal Hilbertianity of Sub-Riemannian Manifolds
Le Donne, E., Lučić, D., & Pasqualetto, E. (2022). Universal Infinitesimal Hilbertianity of Sub-Riemannian Manifolds. Potential Analysis, Early online. https://doi.org/10.1007/s11118-021-09971-8
Published inPotential Analysis
DisciplineAnalyysin ja dynamiikan tutkimuksen huippuyksikköGeometrinen analyysi ja matemaattinen fysiikkaMatematiikkaAnalysis and Dynamics Research (Centre of Excellence)Geometric Analysis and Mathematical PhysicsMathematics
© 2022 the Authors
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.
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Related funder(s)Academy of Finland; European Commission
Funding program(s)Centre of Excellence, AoF; Research post as Academy Research Fellow, AoF; Academy Project, AoF; Research costs of Academy Research Fellow, AoF
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about fundingOpen 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. ...
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