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.
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Acknowledgements
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.
The authors would like to thank Tapio Rajala for the fruitful discussions about the results of Section ??, as well as the anonymous referee for the useful comments.
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Le Donne, E., Lučić, D. & Pasqualetto, E. Universal Infinitesimal Hilbertianity of Sub-Riemannian Manifolds. Potential Anal 59, 349–374 (2023). https://doi.org/10.1007/s11118-021-09971-8
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DOI: https://doi.org/10.1007/s11118-021-09971-8