Sobolev-Kantorovich Inequalities

Michel Ledoux

Analysis and Geometry in Metric Spaces (2015)

  • Volume: 3, Issue: 1, page 157-166, electronic only
  • ISSN: 2299-3274

Abstract

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In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means of heat flows and Harnack inequalities.

How to cite

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Michel Ledoux. "Sobolev-Kantorovich Inequalities." Analysis and Geometry in Metric Spaces 3.1 (2015): 157-166, electronic only. <http://eudml.org/doc/271060>.

@article{MichelLedoux2015,
abstract = {In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means of heat flows and Harnack inequalities.},
author = {Michel Ledoux},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Interpolation inequality; Sobolev norm; Kantorovich distance; heat flow; Harnack inequality; interpolation inequality},
language = {eng},
number = {1},
pages = {157-166, electronic only},
title = {Sobolev-Kantorovich Inequalities},
url = {http://eudml.org/doc/271060},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Michel Ledoux
TI - Sobolev-Kantorovich Inequalities
JO - Analysis and Geometry in Metric Spaces
PY - 2015
VL - 3
IS - 1
SP - 157
EP - 166, electronic only
AB - In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means of heat flows and Harnack inequalities.
LA - eng
KW - Interpolation inequality; Sobolev norm; Kantorovich distance; heat flow; Harnack inequality; interpolation inequality
UR - http://eudml.org/doc/271060
ER -

References

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  11. [11] E. Milman. Isoperimetric and concentration inequalities: equivalence under curvature lower bound. Duke Math. J. 154, 207–239 (2010). [WoS] Zbl1205.53038
  12. [12] F. Otto, C. Villani. Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000).  Zbl0985.58019
  13. [13] C. Villani. Optimal transport. Old and new. Grundlehren der mathematischen Wissenschaften 338. Springer (2009).  
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