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Tytuł artykułu

Sobolev-Kantorovich Inequalities

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EN
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.
Twórcy
  • Institut de Mathématiques de Toulouse, Université de Toulouse–Paul-Sabatier, F-31062 Toulouse, France, and Institut Universitaire de France
Bibliografia
  • [1] D. Bakry, I. Gentil, M. Ledoux. Analysis and geometry of Markov diffusion operators. Grundlehren der mathematischen Wissenschaften 348. Springer (2014).
  • [2] D. Bakry, I. Gentil, M. Ledoux. On Harnack inequalities and optimal transportation Ann. Scuola Norm. Sup. Pisa, to appear (2015).
  • [3] D. Bakry, Z. Qian. Some new results on eigenvectors via dimension, diameter, and Ricci curvature. Adv. Math. 155, 98–153 (2000).
  • [4] V. I. Bogachev, F.-Y. Wang, A. V. Shaposhnikov. Estimates of the Kantorovich norm on manifolds (2015).
  • [5] R. Choksi, S. Conti, R. Kohn, F. Otto. Ground state energy scaling laws during the onset and destruction of the intermediate state in a type I superconductor. Comm. Pure Appl. Math. 6, 595–626 (2008). [WoS]
  • [6] E. Cinti, F. Otto. Interpolation inequalities in pattern formation (2014).
  • [7] E. B. Davies. Heat kernels and spectral theory. Cambridge Tracts in Mathematics 92. Cambridge (1989).
  • [8] M. Ledoux. On improved Sobolev embedding theorems. Math. Res. Letters 10, 659–669 (2003). [Crossref]
  • [9] P. Li, S.-T. Yau. On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986).
  • [10] E. Milman. On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177, 1–43 (2009). [WoS]
  • [11] E. Milman. Isoperimetric and concentration inequalities: equivalence under curvature lower bound. Duke Math. J. 154, 207–239 (2010). [WoS]
  • [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).
  • [13] C. Villani. Optimal transport. Old and new. Grundlehren der mathematischen Wissenschaften 338. Springer (2009).
  • [14] F.-Y. Wang. Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields 109, 417–424 (1997).
  • [15] F.-Y. Wang. Functional inequalities, Markov properties and spectral theory. Science Press (2005).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_agms-2015-0011
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