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Erschienen in:
Markov Semigroups Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

8. Capacity and Isoperimetric-Type Inequalities

verfasst von : Dominique Bakry, Ivan Gentil, Michel Ledoux

Erschienen in: Analysis and Geometry of Markov Diffusion Operators

Verlag: Springer International Publishing

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Abstract

This chapter focuses on inequalities comparing measure and capacity uniformly over a given class of sets as equivalent forms of functional inequalities. The chapter concentrates on the so-called 2-capacities which capture the relevant Dirichlet form information on sets and on the 1-capacities which are related to boundary or surface measures and isoperimetric-type inequalities. The first part of the chapter introduces the basic notions on capacities and co-area formulas to transfer (and back) functional inequalities into measure-capacity inequalities. Sobolev-type, Poincaré and logarithmic Sobolev inequalities are analyzed in this respect. The second part is concerned with measure-capacity inequalities. Together with the heat kernel tools under curvature conditions, measure-capacity inequalities of isoperimetric-type are investigated, leading in particular to the Gaussian isoperimetric inequality as well as to comparison results under curvature conditions. Combination of the heat kernel isoperimetric inequality together with its reverse form further yields an isoperimetric-type Harnack inequality. The last section addresses the relationships between Poincaré and logarithmic Sobolev inequalities and their associated concentration properties.

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Vorheriges Kapitel Generalized Functional Inequalities
Nächstes Kapitel Optimal Transportation and Functional Inequalities
Literatur
14.
C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les Inégalités de Sobolev Logarithmiques. Panoramas et Synthèses, vol. 10 (Société Mathématique de France, Paris, 2000) MATH
35.
D. Bakry, T. Coulhon, M. Ledoux, L. Saloff-Coste, Sobolev inequalities in disguise. Indiana Univ. Math. J. 44(4), 1033–1074 (1995) MathSciNet
37.
38.
D. Bakry, M. Ledoux, Lévy-Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123(2), 259–281 (1996) CrossRefMATHMathSciNet
45.
F. Barthe, P. Cattiaux, C. Roberto, Concentration for independent random variables with heavy tails. Appl. Math. Res. Express 2, 39–60 (2005) CrossRefMathSciNet
46.
F. Barthe, P. Cattiaux, C. Roberto, Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoam. 22(3), 993–1067 (2006) CrossRefMATHMathSciNet
47.
F. Barthe, P. Cattiaux, C. Roberto, Isoperimetry between exponential and Gaussian. Electron. J. Probab. 12(44), 1212–1237 (2007) (electronic) MATHMathSciNet
49.
F. Barthe, C. Roberto, Sobolev inequalities for probability measures on the real line. Stud. Math. 159(3), 481–497 (2003). Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday CrossRefMATHMathSciNet
52.
V. Bayle, A differential inequality for the isoperimetric profile. Int. Math. Res. Not. 7, 311–342 (2004) CrossRefMathSciNet
60.
P. Bérard, G. Besson, S. Gallot, Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov. Invent. Math. 80(2), 295–308 (1985) CrossRefMATHMathSciNet
73.
S.G. Bobkov, An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab. 25(1), 206–214 (1997) CrossRefMATHMathSciNet
75.
S.G. Bobkov, A localized proof of the isoperimetric Bakry-Ledoux inequality and some applications. Teor. Veroâtn. Ee Primen. 47(2), 340–346 (2002) CrossRefMathSciNet
77.
S.G. Bobkov, F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163(1), 1–28 (1999) CrossRefMATHMathSciNet
78.
S.G. Bobkov, C. Houdré, Some connections between isoperimetric and Sobolev-type inequalities. Mem. Am. Math. Soc. 129, 616 (1997), pp. viii+111
80.
S.G. Bobkov, B. Zegarliński, Entropy bounds and isoperimetry. Mem. Am. Math. Soc. 176, 829 (2005)
87.
97.
D. Burago, Y.D. Burago, S. Ivanov, A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33 (American Mathematical Society, Providence, 2001) MATH
98.
Y.D. Burago, V.A. Zalgaller, Geometric Inequalities. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285 (Springer, Berlin, 1988). Translated from the Russian by A.B. Sosinskiĭ, Springer Series in Soviet Mathematics MATH
99.
103.
L.A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities. Commun. Math. Phys. 214(3), 547–563 (2000) CrossRefMATHMathSciNet
114.
G. Carron, Inégalités isopérimétriques de Faber-Krahn et conséquences, in Actes de la Table Ronde de Géométrie Différentielle, Luminy, 1992. Sémin. Congr., vol. 1 (Soc. Math. France, Paris, 1996), pp. 205–232
116.
P. Cattiaux, I. Gentil, A. Guillin, Weak logarithmic Sobolev inequalities and entropic convergence. Probab. Theory Relat. Fields 139(3–4), 563–603 (2007) CrossRefMATHMathSciNet
121.
I. Chavel, Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics, vol. 115 (Academic Press, Orlando, 1984). Including a chapter by Burton Randol, with an appendix by Jozef Dodziuk MATH
122.
I. Chavel, Isoperimetric Inequalities. Cambridge Tracts in Mathematics, vol. 145 (Cambridge University Press, Cambridge, 2001). Differential geometric and analytic perspectives MATH
124.
J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis (Papers Dedicated to Salomon Bochner, 1969) (Princeton Univ. Press, Princeton, 1970), pp. 195–199
162.
J. Dolbeault, I. Gentil, A. Guillin, F.-Y. Wang, L q -functional inequalities and weighted porous media equations. Potential Anal. 28(1), 35–59 (2008) CrossRefMATHMathSciNet
184.
H. Federer, Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153 (Springer, New York, 1969) MATH
193.
S. Gallot, Inégalités isopérimétriques et analytiques sur les variétés Riemanniennes. Astérisque (1988), no. 163–164, pp. 5–6, 31–91, 281 (1989). On the geometry of differentiable manifolds (Rome, 1986)
195.
R.J. Gardner, The Brunn-Minkowski inequality. Bull. Am. Math. Soc. (N.S.) 39(3), 355–405 (2002) CrossRefMATH
207.
N. Gozlan, Poincaré inequalities and dimension free concentration of measure. Ann. Inst. Henri Poincaré Probab. Stat. 46(3), 708–739 (2010) CrossRefMATHMathSciNet
216.
A. Grigor’yan, Isoperimetric inequalities and capacities on Riemannian manifolds, in The Maz’ya Anniversary Collection, Vol. 1, Rostock, 1998. Oper. Theory Adv. Appl., vol. 109 (Birkhäuser, Basel, 1999), pp. 139–153 CrossRef
220.
M. Gromov, Paul Lévy’s isoperimetric inequality. Preprint, 1980
221.
M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, vol. 152 (Birkhäuser, Boston, 1999). Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates MATH
224.
L. Gross, Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975) CrossRef
273.
279.
M. Ledoux, Spectral gap, logarithmic Sobolev constant, and geometric bounds, in Surveys in Differential Geometry, vol. IX (Int. Press, Somerville, 2004), pp. 219–240
280.
M. Ledoux, From concentration to isoperimetry: semigroup proofs, in Concentration, Functional Inequalities and Isoperimetry. Contemp. Math., vol. 545 (Am. Math. Soc., Providence, 2011), pp. 155–166 CrossRef
281.
P. Lévy, Problèmes Concrets d’Analyse Fonctionnelle, 2nd edn. Des Leçons d’Analyse Fonctionnelle (Gauthier-Villars, Paris, 1951) (French). Avec un complément par F. Pellegrino MATH
293.
L. Lovász, M. Simonovits, Random walks in a convex body and an improved volume algorithm. Random Struct. Algorithms 4(4), 359–412 (1993) CrossRefMATH
295.
F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems (Cambridge University Press, Cambridge, 2012) CrossRefMATH
303.
V.G. Maz’ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342 (Springer, Heidelberg, 2011). augmented ed. MATH
312.
313.
E. Milman, A converse to the Maz’ya inequality for capacities under curvature lower bound, in Around the Research of Vladimir Maz’ya. I. Int. Math. Ser. (N. Y.), vol. 11 (Springer, New York, 2010), pp. 321–348 CrossRef
314.
E. Milman, Isoperimetric and concentration inequalities: equivalence under curvature lower bound. Duke Math. J. 154(2), 207–239 (2010) CrossRefMATHMathSciNet
315.
E. Milman, Sharp isoperimetric inequalities and model spaces for curvature-dimension-diameter condition. Preprint, 2012
316.
V.D. Milman, G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces. Lecture Notes in Mathematics, vol. 1200 (Springer, Berlin, 1986). With an appendix by M. Gromov MATH
318.
F. Morgan, Manifolds with density. Not. Am. Math. Soc. 52(8), 853–858 (2005) MATH
321.
B. Muckenhoupt, Hardy’s inequality with weights. Stud. Math. 44, 31–38 (1972). Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I MATHMathSciNet
338.
401.
V.N. Sudakov, B.S. Cirel’son, Extremal properties of half-spaces for spherically invariant measures. Zap. Nauč. Semin. LOMI 41, 14–24, 165 (1974). Problems in the theory of probability distributions, II
428.
F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Relat. Fields 109(3), 417–424 (1997) CrossRefMATH
431.
F.-Y. Wang, Functional Inequalities, Markov Processes, and Spectral Theory (Science Press, Beijing, 2004)
442.
S.-T. Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold. Ann. Sci. Éc. Norm. Super. 8(4), 487–507 (1975) MATH
450.
Metadaten
Titel
Capacity and Isoperimetric-Type Inequalities
verfasst von
Dominique Bakry
Ivan Gentil
Michel Ledoux
Copyright-Jahr
2014
Buch
Analysis and Geometry of Markov Diffusion Operators

Print ISBN: 978-3-319-00226-2

Electronic ISBN: 978-3-319-00227-9

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-319-00227-9_8