A Converse to the Maz’ya Inequality for Capacities under Curvature Lower Bound

  • Emanuel Milman
Part of the International Mathematical Series book series (IMAT, volume 11)


We survey some classical inequalities due to Maz'ya relating iso- capacitary inequalities with their functional and isoperimetric counterparts in a measure-metric space setting, and extend Maz'ya's lower bound for the q-capacity (q > 1) in terms of the 1-capacity (or isoperimetric) profile. We then proceed to describe results by Buser, Bakry, Ledoux and most recently by the author, which show that under suitable convexity assumptions on the measure-metric space, the Maz'ya inequality for capacities may be reversed, up to dimension independent numerical constants: a matching lower bound on 1-capacity may be derived in terms of the q-capacity profile. We extend these results to handle arbitrary q > 1 and weak semiconvexity assumptions, by obtaining some new delicate semigroup estimates.


Sobolev Inequality Isoperimetric Inequality Convexity Assumption Logarithmic Sobolev Inequality Young Function 
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  1. 1.
    Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/84. Lect. Notes Math. 1123, pp. 177–206. Springer (1985)Google Scholar
  2. 2.
    Bakry, D., Ledoux, M.: Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123, no. 2, 259–281 (1996)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Barthe, F., Cattiaux, P., Roberto, C.: Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoamericana 22, no. 3, 993–1067 (2006)MATHMathSciNetGoogle Scholar
  4. 4.
    Barthe, F., Kolesnikov, A. V.: Mass transport and variants of the logarithmic Sobolev inequality. J. Geom. Anal. 18, no. 4, 921–979 (2008)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Barthe, F., Maurey, B.: Some remarks on isoperimetry of Gaussian type. Ann. Inst. H. Poincaré Probab. Statist. 36, no. 4, 419–434 (2000)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bobkov, S.G.: A functional form of the isoperimetric inequality for the Gaussian measure. J. Funct. Anal. 135, no. 1, 39–49 (1996)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bobkov, S.G.: An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab. 25, no. 1, 206–214 (1997)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bobkov, S.G., Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163, no. 1, 1–28 (1999)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bobkov, S.G., Houdré, C.: Isoperimetric constants for product probability measures. Ann. Probab. 25, no. 1, 184–205 (1997)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Bobkov, S.G., Houdré, C.: Some connections between isoperimetric and Sobolev type inequalities. Mem. Am. Math. Soc. 129, no. 616 (1997)Google Scholar
  11. 11.
    Bobkov, S.G., Zegarlinski, B.: Entropy bounds and isoperimetry. Mem. Am. Math. Soc. 176, no. 829 (2005)MathSciNetGoogle Scholar
  12. 12.
    Borell, Ch.: Convex measures on locally convex spaces. Ark. Mat. 12, 239–252 (1974)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Buser, P.: A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4) 15, no. 2, 213–230 (1982)MATHMathSciNetGoogle Scholar
  14. 14.
    Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Gunning, R. (ed.), Problems in Analysis, pp. 195–199. Princeton Univ. Press, Princeton, NJ (1970)Google Scholar
  15. 15.
    Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge (1989)MATHCrossRefGoogle Scholar
  16. 16.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, no. 418–491 (1959)MATHMathSciNetGoogle Scholar
  17. 17.
    Federer, H., Fleming, W.H.: Normal and integral currents. Ann. Math. 72, 458–520 (1960)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fleming, W.H.: Functions whose partial derivatives are measures. Illinois J. Math. 4, 52–478 (1960)MathSciNetGoogle Scholar
  19. 19.
    Hsu, E.P.: Multiplicative functional for the heat equation on manifolds with boundary. Michigan Math. J. 50, no. 2, 351–367 (2002)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Ledoux, M.: A simple analytic proof of an inequality by P. Buser. Proc. Am. Math. Soc. 121, no. 3, 951–959 (1994)MATHMathSciNetGoogle Scholar
  21. 21.
    Ledoux, M.: The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6) 9, no. 2, 305–366 (2000)MATHMathSciNetGoogle Scholar
  22. 22.
    Ledoux, M.: Spectral gap, logarithmic Sobolev constant, and geometric bounds. In: Surveys in Differential Geometry. IX, pp. 219–240. Int. Press, Somerville, MA (2004)Google Scholar
  23. 23.
    Maz'ya, V.G.: Classes of domains and imbedding theorems for function spaces (Russian). Dokl. Akad. Nauk SSSR 3, 527–530 (1960); English transl.: Sov. Math. Dokl. 1, 882–885 (1961)Google Scholar
  24. 24.
    Maz'ya, V.G.: p-Conductivity and theorems on imbedding certain functional spaces into a C-space (Russian). Dokl. Akad. Nauk SSSR 140, 299–302 (1961); English transl.: Sov. Math. Dokl. 2, 1200–1203 (1961)Google Scholar
  25. 25.
    Maz'ya, V.G.: The negative spectrum of the higher-dimensional Schrödinger operator (Russian). Dokl. Akad. Nauk SSSR 144, 721–722 (1962); English transl.: Sov. Math. Dokl. 3, 808–810 (1962)Google Scholar
  26. 26.
    Maz'ya, V.G.: On the solvability of the Neumann problem (Russian). Dokl. Akad. Nauk SSSR 147, 294–296 (1962); English transl.: Sov. Math. Dokl. 3, 1595–1598 (1962)Google Scholar
  27. 27.
    Maz'ya, V.G.: Sobolev spaces. Springer, Berlin (1985)Google Scholar
  28. 28.
    Milman, E.: On the role of convexity in functional and isoperimetric inequalities. to appear in the Proc. London Math. Soc.,, 2008.Google Scholar
  29. 29.
    Milman, E.: On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177, no. 1, 1–43 (2009)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Milman, E.: Isoperimetric and concentration inequalities - equivalence under curvature lower bound. (2009)Google Scholar
  31. 31.
    Milman, E., Sodin, S.:. An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies. J. Funct. Anal. 254, no. 5, 1235–1268 (2008) Scholar
  32. 32.
    Muckenhoupt, B.: Hardy's inequality with weights. Studia Math. 44, 31–38 (1972)MATHMathSciNetGoogle Scholar
  33. 33.
    Qian, Z.: A gradient estimate on a manifold with convex boundary. Proc. Roy. Soc. Edinburgh Sect. A 127, no. 1, 171–179 (1997)MATHMathSciNetGoogle Scholar
  34. 34.
    Roberto, C., Zegarliński, B.: Orlicz–Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semigroups. J. Funct. Anal. 243, no. 1, 28–66 (2007)MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Sodin, S.: An isoperimetric inequality on the ∞p balls. Ann. Inst. H. Poincaré Probab. Statist. 44, no. 2, 362–373 (2008)MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Wang, F.-Y.: Gradient estimates and the first Neumann eigenvalue on mani-folds with boundary. Stochastic Process. Appl. 115, no. 9, 1475–1486 (2005)MATHMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of Mathematics, Institute for Advanced StudyPrincetonUSA

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