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Isoperimetric Inequalities for Non-Local Dirichlet Forms

  • Feng-Yu Wang
  • Jian WangEmail author
Article
  • 6 Downloads

Abstract

Let \((E,{\mathscr{E}} F,\mu )\) be a σ-finite measure space. For a non-negative symmetric measurable function J(x,y) on E × E, consider the quadratic form

$$ \mathscr{E}(f,f):= \frac{1}{2}{\int}_{E\times E} (f(x)-f(y))^{2} J(x,y) \mu(\text{\text{d}} x) \mu(\text{\text{d}} y) $$

in L2(μ). We characterize the relationship between the isoperimetric inequality and the super Poincaré inequality associated with \({\mathscr{E}}\). In particular, sharp Orlicz-Sobolev type and Poincaré type isoperimetric inequalities are derived for stable-like Dirichlet forms on \(\mathbb {R}^{n}\), which include the existing fractional isoperimetric inequality as a special example.

Keywords

Isoperimetric inequality Non-local Dirichlet form Super Poincaré inequality Orlicz norm 

Mathematics Subject Classification (2010)

47G20 47D62 

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Notes

Acknowledgments

Supported in part by NNSFC (11431014, 11522106, 11626245, 11626250, 11771326, 11831014), the Program for Probability and Statistics: Theory and Application (No. IRTL1704) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ). The authors would like to thank Professor Takashi Kumagai and the referee for their helpful comments.

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., De Philippis, G., Martinazzi, L.: Gamma-convergence of nonlocal perimeter functionals. Manuscripta Math. 134, 377–403 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Barthe, F., Cattiaux, P., Roberto, C.: Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and application to isoperimetry. Revista Mat. Iberoamer. 22, 993–1066 (2006)zbMATHCrossRefGoogle Scholar
  4. 4.
    Barthe, F., Cattiaux, P., Roberto, C.: Isoperimetry between exponential and Gaussian. Electron. J. Proba. 12, 1212–1237 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bakry, D., Coulhon, T., Ledoux, M., Saloff-Coste, L.: Sobolev inequalities in disguise. Ind. Univ. Math. J. 44, 1033–1074 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bendikov, A., Saloff-Coste, L.: Random walks on groups and discrete subordination. Math. Nachr. 285, 580–605 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Besov, O.V.: On a certain family of functional spaces. Imbedding and continuation theorems. Dokl. Akad. Nauk SSSR 126, 1163–1165 (1959)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Besov, O.V.: On some conditions of membership in l p for derivatives of periodic functions. Naučn. Dokl. Vyss. Skoly. Fiz.-Mat. Nauki 1, 13–17 (1959)Google Scholar
  9. 9.
    Bobkov, S.G., Houdré, C.: Isoperimetric constants for product probability measures. Ann. Probab. 25, 184–205 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bourgain, J., Brezis, H., Mironescu, P.: Limiting embedding theorems for W s,p when s 1 and applications. J. Anal. Math. 87, 77–101 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Buser, P.: A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. 15, 213–230 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Caffarelli, L., Roquejoffre, J.M., Savin, O.: Non-local minimal surfaces. Comm. Pure Appl. Math. 63, 1111–1144 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Caffarelli, L., Valdinoci, E.: Uniform estimates and limiting arguments for nonlocal minimal surfaces. Calc. Var. Partial Differ. Equ. 41, 203–240 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Cheeger, J.: A Lower Bound for the Smallest Eigenvalue of the Laplacian, In: Problem in Analysis, a Symposium in Honor of S. Bochner, pp. 195–199. Princeton Uinversity Press, Princeton (1970)Google Scholar
  15. 15.
    Chen, M.-F.: Nash inequalities for general symmetric forms. Acta Math. Sin. Engl. Ser. 15, 353–370 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Chen, M.-F.: Logarithmic Sobolev inequality for symmetric forms. Sci. in China(A) 43, 601–608 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Chen, M.-F., Wang, F.-Y.: Cheeger’s inequalities for general symmetric forms and existence criteria for spectral gap. Ann.Probab. 28, 235–257 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Chen, X., Wang, F.-Y., Wang, J.: Functional Inequalities for Pure-Jump Dirichlet Forms. In: Chen, Z.-Q., Jacob, N., Takeda, M., Uemura, T. (eds.) Festschrift Masatoshi Fukushima, pp 143–162. World Scientific, New Jersey (2015)Google Scholar
  19. 19.
    Chen, Z.-Q., Kim, P., Kumagai, T.: Weighted Poincare inequality and heat kernel estimates for finite range jump processes. Math. Ann. 342, 833–883 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on d-sets. Stochastic Process Appl. 108, 27–62 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Coulhon, T.: Heat Kernel and Isoperimetry on Non-Compact Riemannian Manifolds. In: Contemporary Mathematics, vol. 338, pp 65–99. American Mathematical Society, Providence (2003)Google Scholar
  22. 22.
    Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)Google Scholar
  23. 23.
    Dávila, J.: On an open question about functions of bounded variation. Calc. Var. Partial Differ. Equ. 15, 517–529 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Dipierro, S., Figalli, A., Palatucci, G., Valdinoci, E.: Asymptotics of the s-perimeter as s ↘ 0. Discret. Contin Dyn. Syst. 33, 2777–2790 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Erdélyi, A., Tricomi, F.G.: The asymptotic expansion of a ratio of gamma functions. Pacific J. Math. 1, 133–142 (1951)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Figalli, A., Fusco, N., Maggi, F., Millot, V., Morini, M.: Isoperimetry and stability properties of balls with respect to nonlocal energies. Commun. Math. Phys. 336, 441–507 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Frank, R.L., Seiringer, R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255, 3407–3430 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Fusco, N., Millot, V., Morini, M.: A quantitative isoperimetric inequality for fractional perimeters. J. Funct. Anal. 261, 697–715 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Hebisch, W., Saloff-Coste, L.: Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21, 673–709 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Lawler, G.F., Sokal, A.D.: Bounds on the l 2 spectrum for Markov chain and Markov processes: a generalization of Cheeger’s inequality. Trans. Amer. Math. Soc. 309, 557–580 (1998)zbMATHGoogle Scholar
  31. 31.
    Ledoux, M.: A simple proof of an inequality by P. Buser. Proc. Amer. Math. Soc. 121, 951–958 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Mao, Y.H.: General Sobolev type inequalities for symmetric forms. J. Math. Anal. Appl. 338, 1092–1099 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Mao, Y.H.: L p-poincaré inequality for general symmetric forms. Acta Math. Sin. Engl. Ser. 25, 2055–2064 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Maz’ja, V.G.: Sobolev spaces. Springer Series in Soviet Mathematics, Springer, Berlin (1985)Google Scholar
  35. 35.
    Maz’ya, V.: Lectures on Isoperimetric and Isocapacitary Inequalities in the Theory of Sobolev Spaces. In: Contemporary Mathematics, vol. 338, pp 307–340. American Mathematical Society, Providence (2003)Google Scholar
  36. 36.
    Maz’ya, V., Shaposhnikova, T.: On the Bourgain, Brezis, and Mironesce theorem concerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195, 230–238 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Maz’ya, V., Shaposhnikova, T.: . Erratum. J. Funct. Anal. 201, 298–300 (2003)CrossRefGoogle Scholar
  38. 38.
    Milman, E.: On the role of convexity in functional and isoperimetric inequalities. Proc. Lond. Math. Soc. 99, 32–66 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Mimica, A.: On subordinate random walks. Forum Math. 29, 653–664 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Murugan, M., Saloff-Coste, L.: Transition probability estimates for long range random walks. NY J. Math. 21, 723–757 (2015)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Rao, M.M., Ren, Z.D.: Applications of Orlicz Spaces. Marcel Dekker, New York (2002)zbMATHGoogle Scholar
  43. 43.
    Röckner, M., Wang, F.-Y.: Weak Poincaré inequalities and L 2-convergence rates of Markov semigroups. J. Funct. Anal. 185, 564–603 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Saloff-Coste, L.: Lectures on Finite Markov Chains. In: Ecole D’eté De ProbabilitéS De Saint-Flour XXVI-1996. Lecture Notes in Mathematics, vol. 1665, pp 301–413. Springer, Berlin (1997)Google Scholar
  45. 45.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  46. 46.
    Schilling, R.L., Sztonyk, P., Wang, J.: Coupling property and gradient estimates of lévy processes via the symbol. Bernoulli 18, 1128–1149 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces, Lect. Notes Unione Mat. Ital., vol. 3. Springer, Berlin (2007)Google Scholar
  48. 48.
    Talagrand, M.: A new isoperimetric inequality and the concentration of measure phenomenon, Israel Seminar (GAFA). Lecture Notes in Math., vol. 1469, pp 94–124. Springer, Berlin (1991)Google Scholar
  49. 49.
    Telcs, A.: The art of random walks, Lecture Notes in Mathematics, vol 1855. Springer, Berlin (2006)Google Scholar
  50. 50.
    Visintin, A.: Nonconvex functionals related to multiphase systems. SIAM J. Math. Anal. 21, 1281–1304 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Wang, F., Zhang, Y.-H.: F-sobolev inequality for general symmetric forms. Northeast. Math. J. 19, 133–138 (2003)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Wang, F.-Y.: Functional inequalities for empty essential spectrum. J. Funct. Anal. 170, 219–245 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Wang, F.-Y.: Functional inequalities, semigroup properties and spectrum estimates. Infin. Dimens. Anal. Quant. Probab. Relat. Top. 3, 263–295 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Wang, F.-Y.: Sobolev type inequalities for general symmetric forms. Proc. Amer. Math. Soc 128, 3675–3682 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Wang, F.-Y.: Functional Inequalities, Markov Processes and Spectral Theory. Science Press, Beijing (2005)Google Scholar
  56. 56.
    Wang, F.-Y., Wang, J.: Functional inequalities for stable-like Dirichlet forms. J. Theor. Probab. 28, 423–448 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    žnidarič, M.: Asymptotic expansion for inverse moments of Binomial and Poisson distributions. Open Stat. Probab. J. 1, 7–10 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Center of Applied MathematicsTianjin UniversityTianjinChina
  2. 2.Department of MathematicsSwansea UniversitySingleton ParkUK
  3. 3.College of Mathematics and Informatics & Fujian Key Laboratory of Mathematical Analysis and Applications (FJKLMAA)Fujian Normal UniversityFuzhouChina

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