Revista Matemática Complutense

, Volume 28, Issue 2, pp 359–392 | Cite as

Integral isoperimetric transference and dimensionless Sobolev inequalities

Article

Abstract

We introduce the concept of Gaussian integral isoperimetric transfer and show how it can be applied to obtain a new class of sharp Sobolev-Poincaré inequalities with constants independent of the dimension. In the special case of \(L^{q}\) spaces on the unit \(n\)-dimensional cube our results extend the recent inequalities that were obtained in Fiorenza et al. (2012) using extrapolation.

Keywords

Sobolev inequalities Symmetrization Isoperimetric inequalities Extrapolation 

Mathematics Subject Classification (2000)

46E30 26D10 

Notes

Acknowledgments

We are very grateful to the referees for their comments and suggestions to improve the quality of the paper.

References

  1. 1.
    Alvino, A.: Sulla diseguaglianza di Sobolev in spazi di Lorentz. Boll. Un. Mat. Ital. 5(14–A), 148–156 (1977)MathSciNetGoogle Scholar
  2. 2.
    Astashkin, S.V., Lykov, K.V.: Strong extrapolation spaces and interpolation. Siberian Math. J. 50, 199–213 (2009)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Barthe, F.: Log-concave and spherical models in isoperimetry. Geom. Funct. Anal. 12, 32–55 (2002)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. SpringerVerlag, Berlin-Heidelberg (1976)CrossRefMATHGoogle Scholar
  5. 5.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)MATHGoogle Scholar
  6. 6.
    Bobkov, S.G., Houdré, C.: Some connections between isoperimetric and Sobolev type inequalities. Mem. Amer. Math. Soc. 129, 616 (1997)Google Scholar
  7. 7.
    Boyd, D.W.: Indices of function spaces and their relationship to interpolation. Can. J. Math. 21, 1245–1254 (1969)CrossRefMATHGoogle Scholar
  8. 8.
    Burago, Y.D., Mazja, V.G.: Certain questions of potential theory and function theory for regions with irregular boundaries. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 3, 152 (1967). (English translation: Potential theory and function theory for irregular regions. Seminars in Mathematics. Steklov, V.A. Mathematical Institute, Leningrad, Vol. 3, Consultants Bureau, New York, 1969)MathSciNetGoogle Scholar
  9. 9.
    Cordero-Erausquin, D., Nazaret, B., Villani, C.: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 307–332 (2004)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Fiorenza, A.: Duality and reflexivity in grand Lebesgue spaces. Collect. Math. 51, 131–148 (2000)MATHMathSciNetGoogle Scholar
  11. 11.
    Fiorenza, A., Karadzhov, G.E.: Grand and Small Lebesgue Spaces and their analogs. Z. Anal. Anwendungen 23, 657–681 (2004)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Fiorenza, A., Krbec, M., Schmeisser, H.J.: An improvement of dimension-free Sobolev imbeddings in r.i. spaces. J. Funct. Anal (2012) (to appear)Google Scholar
  13. 13.
    Garsia, A., Rodemich, E.: Monotonicity of certain functionals under rearrangements. Ann. Inst. Fourier (Grenoble) 24, 67–116 (1974)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Griebel, M.: Sparse grids and related approximation schemes for higher dimensional problems. In: Pardo, L., Pinkus, A., Suli, E., Todd, M. (eds.) Foundations of Computational Mathematics (FoCM05), Santander, pp. 106–161. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  15. 15.
    Iwaniec, T., Sbordone, C.: On the integrability of the Jacobian under minimal hypotheses. Arch. Rational Mech. Anal. 119, 129–143 (1992)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Karadzhov, G.E., Milman, M.: Extrapolation theory: new results and applications. J. Approx. Theory 133, 38–99 (2005)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Krbec, M., Schmeisser, H.J.: On dimension-free Sobolev imbeddings I. J. Math. Anal. Appl. 387, 114–125 (2012)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Krbec, M., Schmeisser, H.J.: On dimension-free Sobolev imbeddings II. Rev. Mat. Complutense 25, 247–265 (2012)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Martin, J., Milman, M.: Pointwise symmetrization inequalities for Sobolev functions and applications. Adv. Math. 225, 121–199 (2010)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Martin, J., Milman, M.: Fractional Sobolev inequalities: symmetrization, isoperimetry and interpolation (2012)Google Scholar
  21. 21.
    Martin, J., Milman, M.: Sobolev inequalities, rearrangements, isoperimetry and interpolation spaces. Contemp. Math. 545, 167–193 (2011)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Martin, J., Milman, M., Pustylnik, E.: Sobolev inequalities: symmetrization and self improvement via truncation. J. Funct. Anal. 252, 677–695 (2007)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Maz’ya, V.G.: Sobolev Spaces. Springer-Verlag, New York (1985)Google Scholar
  24. 24.
    Milman, E.: A converse to the Maz’ya inequality for capacities under curvature lower bound. In: Laptev, A. (ed.) Around the Research of Vladimir Maz’ya I: Function Spaces, pp. 321–348. Springer, New York (2010)CrossRefGoogle Scholar
  25. 25.
    Milman, E., Rotem, L.: Complemented Brunn-Minkowski Inequalities and Isoperimetry for Homogeneous and Non-Homogeneous Measures ( arXiv:1308.5695)
  26. 26.
    Milman, M.: The computation of the \(K\)-functional for couples of rearrangement invariant spaces. Results Math. 5, 174–176 (1982)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Rakotoson, J.M., Simon, B.: Relative Rearrangement on a finite measure space spplication to the regularity of weighted monotone rearrangement (Part 1). Rev. R. Acad. Cienc. Exact. Fis. Nat. 91, 17–31 (1997)MATHMathSciNetGoogle Scholar
  28. 28.
    Ros, A.: The isoperimetric problem. In: Global Theory of Minimal Surfaces. Clay Math. Proc. vol. 2, pp. 175–209. Am. Math. Soc., Providence (2005)Google Scholar
  29. 29.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Math. Pura Appl. 110, 353–372 (1976)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Triebel, H.: Tractable Embeddings of Besov Spaces into Zygmund Spaces, Function Spaces, vol. IX, pp 361–377. Banach Center Publ. 92, Polish Acad. Sci. Inst. Math., Warsaw (2011)Google Scholar
  31. 31.
    Triebel, H.: Tractable Embeddings. University of Jena, Jena (2012)Google Scholar

Copyright information

© Universidad Complutense de Madrid 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversitat Autònoma de BarcelonaBarcelonaSpain

Personalised recommendations