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Potential Analysis

, Volume 38, Issue 3, pp 805–841 | Cite as

Self-improvement of Poincaré Type Inequalities Associated with Approximations of the Identity and Semigroups

  • Ana Jiménez-del-Toro
  • José María Martell
Article

Abstract

The purpose of this paper is to present a general method that allows us to study self-improving properties of generalized Poincaré inequalities. When measuring the oscillation in a given cube, we replace the average by an approximation of the identity or a semigroup scaled to that cube and whose kernel decays fast enough. We apply the method to obtain self-improvement in the scale of Lebesgue spaces of Poincaré type inequalities. In particular, we propose some expanded Poincaré estimates that take into account the lack of localization of the approximation of the identity or the semigroup. As a consequence of this method we are able to obtain global pseudo-Poincaré inequalities.

Keywords

Semigroups Heat kernels Self-improving properties Generalized Poincaré–Sobolev and Hardy inequalities Pseudo-Poincaré inequalities Dyadic cubes Weights Good-λ inequalities 

Mathematics Subject Classifications (2010)

46E35 (47D06, 46E30, 42B25) 

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References

  1. 1.
    Auscher, P.: On necessary and sufficient conditions for L p estimates of Riesz transform associated elliptic operators on ℝn and related estimates. Mem. Am. Math. Soc. 186(871), (2007)Google Scholar
  2. 2.
    Auscher, P., Martell, J.M.: Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators. J. Funct. Anal. 241, 703–746 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Badr, N., Jiménez-del-Toro, A., Martell, J.M.: L p self-improvement of generalized Poincaré inequalities in spaces of homogeneous type. J. Funct. Anal. 260(11), 3147–3188 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bakry, D., Coulhon, T., Ledoux, M., Saloff-Coste, L.: Sobolev inequalities in disguise. Indiana J. Math. 44, 1033–1074 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Benett, C., Sharpley, R.C.: Interpolation of operators. Pure Appl. Math. 129 (1988)Google Scholar
  6. 6.
    Burkholder, D.L., Gundy, R.F.: Extrapolation and interpolation of quasilinear operators on martingales. Acta Math. 124, 249–304 (1970)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Deng, D., Duong, X.T., Yan, L.: A characterization of the Morrey–Campanato spaces. Math. Z. 250(3), 641–655 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Duong, X.T., Yan, L.: New function spaces of BMO type, the John–Nirenberg inequality, interpolation, and applications. Comm. Pure Appl. Math. 58(10), 1375–1420 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Franchi, B., Pérez, C., Wheeden, R.L.: Self-improving properties of John–Nirenberg and Poincaré inequalities on space of homogeneous type. J. Funct. Anal. 153(1), 108–146 (1998)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    García-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics, vol. 116. North Holland Math. Studies, North Holland, Amsterdam (1985)Google Scholar
  11. 11.
    Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, Prentice Hall (2004)MATHGoogle Scholar
  12. 12.
    Heinonen, J., Koskela, P.: From local to global in quasiconformal estructures. Proc. Natl. Acad. Sci. U.S.A. 93(2), 554–556 (1996)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181(1), 1–61 (1998)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hofmann, S., Martell, J.M.: L p bounds for Riesz transforms and square roots associated to second order elliptic operators. Publ. Mat. 47, 497–515 (2003)MathSciNetMATHGoogle Scholar
  15. 15.
    Jiménez-del-Toro, A.: Exponential self-improvement of generalized Poincaré inequalities associated with approximations of the identity and semigroups. Trans. Am. Math. Soc. 364(2), 637–660 (2012)MATHCrossRefGoogle Scholar
  16. 16.
    Ledoux, M.: On improved Sobolev embedding theorems. Math. Res. Lett. 10(5–6), 659–669 (2003)MathSciNetMATHGoogle Scholar
  17. 17.
    Martell, J.M.: Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications. Studia Math. 161, 113–145 (2004)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Martin, J., Milman, M.: Sharp Gagliardo-Nirenberg inequalities via symmetrization. Math. Res. Lett. 14(1), 49–62 (2007)MathSciNetMATHGoogle Scholar
  19. 19.
    Rao, M.M., Ren, Z.D.: Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146. Marcel Dekker Inc. (1991)Google Scholar
  20. 20.
    Saloff-Coste, L.: Aspects of Sobolev-type inequalities. London Math. Soc. Lecture Notes Series, vol. 289. Cambridge University Press (2002)Google Scholar
  21. 21.
    Spanne, S.: Some function spaces defined using the mean oscillation over cubes. Ann. Sc. Norm Super. Pisa 19, 593–608 (1965)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  2. 2.Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCMConsejo Superior de Investigaciones CientíficasMadridSpain

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