The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1763–1810 | Cite as

A maximal Function Approach to Two-Measure Poincaré Inequalities

  • Juha Kinnunen
  • Riikka Korte
  • Juha Lehrbäck
  • Antti V. VähäkangasEmail author


This paper extends the self-improvement result of Keith and Zhong in  Keith and Zhong (Ann. Math. 167(2):575–599, 2008) to the two-measure case. Our main result shows that a two-measure (pp)-Poincaré inequality for \(1<p<\infty \) improves to a \((p,p-{\varepsilon })\)-Poincaré inequality for some \({\varepsilon }>0\) under a balance condition on the measures. The corresponding result for a maximal Poincaré inequality is also considered. In this case the left-hand side in the Poincaré inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincaré inequalities is used to characterize the self-improvement of two-measure Poincaré inequalities. Examples are constructed to illustrate the role of the assumptions. Harmonic analysis and PDE techniques are used extensively in the arguments.


Poincaré inequality Self-improvement Geodesic two-measure space 

Mathematics Subject Classification

31E05 35A23 46E35 


  1. 1.
    Björn, J.: Poincaré inequalities for powers and products of admissible weights. Ann. Acad. Sci. Fenn. Math. 26(1), 175–188 (2001)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces. EMS Tracts in Mathematics, vol. 17. European Mathematical Society (EMS), Zürich (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    Chanillo, S., Wheeden, R.L.: \(L^p\)-estimates for fractional integrals and Sobolev inequalities with applications to Schrödinger operators. Commun. Partial Differ. Equ. 10(9), 1077–1116 (1985)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chanillo, S., Wheeden, R.L.: Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions. Am. J. Math. 107(5), 1191–1226 (1985)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chanillo, S., Wheeden, R.L.: Poincaré inequalities for a class of non-\(A_p\) weights. Indiana Univ. Math. J. 41(3), 605–623 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cruz-Uribe, D., Rodney, S., Rosta, E.: Poincare inequalities and Neumann problems for the \(p\)-Laplacian. Canad. Math. Bull. (2018). Scholar
  8. 8.
    Cruz-Uribe, D., Rodney, S., Rosta, E.: Global Sobolev inequalities and degenerate \(p\)-Laplacian equations. arXiv:1801.09610 [math.AP] (2018)
  9. 9.
    DeJarnette, N.R.: Self improving Orlicz–Poincaré inequalities. Ph.D. thesis, University of Illinois at Urbana-Champaign (2014)Google Scholar
  10. 10.
    Di Marino, S., Speight, G.: The \(p\)-weak gradient depends on \(p\). Proc. Am. Math. Soc. 143(12), 5239–5252 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Eriksson-Bique, S.: Alternative proof of Keith–Zhong self-improvement and connectivity. arXiv:1610.02129 [math.MG] (2016)
  12. 12.
    Franchi, B., Lu, G., Wheeden, R.L.: Representation formulas and weighted Poincaré inequalities for Hörmander vector fields. Ann. Inst. Fourier (Grenoble) 45(2), 577–604 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Franchi, B., Pérez, C., Wheeden, R.L.: Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type. J. Funct. Anal. 153(1), 108–146 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Franchi, B., Pérez, C., Wheeden, R.L.: A sum operator with applications to self-improving properties of Poincaré inequalities in metric spaces. J. Fourier Anal. Appl. 9(5), 511–540 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hajłasz, P., Kinnunen, J.: Hölder quasicontinuity of Sobolev functions on metric spaces. Rev. Mat. Iberoam. 14(3), 601–622 (1998)CrossRefzbMATHGoogle Scholar
  16. 16.
    Hajłasz, P., Koskela, P.: Sobolev meets Poincaré. C. R. Acad. Sci. Paris Sér. I Math. 320(10), 1211–1215 (1995)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer, New York (2001)Google Scholar
  18. 18.
    Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.T.: Sobolev spaces on metric measure spaces: an approach based on upper gradients, New Mathematical Monographs, vol. 27. Cambridge University Press, Cambridge (2015)Google Scholar
  19. 19.
    Keith, S.: Modulus and the Poincaré inequality on metric measure spaces. Math. Z. 245(2), 255–292 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Keith, S., Zhong, X.: The Poincaré inequality is an open ended condition. Ann. Math. 167(2), 575–599 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kinnunen, J., Lehrbäck, J., Vähäkangas, A.V., Zhong, X.: Maximal function estimates and self-improvement results for Poincaré inequalities. Manuscripta Math. (2018).
  22. 22.
    Lerner, A.K., Pérez, C.: A new characterization of the Muckenhoupt \(A_p\) weights through an extension of the Lorentz-Shimogaki theorem. Indiana Univ. Math. J. 56(6), 2697–2722 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    MacManus, P., Pérez, C.: Generalized Poincaré inequalities: sharp self-improving properties. Internat. Math. Res. Notices 2, 101–116 (1998)CrossRefzbMATHGoogle Scholar
  24. 24.
    Strömberg, J.-O., Torchinsky, A.: Weighted Hardy Spaces. Lecture Notes in Mathematics, vol. 1381. Springer, Berlin (1989)Google Scholar
  25. 25.
    Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Pure and Applied Mathematics, vol. 123. Academic Press Inc., Orlando, FL (1986)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsAalto UniversityEspooFinland
  2. 2.Department of Mathematics and StatisticsUniversity of JyvaskylaJyvaskylaFinland

Personalised recommendations