Advertisement

manuscripta mathematica

, Volume 158, Issue 1–2, pp 119–147 | Cite as

Maximal function estimates and self-improvement results for Poincaré inequalities

  • Juha Kinnunen
  • Juha Lehrbäck
  • Antti V. VähäkangasEmail author
  • Xiao Zhong
Article

Abstract

Our main result is an estimate for a sharp maximal function, which implies a Keith–Zhong type self-improvement property of Poincaré inequalities related to differentiable structures on metric measure spaces. As an application, we give structure independent representation for Sobolev norms and universality results for Sobolev spaces.

Mathematics Subject Classification

42B25 35A23 46E35 31E05 30L99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

Funding was provided by Academy of Finland.

References

  1. 1.
    Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces, Volume 17 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich (2011)CrossRefGoogle Scholar
  2. 2.
    Eriksson-Bique, S.: Alternative proof of Keith–Zhong self-improvement (2016). arXiv:1610.02129
  3. 3.
    García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics, Volume 116 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam (1985)zbMATHGoogle Scholar
  4. 4.
    Gogatishvili, A., Koskela, P., Zhou, Y.: Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces. Forum Math. 25(4), 787–819 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gol’dshtein, V., Troyanov, M.: Axiomatic theory of Sobolev spaces. Expos. Math. 19(4), 289–336 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hajłasz, P.: Sobolev spaces on metric-measure spaces. In: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), Volume 338 of Contemporary Mathematics, pp. 173–218. American Mathematical Society, Providence (2003)Google Scholar
  7. 7.
    Hajłasz, P., Kinnunen, J.: Hölder quasicontinuity of Sobolev functions on metric spaces. Rev. Mat. Iberoam. 14(3), 601–622 (1998)CrossRefzbMATHGoogle Scholar
  8. 8.
    Hajłasz, P., Koskela, P.: Sobolev meets Poincaré. C. R. Acad. Sci. Paris Sér. I Math. 320(10), 1211–1215 (1995)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  10. 10.
    Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.T.: Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients, Volume 27 of New Mathematical Monographs. Cambridge University Press, Cambridge (2015)CrossRefzbMATHGoogle Scholar
  11. 11.
    Jiang, R., Shanmugalingam, N., Yang, D., Yuan, W.: Hajłasz gradients are upper gradients. J. Math. Anal. Appl. 422(1), 397–407 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Keith, S., Zhong, X.: The Poincaré inequality is an open ended condition. Ann. Math. (2) 167(2), 575–599 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1989)zbMATHGoogle Scholar
  14. 14.
    Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoam. 16(2), 243–279 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shanmugalingam, N.: A universality property of Sobolev spaces in metric measure spaces. In: Sobolev Spaces in Mathematics I, volume 8 of The International Mathematical Series, pp. 345–359. Springer, New York (2009)Google Scholar
  16. 16.
    Shanmugalingam, N., Yang, D., Yuan, W.: Newton–Besov spaces and Newton–Triebel–Lizorkin spaces on metric measure spaces. Positivity 19(2), 177–220 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsAalto UniversityEspooFinland
  2. 2.Department of Mathematics and StatisticsUniversity of JyvaskylaJyväskyläFinland
  3. 3.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

Personalised recommendations