Skip to main content
SpringerLink
Log in

Some Boundary Harnack Principles with Uniform Constants

Potential Analysis Aims and scope Submit manuscript

Cite this article

Abstract

We prove two versions of a boundary Harnack principle in which the constants do not depend on the domain by using probabilistic methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Aikawa, H.: Boundary Harnack inequality and Martin boundary for a uniform domain. J. Math. Soc. Jpn. 53, 119–145 (2001)

    Article  Google Scholar 

  2. Aikawa, H.: Equivalence between the Boundary Harnack Principle and the Carleson estimate. Math. Scand. 103(1), 61–76 (2008)

    Article  MathSciNet  Google Scholar 

  3. Aikawa, H., Hirata, K., Lundh, T.: Martin boundary points of a John domain and unions of convex sets. J. Math. Soc. Jpn. 1, 58 (2006)

    MATH  Google Scholar 

  4. Ancona, A.: Principle de Harnack á la frontiére et theoreme de Fatou pour un opérateur elliptique dans un domain lipschitzien. Ann. Inst. Fourier (Grenoble) 28(4), 169–213 (1978)

    Article  MathSciNet  Google Scholar 

  5. Bass, R.F.: Probabilistic Techniques in Analysis. Springer, New York (1995)

    MATH  Google Scholar 

  6. Bass, R.F., Burdzy, K.: A boundary Harnack principle in twisted Hölder domains. Ann. of Math. (2) 134(2), 253–276 (1991)

    Article  MathSciNet  Google Scholar 

  7. Banuelos, R., Bass, R.F., Burdzy, K.: Hölder domains and the boundary Harnack principle. Duke Math. J. 64, 195–200 (1991)

    Article  MathSciNet  Google Scholar 

  8. Bogdan, K., Kulczycki, T., Kwaśnicki, M.: Estimates and structure of α-harmonic functions. Probab. Theory Relat. Fields 140(3–4), 345–381 (2008)

    Article  MathSciNet  Google Scholar 

  9. Dahlberg, B.E.: Estimates of harmonic measure. Arch. Rational Mech. Anal. 65, 275–282 (1977)

    Article  MathSciNet  Google Scholar 

  10. Jerison, D.S., Kenig, C.E.: Boundary behavior of harmonic functions in non-tangentially accessible domains. Adv. Math. 46, 80–147 (1982)

    Article  MathSciNet  Google Scholar 

  11. Lierl, J.: Scale-invariant boundary Harnack principle in inner uniform domains in fractal-type spaces. Potential Anal. 43(4), 717–747 (2015)

    Article  MathSciNet  Google Scholar 

  12. Lierl, J., Saloff-Coste, L.: Scale invariant boundary Harnack principle in inner uniform domains. Osaka J. Math. 51, 619–656 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Masson, R: The growth exponent for planar loop-erased random walk. Electron. J. Probab. 14(36), 1012–1073 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Mörters, P., Peres, Y.: Brownian Motion (Cambridge Series in Statistical and Probabilistic Mathematics). Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  15. Wu, J.-M.G.: Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains. Ann. Inst. Fourier (Grenoble) 28(4), 147–167 (1978)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first named author, Martin T. Barlow, was partially supported by NSERC (Canada). The second named author, Deniz Karlı, was partially supported by NSERC (Canada) and partially by the BAP grant, numbered 20A101, at the Işık University, Istanbul, Turkey. We thank our referees for their comments, and in particular one referee for suggesting a considerable simplification of our proof of Theorem 2. We also thank Pınar KarlıAkgün for drawing the figures in this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2

    Martin T. Barlow

  2. Department of Mathematics, Işık University, Şile, Istanbul, Turkey, 34980

    Deniz Karli

Authors
  1. Martin T. Barlow
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Deniz Karli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Deniz Karli.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research partially supported by NSERC (Canada) and BAP 20A101 Grant of Işık University (Turkey).

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barlow, M.T., Karli, D. Some Boundary Harnack Principles with Uniform Constants. Potential Anal 57, 433–446 (2022). https://doi.org/10.1007/s11118-021-09922-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-021-09922-3

Keywords

Mathematics Subject Classification (2010)

search

Navigation