Boundary Harnack Principle and the Quasihyperbolic Boundary Condition

Part of the International Mathematical Series book series (IMAT, volume 9)


We discuss the global boundary Harnack principle for domains in R n (n ≥ 2) satisfying some conditions related to the quasihyperbolic metric. For this purpose, we reformulate the global boundary Harnack principle and the global Carleson estimate in terms of the Green function. Based on our other result asserting the equivalence between the global boundary Harnack principle and the global Carleson estimate, we obtain four equivalent con ditions. Using the box argument for the estimate of the harmonic measure, we obtain the global Carleson estimate in terms of the Green function (and thus the global boundary Harnack principle) for a domain satisfying a con dition on the quasihyperbolic metric and the capacity density condition, as well as for a Holder domain whose boundary is locally given by the graph of a Holder continuous function in R n−1. Our argument is purely analytic and elementary, unlike the probabilistic approach of Bass-Burdzy-Banuelos.


Green Function Lipschitz Domain Harmonic Measure Harnack Inequality Logarithmic Capacity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aikawa, H.: Boundary Harnack principle and Martin boundary for a uniform domain. J. Math. Soc. Japan 53, no. 1, 119–145 (2001)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aikawa, H.: Equivalence between the boundary Harnack principle and the Carleson estimate, Math. Scad. [To appear]Google Scholar
  3. 3.
    Ancona, A.: Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien. Ann. Inst. Fourier (Grenoble) 28, no. 4, 169– 213, x (1978)MATHMathSciNetGoogle Scholar
  4. 4.
    Ancona, A.: First eigenvalues and comparison of Green's functions for elliptic operators on manifolds or domains. J. Anal. Math. 72, 45–92 (1997)MATHMathSciNetGoogle Scholar
  5. 5.
    Armitagem, D.H., Gardiner, S.J.: Classical Potential Theory. Springer, London (2001)Google Scholar
  6. 6.
    Bañuelos, R.: Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators. J. Funct. Anal. 100, no. 1, 181–206 (1991)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bañuelos, R., Bass, R.F., Burdzy, K.: Hölder domains and the boundary Harnack principle. Duke Math. J. 64, no. 1, 195–200 (1991)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bass, R.F., Burdzy, K.: A boundary Harnack principle in twisted Hölder domains. Ann. Math. (2) 134, no. 2, 253–276 (1991)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bass, R.F., Burdzy, K.: Lifetimes of conditioned diffusions. Probab. Theory Related Fields 91, no. 3–4, 405–443 (1992)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Caffarelli, L., Fabes, E., Mortola, S., Salsa, S.: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana Univ. Math. J. 30, no. 4, 621–640 (1981)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dahlberg, B.E.J.: Estimates of harmonic measure. Arch. Ration. Mech. Anal. 65, no. 3, 275–288 (1977)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Fabes, E.B., Garofalo, N., Salsa, S.: A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations. Illinois J. Math. 30, no. 4, 536–565 (1986)MATHMathSciNetGoogle Scholar
  13. 13.
    Jerison, D.S., Kenig, C.E.: Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. Math. 46, no. 1, 80–147 (1982)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Smith, W., Stegenga, D.A.: Hölder domains and Poincaré domains. Trans. Am. Math. Soc. 319, no. 1, 67–100 (1990)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Smith, W., Stegenga, D.A.: Exponential integrability of the quasi-hyperbolic metric on Hölder domains. Ann. Acad. Sci. Fenn. Ser. A I Math. 16, no. 2, 345–360 (1991)MathSciNetGoogle Scholar
  16. 16.
    Wu, J.M.G.: Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains. Ann. Inst. Fourier (Grenoble) 28, no. 4, 147–167, vi (1978)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Hokkaido UniversityJapan

Personalised recommendations