Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T16:38:22.074Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimates for the Products of the Green Function and the Martin Kernel

Published online by Cambridge University Press:  11 January 2016

Kentaro Hirata
Kentaro Hirata*
Affiliation:
Faculty of Education and Human Studies, Akita University, Akita 010-8502, Japan, hirata@math.akita-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Ω be a proper subdomain of ℝn, n ≥ 2, and let x0∈ Ω be fixed. By GΩ and KΩ we denote the Green function and the Martin kernel for Ω, respectively. Under a certain assumption on Ω near a boundary point ξ, we show that the product GΩ(x, x0)KΩ(x, ξ) is comparable to |x - ξ|2-n for x in a nontangential cone with vertex at ξ. We also give an estimate for the product KΩ(x, ξ)KΩ(x,η) in a uniform domain, where η is another boundary point.

Keywords

31B25
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 188 , December 2007 , pp. 1 - 18
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

References

[1] Aikawa, H., Norm estimate of Green operator, perturbation of Green function and integrability of superhamonic functions, Math. Ann., 312 (1998), no. 2, 289–318.CrossRefGoogle Scholar
[2] Aikawa, H., Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan, 53 (2001), no. 1, 119–145.CrossRefGoogle Scholar
[3] Aikawa, H., Holder continuity of the Dirichlet solution for a general domain, Bull. London Math. Soc., 34 (2002), no. 6, 691–702.CrossRefGoogle Scholar
[4] Aikawa, H., Hirata, K. and Lundh, T., Martin boundary points of a John domain and unions of convex sets, J. Math. Soc. Japan, 58 (2006), no. 1, 247–274.Google Scholar
[5] Aikawa, H. and Lundh, T., The 3G inequality for a uniformly John domain, Kodai Math. J., 28 (2005), no. 2, 209–219.CrossRefGoogle Scholar
[6] Armitage, D. H. and Gardiner, S. J., Classical potential theory, Springer-Verlag London Ltd., 2001.CrossRefGoogle Scholar
[7] Bass, R. F. and Burdzy, K., Conditioned Brownian motion in planar domains, Probab. Theory Related Fields, 101 (1995), no. 4, 479–493.CrossRefGoogle Scholar
[8] Bogdan, K., Sharp estimates for the Green function in Lipschitz domains, J. Math. Anal. Appl., 243 (2000), no. 2, 326–337.CrossRefGoogle Scholar
[9] Burdzy, K., Brownian excursions and minimal thinness. Part III: Applications to the angular derivative problem, Math. Z., 192 (1986), no. 1, 89–107.CrossRefGoogle Scholar
[10] Burdzy, K., Multidimensional Brownian excursions and potential theory, Pitman Re search Notes in Mathematics Series, 164, 1987.Google Scholar
[11] Carroll, T., A classical proof of Burdzy’s theorem on the angular derivative, J. London Math. Soc. (2), 38 (1988), no. 3, 423–441.CrossRefGoogle Scholar
[12] Carroll, T., Boundary behaviour of positive harmonic functions on Lipschitz domains, Ann. Acad. Sci. Fenn. Math., 27 (2002), no. 1, 231–236.Google Scholar
[13] Cranston, M., Fabes, E. and Zhao, Z., Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc, 307 (1988), no. 1, 171–194.Google Scholar
[14] Chung, K. L. and Zhao, Z., From Brownian motion to Schrodinger’s equation, Springer-Verlag, 1995.CrossRefGoogle Scholar
[15] Gardiner, S. J., A short proof of Burdzy’s theorem on the angular derivative, Bull. London Math. Soc, 23 (1991), no. 6, 575–579.CrossRefGoogle Scholar
[16] Gehring, F. W. and Osgood, B. G., Uniform domains and the quasihyperbolic metric, J. Analyse Math., 36 (1979), 50–74.CrossRefGoogle Scholar
[17] Hansen, W., Uniform boundary Harnack principle and generalized triangle property, J. Funct. Anal., 226 (2005), no. 2, 452–484.CrossRefGoogle Scholar
[18] Hirata, K., Boundary behaviour of quotients of Martin kernels, Proc Edinb. Math. Soc. (2), 50 (2007), no. 2, 377–388.CrossRefGoogle Scholar