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3G-inequality for planar domains
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  • Published: 03 November 2011

3G-inequality for planar domains

  • Wolfhard Hansen1 

Probability Theory and Related Fields volume 152, pages 357–366 (2012)Cite this article

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Abstract

The 3G-inequality for Green functions g D on arbitrary bounded domains in \({{\mathbb{R}}^2}\) , which Bass and Burdzy (Probab Theory Relat Fields 101(4):479–493, 1995) obtained by a genuinely probabilistic proof (using loops of Brownian motion around the origin), is proven (in a more precise form) employing elementary properties of harmonic measures only. Since harmonic measures are hitting distributions of Brownian motion, this purely analytic proof can be viewed as well as being probabilistic. A spin-off is an upper estimate of g D on subdisks B′ of an open disk B in terms of g B divided by the capacity of \({B'\setminus D}\) with respect to B.

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References

  1. Aikawa H., Lundh T.: The 3G inequality for a uniformly John domain. Kodai Math. J. 28, 209–219 (2005)

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  2. Armitage, D.H., Gardiner, S.J.: Classical Potential Theory. In: Springer Monographs in Mathematics. Springer-Verlag London Ltd., London (2001)

  3. Bass R.F., Burdzy K.: Conditioned Brownian motion in planar domains. Probab. Theory Relat. Fields 101(4), 479–493 (1995)

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  4. Hansen W.: Simple counterexamples to the 3G-inequality. Expo. Math. 24(1), 97–102 (2006)

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  5. Helms, L.L.: Introduction to Potential Theory. In: Pure and Applied Mathematics, vol. XXII. Wiley-Interscience, New York (1969)

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, 33501, Bielefeld, Germany

    Wolfhard Hansen

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  1. Wolfhard Hansen
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Correspondence to Wolfhard Hansen.

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Cite this article

Hansen, W. 3G-inequality for planar domains. Probab. Theory Relat. Fields 152, 357–366 (2012). https://doi.org/10.1007/s00440-010-0335-2

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  • Received: 08 February 2010

  • Revised: 25 August 2010

  • Published: 03 November 2011

  • Issue Date: February 2012

  • DOI: https://doi.org/10.1007/s00440-010-0335-2

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Keywords

  • Green function
  • Planar domain
  • Capacity
  • Harmonic measure
  • Brownian motion
  • Loop
  • 3G-inequality
  • Greenian domain

Mathematics Subject Classification (2000)

  • 31A15
  • 60J45
  • 60J65
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