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.
References
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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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DOI: https://doi.org/10.1007/s00440-010-0335-2
Keywords
- Green function
- Planar domain
- Capacity
- Harmonic measure
- Brownian motion
- Loop
- 3G-inequality
- Greenian domain
Mathematics Subject Classification (2000)
- 31A15
- 60J45
- 60J65