Illinois Journal of Mathematics

Symmetrization and harmonic measure

Dimitrios Betsakos

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We prove the equality statements for the classical symmetrization estimates for harmonic measure. In fact, we prove more general results for $\alpha$-harmonic measure.\break The $\alpha$-harmonic measure is the hitting distribution of symmetric $\alpha$-stable processes upon exiting an open set in $\mathbb{R^n}$ ($0<\alpha<2$, $n\geq2$). It can also be defined in the context of Riesz potential theory and the fractional Laplacian. We prove polarization and symmetrization inequalities for $\alpha$-harmonic measure. We give a complete description of the corresponding equality cases. The proofs involve analytic and probabilistic arguments.

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Illinois J. Math., Volume 52, Number 3 (2008), 919-949.

First available in Project Euclid: 1 October 2009

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Primary: 30C85: Capacity and harmonic measure in the complex plane [See also 31A15] 31B15: Potentials and capacities, extremal length 31C05: Harmonic, subharmonic, superharmonic functions 60G52: Stable processes 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]


Betsakos, Dimitrios. Symmetrization and harmonic measure. Illinois J. Math. 52 (2008), no. 3, 919--949. doi:10.1215/ijm/1254403722.

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