## Illinois Journal of Mathematics

### Symmetrization and harmonic measure

Dimitrios Betsakos

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 1 October 2009

https://projecteuclid.org/euclid.ijm/1254403722

Digital Object Identifier
doi:10.1215/ijm/1254403722

Mathematical Reviews number (MathSciNet)
MR2546015

Zentralblatt MATH identifier
1180.31009

#### Citation

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

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