Symmetrization and harmonic measure
type: Item
instance of: scholarly article
title: Symmetrization and harmonic measure
zbMATH Open document ID: 1180.31009
author: Dimitrios Betsakos
published in: Illinois Journal of Mathematics
publication date: 2009-10-15T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ijm/1254403722
review text: This paper is concerned with symmetrization inequalities for \(\alpha \)-harmonic measures \((0<\alpha <2)\) associated with domains in \(\mathbb{R}^{n}\). The main results establish when equality occurs. They are formulated in terms of \(1\)-dimensional Steiner symmetrization, and the cases of equality essentially involve symmetry about a hyperplane. The proofs are carried through by reducing symmetrization results to polarization results, and then using probabilistic methods to establish the latter. In the classical case where \(\alpha =2\) equality statements had been previously been established when \(n=2\) by \textit{M. Essén} and \textit{D. F. Shea} [Ann. Acad. Sci. Fenn., Ser. A I 4, 311--340 (1979; Zbl 0392.31001)], and in all dimensions by \textit{A. Yu. Solynin} [Algebra Anal. 8, No. 6, 148--185 (1996; Zbl 0890.31004)], under additional regularity assumptions with which the author can now dispense.
reviewed by: Stephen J. Gardiner
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 60G52
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 30C85
zbMATH DE Number: 5615676
zbMATH Keywords: alpha-harmonic measure
zbMATH Keywords: Steiner symmetrization
zbMATH Keywords: polarization
zbMATH Keywords: symmetric stable process
MaRDI profile type: Publication
cites work: Integral means, univalent functions and circular symmetrization
cites work: Q4312128
cites work: Q4673516
cites work: Spherical rearrangements, subharmonic functions, and \(\ast\)-functions in \(n\)-space
cites work: A Brascamp-Lieb-Luttinger-type inequality and applications to symmetric stable processes
cites work: Q4400428
cites work: Symmetrization, symmetric stable processes, and Riesz capacities
cites work: Some properties of α-harmonic measure
cites work: Q3517457
cites work: Potential theory. An analytic and probabilistic approach to balayage
cites work: The boundary Harnack principle for the fractional Laplacian
cites work: Representation of \(a\)-harmonic functions in Lipschitz domains
cites work: Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains
cites work: An approach to symmetrization via polarization
cites work: Estimates on Green functions and Poisson kernels for symmetric stable processes
cites work: Martin boundary and integral representation for harmonic functions of symmetric stable processes
cites work: Capacities and geometric transformations of subsets in \(n\)-space
cites work: Symmetrization in the geometric theory of functions of a complex variable
cites work: Q5342182
cites work: On some questions of uniqueness in the theory of symmetrization
cites work: Q4692700
cites work: Q5666837
cites work: Q5657015
cites work: Universal approximation of symmetrizations by polarizations
cites work: Q3127994
cites work: Boundary Harnack principle for symmetric stable processes
cites work: Harmonic measures for symmetric stable processes
description: scientific article; zbMATH DE number 5615676