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7 - Symmetrization on Spheres, and Hyperbolic and Gauss Spaces

Published online by Cambridge University Press:  22 February 2019

Albert Baernstein II
Albert Baernstein II
Affiliation:
Washington University, St Louis
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Summary

Chapter 7 covers symmetrization in the sphere,hyperbolic space, and Gauss space, and includes as an application a landmark theorem of Gehring on quasiconformal mappings. Spheres and hyperbolic spaces have a canonical distance and measure, and possess rich isometry groups of measure preserving mappings. There are plenty of hyperplanes in which to polarize, and so most of the theoryfrom Chapters 2 and 6 can be extended.Sphericaland hyperbolic analogs of inequalities from Chapters 1 and 2 are developed., including the basic polarization inequalityand the foundational inequality for integrals of functions on the sphere under symmetric decreasing rearrangement.We also find a discussion on (k,n)-caps symmetrization.

Keywords

Spherical symmetrizationHyperbolic space symmetrizationGauss space symmetrizationquasiconcformal mappingsspherical polarizationhyperbolic space polarizationDirichlet integrals
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Chapter
Information
Symmetrization in Analysis , pp. 216 - 253
Publisher: Cambridge University Press
Print publication year: 2019

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