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Published online by Cambridge University Press: 22 February 2019
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.
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