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Published online by Cambridge University Press: 22 February 2019
Chapter 8 studies symmetrization and convolution.The Riesz-Sobolev convolution theorem is first proved for functions in the unit circle, and then the real line, and finally in n-dimensional space. The Brunn-Minkowski inequality is proved as an application. The Brascamp-LIeb-Luttinger inequality, which extends the Riesz-Sobolev inequality to multiple integrals,is proved too. It implies that the Dirichlet heat kernel increases under symmetrization of the domain.The chapter includes a variation of the sharp Hardy-Littlewood-Sobolev inequality that implies Beckner's logarithmic Sobolev inequality. The latter result is used to establish hypercontractivity of the Poisson semigroup.
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