Convolution and Beyond

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.