Search

In this paper, we prove the equivalence between logarithmic Sobolev inequality and hypercontractivity of a class of quantum Markov semigroup and its associated Dirichlet form based on a probability gage space.

Summary

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.

Search Results

Refine search

Refine search

Actions for selected content:

2 results

Equivalence between logarithmic Sobolev inequality and hypercontractivity in a probability gage space

8 - Convolution and Beyond

Summary

Search Results

Refine search

Refine search

Actions for selected content:

Save Search

2 results

Equivalence between logarithmic Sobolev inequality and hypercontractivity in a probability gage space

8 - Convolution and Beyond

Summary