Dimension dependent hypercontractivity for Gaussian kernels

TL;DR

This paper establishes sharp, dimension-dependent hypercontractive bounds for Gaussian kernels and diffusion semigroups, revealing refined properties and implications for classical inequalities and non-diffusive processes.

Contribution

It introduces dimension-dependent hypercontractive bounds that capture refined kernel properties, extending classical results and analyzing non-diffusive Levy-driven semigroups.

Findings

Derived sharp, local, dimension-dependent hypercontractive bounds.

Established implications for classical bounds and inequalities.

Analyzed hypercontractivity for Levy-driven Ornstein-Uhlenbeck semigroups.

Abstract

We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace estimates. They imply classical bounds on the Ornstein-Uhlenbeck semigroup and a dimensional and refined (transportation) Talagrand inequality when applied to the Hamilton-Jacobi equation. Hypercontractive bounds on the Ornstein-Uhlenbeck semigroup driven by a non-diffusive L\'evy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.