Dimension dependent hypercontractivity for Gaussian kernels

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 Lévy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.

24 pages

References in corpus (2)

• Subelliptic Li-Yau estimates on three dimensional model spaces
• Weighted Nash Inequalities

Cited by in corpus (7)

• $H^{\infty }$-calculus for semigroup generators on BMO
• An H1-BMO duality theory for semigroups of operators
• Information Constrained Optimal Transport: From Talagrand, to Marton, to Cover
• New Brunn--Minkowski and functional inequalities via convexity of entropy
• Generalized Talagrand Inequality for Sinkhorn Distance using Entropy Power Inequality
• Weyl calculus with respect to the Gaussian measure and restricted $L^p$-$L^q$ boundedness of the Ornstein-Uhlenbeck semigroup in complex time
• Dimension-independent estimates for heat operators and harmonic functions