Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality

TL;DR

This paper establishes deficit estimates for key functional inequalities like hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality, under semi-log conditions, using flow monotonicity and optimal transport methods.

Contribution

It introduces a unified framework for deriving deficit estimates for several inequalities under semi-log assumptions, extending and improving recent results.

Findings

Deficit estimates for hypercontractivity using flow monotonicity.

Complementary and improved deficit estimates for the logarithmic Sobolev inequality.

Enhanced deficit bounds for Talagrand's transportation cost inequality.

Abstract

We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or semi-log-concave. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Similarly, our result on Talagrand's transportation cost inequality complements and, for a large class of semi-log-concave inputs, improves a deficit estimate recently proved by Mikulincer. Our deficit estimates for hypercontractivity will be obtained by using a flow monotonicity scheme built on the Fokker--Planck equation, and our deficit estimates for the logarithmic Sobolev inequality will be derived as a corollary. For Talagrand's inequality, we use an optimal transportation argument. An appealing feature of our framework is robustness and this allows us to derive deficit estimates for the hypercontracivity inequality associated with the Hamilton--Jacobi equation, the Poincar\'e inequality, and for Beckner's inequality.