# Sharp stability in hypercontractivity estimates and logarithmic Sobolev inequalities

## TL;DR

This paper establishes sharp stability results for hypercontractivity estimates and logarithmic Sobolev inequalities in Euclidean and Gaussian settings, using recent stability results for the Prékopa--Leindler inequality.

## Contribution

It introduces new stability results for hypercontractivity and logarithmic Sobolev inequalities, achieving sharpness with an optimal exponent, and applies recent inequalities to these classical functional inequalities.

## Key findings

- Stability results are sharp with an exponent of 1/2.
- Applicable to both Euclidean and Gaussian settings.
- Uses recent stability results for the Prékopa--Leindler inequality.

## Abstract

We prove stability results in hypercontractivity estimates for the Hopf--Lax semigroup in $\mathbb R^n$ and apply them to deduce stability results for the Euclidean $L^p$-logarithmic Sobolev inequality for any $p>1$. As a main tool, we use recent stability results for the Pr\'ekopa--Leindler inequality, due to B\"or\"oczky and De (2021), Figalli and Ramos (2024) and Figalli, van Hintum, and Tiba (2025). Under mild assumptions on the functions, most of our stability results turn out to be sharp, as they are reflected in the optimal exponent $1/2$ both in the hypercontractivity and $L^p$-logarithmic Sobolev deficits, respectively. This approach also works for establishing stability of Gaussian hypercontractivity estimates and Gaussian logarithmic Sobolev inequality, respectively.