Revised logarithmic Sobolev inequalities of fractional order

TL;DR

This paper establishes revised logarithmic Sobolev inequalities involving fractional derivatives on Euclidean space, providing explicit constants and analyzing their asymptotic behavior for large dimensions, extending classical results.

Contribution

It introduces new fractional order logarithmic Sobolev inequalities with explicit constants and explores their asymptotic sharpness in high dimensions.

Findings

Explicit constant C(n,s,a) derived for fractional Sobolev inequalities.

Asymptotic behavior of constants matches known sharp constants for large n.

Extended inequalities to functions in L^q and Sobolev spaces with specific p and q ranges.

Abstract

In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on $\mathbb{R}^n$ with an explicit expression for the constant. Namely, we show that for all $0<s<\frac{n}{2}$ and $a>0$ we have the inequality \[ \int_{\mathbb{R}^n}|f(x)|^2 \log \left( \frac{|f(x)|^2}{\|f\|^{2}_{L^2(\mathbb{R}^n)}}\right)\,dx+\frac{n}{s}(1+\log a)\|f\|_{L^2(\mathbb{R}^n)}^{2}\leq C(n,s,a)\|(-\Delta)^{s/2}f\|^{2}_{L^2(\mathbb{R}^n)} \] with an explicit $C(n,s,a)$ depending on $a$, the order $s$, and the dimension $n$, and investigate the behaviour of $C(n,s,a)$ for large $n$. Notably, for large $n$ and when $s=1$, the constant $C(n,1,a)$ is asymptotically the same as the sharp constant of Lieb and Loss. Moreover, we prove a similar type inequality for functions $f \in L^q(\mathbb{R}^n)\cap W^{1,p}(\mathbb{R}^n)$ whenever $1<p<n$ and $p<q\leq \frac{p(n-1)}{n-p}$.