Revised logarithmic Sobolev inequalities of fractional order

Abstract

In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on $R^n$ 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$\underset{R^n}{\int }\left|f\right(x){|}^2\log \left(\frac{\left|f\right(x){|}^2}{{\Vert f\Vert }_{L^2\left(R^n\right)}^2}\right)dx+\frac{n}{s}(1+\log a){\Vert f\Vert }_{L^2\left(R^n\right)}^2\le C(n,s,a){\Vert {(-\Delta )}^{s/2}f\Vert }_{L^2\left(R^n\right)}^2$ 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 that was computed in [13]. Moreover, we prove a similar type inequality for functions $f\in L^q\left(R^n\right)\cap W^{1,p}\left(R^n\right)$ whenever $1<p<n$ and $p<q\le \frac{p(n-1)}{n-p}$.