Revised logarithmic Sobolev inequalities of fractional order

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 $${\int }_{{\mathbb{R}}^n}\mathrm{\mid }f(x){\mathrm{\mid }}^2\log \left(\frac{\mathrm{\mid }f(x){\mathrm{\mid }}^2}{\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2({\mathbb{R}}^n)}^2}\right)dx+\frac{n}{s}(1+\log a)\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2({\mathbb{R}}^n)}^2\le C(n,s,a)\mathrm{\parallel }(-\mathrm{\Delta }{)}^{s\mathrm{/}2}f{\mathrm{\parallel }}_{L^2({\mathbb{R}}^n)}^2$$ 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. 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 $p<q\le \frac{p(n-1)}{n-p}$.