Revised logarithmic Sobolev inequalities of fractional order

In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on Rn with an explicit expression for the constant. Namely, we show that for all 0<s<n2 and a>0 we have the inequality integral(n)(R)|f(x)|(2)log (|f(x)|(2)||f|| (2)L2(R-n))dx+n/s(1+log a)||f||L-2(R-n)<= C(n,s,a)||(-Delta)(s/2)f|| L-2(2)(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 that was computed in [13]. Moreover, we prove a similar type inequality for functions f is an element of L-q(Rn)boolean AND W1,p(R-n) whenever 1<p<n and p<q <= p(n-1)/n-p. (c) 2024 Elsevier rights are reserved, including those for text and data mining, Al training, and similar technologies.