On logarithmic Sobolev inequalities for higher order fractional derivatives

On R, we prove the existence of sharp logarithmic Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. Any function f epsilon H-s(R-n) satisfies [GRAPHICS] with alpha > 0 be any number and where the operators (-Delta)(s/2) in Fourier spaces are defined by (-Delta)sl2f(k) := (2pi\k\)(s)(f) over cap (k). To cite this article: A. Cotsiolis, NX Tavoularis, C R. Acad. Sci. Paris, Ser. 1340 (2005). (C) 2004 Academie des sciences. Published by Elsevier SAS. All rights reserved.