Some remarks on weighted logarithmic Sobolev inequality Patrick CattiauxArnaud GuillinLi-Ming Wu 26D1047D0760G1060J60Lyapunov functionsTalagrand transportation information inequalitylogarithmic Sobolev inequality We give here a simple proof of weighted logarithmic Sobolev inequality, for example for Cauchy type measures, with optimal weight, sharpening results of Bobkov-Ledoux [S.G. Bobkov and M. Ledoux, \textit{Weighted Poincar\'e-type inequalities for Cauchy and other convex measures}, Ann. Probab. \textbf{37} (2009), no. 2, 403--427]. Some consequences are also discussed. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4405 10.1512/iumj.2011.60.4405 en Indiana Univ. Math. J. 60 (2011) 1885 - 1904 state-of-the-art mathematics http://iumj.org/access/