A-Sharp-analog-of-Young’s-Inequality-on-SN-and-Related-Entropy-Inequalities.pdfVIP

A Sharp analog of Young’s Inequality on SN and Related Entropy Inequalities E. A. Carlen1 E. H. Lieb2 M. Loss1 1. School of Mathematics, Georgia Tech, Atlanta GA 30332 2. Departments of Mathematics and Physics, Jadwin Hall, Princeton University, P.O. Box 708, Princeton NJ 08544 Abstract We prove a sharp analog of Young’s inequality on SN , and deduce from it certain sharp entropy inequalities. The proof turns on constructing a nonlinear heat flow that drives trial functions to optimizers in a monotonic manner. This strategy also works for the generalization of Young’s inequality on RN to more than three functions, and leads to significant new information about the optimizers and the constants. Math reviews Classification Numbers: 43A15, 52A40, 82C40 Key words: Inqualities, entropy, optimizers, best constants 1 Work partially supported by U.S. National Science Foundation grant DMS 03-00349. 2 Work partially supported by U.S. National Science Foundation grant PHY-0139984. c 2004 by the authors. Reproduction of this article, in its entirety, by any means is permitted for non-commercial purposes. 14/june/2006; 12:24 1 1. Introduction This paper concerns further generalizations of the generalized Young’s inequality due to Brascamp and Lieb [5], which we now recall. For any N ≥ M , let a ,a , . . .,a be any set of N non zero vectors spanning IRM . Let 1 2 N f , f , . . ., f be any set on N non negative measurable functions on IR. Given numbers 1 2 N pj with 1 ≤ pj ≤ ∞ for j = 1, 2 . . ., N, form the vector p = (1/p , 1/p , . . ., 1/p ) , (1.1)