We prove a sharp analog of Young’s inequality on SN , and deduce from it certain sharp entropy inequalities. The proof consists of constructing a nonlinear heat flow that drives the inequality to its sharp value, while transporting the functions to the optimizers. This strategy also works for the general Young inequality on R N and yields a fairly simple

proof of the fact (due to Brascamp and Lieb) that it suffices to optimize over Gaussian functions in order to find the sharp constant.