A Sharp analog of Young's Inequality on $S^N$ and Related Entropy   Inequalities
  We prove a sharp analog of Young's inequality on $S^N$, 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 $R^N$ to more than three functions, and leads to significant new information about the optimizers and the constants. 
Eric Carlen, Elliott Lieb, Michael Loss