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.