2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72089We 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.Functional AnalysisProbability43A15; 52A40; 82C40text