A Sharp analog of Young's Inequality on $S^N$ and Related Entropy Inequalities

TL;DR

This paper establishes a sharp version of Young's inequality on the sphere $S^N$, deriving related entropy inequalities through a nonlinear heat flow approach that identifies optimizers and constants.

Contribution

It introduces a novel nonlinear heat flow method to prove sharp inequalities on $S^N$ and extends these results to multiple functions on $R^N$, revealing new optimizer insights.

Findings

Proved a sharp Young's inequality on $S^N$.

Derived new entropy inequalities from the main result.

Extended the inequality to multiple functions on $R^N$.

Abstract

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.