On Young's inequality for Heisenberg groups

TL;DR

This paper characterizes near-extremizers of Young's convolution inequality on Heisenberg groups, revealing that optimal constants match Euclidean space but extremizers do not exist, through analysis of approximate solutions to functional equations.

Contribution

It provides a characterization of near-extremizers for Young's inequality on Heisenberg groups and develops a new approach using approximate solutions to functional equations.

Findings

Optimal constants match Euclidean space for Heisenberg groups

No extremizing functions exist for the inequality on Heisenberg groups

Characterization of near-extremizers through approximate solutions

Abstract

Young's convolution inequality provides an upper bound for the convolution of functions in terms of $L^p$ norms. It is known that for certain groups, including Heisenberg groups, the optimal constant in this inequality is equal to that for Euclidean space of the same topological dimension, yet no extremizing functions exist. For Heisenberg groups we characterize ordered triples of functions that nearly extremize the inequality. The analysis relies on a characterization of approximate solutions of a certain class of functional equations. A result of this type is developed for a class of such equations.