# A Quantitative Stability Theorem for Convolution on the Heisenberg Group

## TL;DR

This paper establishes a quantitative stability result for convolution on the Heisenberg group, showing near-maximizers are close to Gaussian triples, extending understanding of convolution inequalities in non-commutative settings.

## Contribution

It provides a quantitative stability theorem for convolution on the Heisenberg group, characterizing near-maximizers as close to Gaussian triples, which was previously only qualitatively understood.

## Key findings

- Near-maximizers are close to Gaussian triples after symmetry adjustments.
- The expansion method effectively quantifies the stability of convolution inequalities.
- The result parallels known Euclidean space results in a non-commutative setting.

## Abstract

Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this paper, we use the expansion method to prove a quantitative version of this characterization.