A quantitative stability theorem for convolution on the Heisenberg group

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