Although the convolution operators on Euclidean space and the Heisenberg group satisfy the same 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.
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Kevin O'Neill, A quantitative stability theorem for convolution on the Heisenberg group. Rev. Mat. Iberoam. 37 (2021), no. 5, pp. 1861–1884DOI 10.4171/RMI/1250