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Revista Matemática Iberoamericana

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Volume 37, Issue 5, 2021, pp. 1861–1884
DOI: 10.4171/rmi/1250

Published online: 2021-03-15

A quantitative stability theorem for convolution on the Heisenberg group

Kevin O'Neill[1]

(1) University of California Davis, USA

Although the convolution operators 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.

Keywords: Heisenberg group, quantitative stability, sharp constants

O'Neill Kevin: A quantitative stability theorem for convolution on the Heisenberg group. Rev. Mat. Iberoam. 37 (2021), 1861-1884. doi: 10.4171/rmi/1250