A Quantitative Stability Theorem for Convolution on the Heisenberg Group

2019/07/27 by Kevin O'Neill, O'Neill, Kevin
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA

paper · pdf · doi:10.48550/arxiv.1907.11986

22 pages

arxiv created 2019/07/27 · arxiv updated 2019/07/30

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.

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