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Any pair of functions that nearly realizes equality in Young's convolution inequality with sharp constant, for Euclidean groups, is close in norm to a pair that realizes equality.
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Michael Christ, Near-extremizers of Young’s inequality for Euclidean groups. Rev. Mat. Iberoam. 35 (2019), no. 7, pp. 1925–1972DOI 10.4171/RMI/1055