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Inequalities of Riesz-Sobolev type for compact connected abelian groups
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 144, Number 5, October 2022
- pp. 1367-1435
- 10.1353/ajm.2022.0032
- Article
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abstract:
An analogue of the Riesz-Sobolev convolution inequality is formulated and proved for arbitrary compact connected Abelian groups. Maximizers are characterized, and a quantitative stability theorem is proved, under natural hypotheses. A corresponding stability theorem for sets whose sumset has nearly minimal measure is also proved, sharpening recent results of other authors. For the special case of the group $\Bbb{R}/\Bbb{Z}$, a continuous deformation of sets is developed, under which a scaled Riesz-Sobolev functional is shown to be nondecreasing.