Inequalities of Riesz-Sobolev type for compact connected abelian groups

Inequalities of Riesz-Sobolev type for compact connected abelian groups

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...

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Veröffentlicht in:American journal of mathematics 2022-10, Vol.144 (5), p.1367-1435
Hauptverfasser: Christ, Michael, Iliopoulou, Marina
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Group theory
Stability
Theorems
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Iliopoulou, Marina
description 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.
doi_str_mv 10.1353/ajm.2022.0032
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subjects Group theory
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Theorems
title Inequalities of Riesz-Sobolev type for compact connected abelian groups
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