Inequalities of Riesz-Sobolev type for compact connected Abelian groups

TL;DR

This paper extends the Riesz-Sobolev convolution inequality to compact connected Abelian groups, characterizes maximizers, and proves stability results, including for the circle group, with implications for sumset measure minimization.

Contribution

It formulates and proves a generalized Riesz-Sobolev inequality for Abelian groups, characterizes maximizers, and establishes stability theorems, including a continuous deformation approach for the circle group.

Findings

Riesz-Sobolev inequality is valid for all compact connected Abelian groups.

Maximizers of the inequality are characterized.

A quantitative stability theorem is established.

Abstract

A version 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 $\mathbb{R}/\mathbb{Z}$, a continuous deformation of sets is developed, under which an appropriately scaled Riesz-Sobolev functional is shown to be nondecreasing.