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 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.