Some weak geometric inequalities for the Riesz potential

In the present paper, we prove that the first eigenvalue of the Riesz potential is weakly maximised in a quasi-ball among all Haar measurable sets on homogeneous Lie groups. It is an analogue of the classical Rayleigh–Faber–Krahn inequality for the Riesz potential. We also prove a weak version of the Hong–Krahn–Szegö inequality for the Riesz potential on homogeneous Lie groups.