Fine P\'olya-Szeg\H{o} rearrangement inequalities in metric spaces and applications

TL;DR

This paper develops advanced rearrangement inequalities in metric spaces with applications to geometric and functional inequalities, especially under Ricci curvature bounds, leading to new results like a Faber-Krahn theorem and eigenvalue bounds.

Contribution

It introduces a unified framework for Pólya-Szegő inequalities in metric measure spaces with Ricci bounds and derives several new geometric and spectral inequalities.

Findings

New rearrangement inequalities in metric spaces supporting isoperimetric inequalities.

Characterization of equality cases under Ricci lower bounds.

Applications include a Faber-Krahn theorem and bounds for Neumann eigenvalues.

Abstract

We study fine P\'olya-Szeg\H{o} rearrangement inequalities into weighted intervals for Sobolev functions and functions of bounded variation defined on metric measure spaces supporting an isoperimetric inequality. We then specialize this theory to spaces with synthetic Ricci lower bounds and characterize equality cases under minimal assumptions. As applications of our theory, we show new results around geometric and functional inequalities under Ricci lower bounds answering also questions raised in the literature. Finally, we study further settings and deduce a Faber-Krahn theorem on Euclidean spaces with radial log-convex densities, a boosted P\'olya-Szeg\H{o} inequality with asymmetry reminder on weighted convex cones, the rigidity of Sobolev inequalities on Euclidean spaces outside a convex set and a general lower bound for Neumann eigenvalues on open sets in metric spaces.