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arXiv · 2409.14182

Fine Pólya-Szegő rearrangement inequalities in metric spaces and applications

Francesco Nobili·Ivan Yuri Violo
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Abstract

We study fine Pólya-Szegő 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 Pólya-Szegő 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.

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Francesco Nobili, Ivan Yuri Violo. 2025-10-13. Fine Pólya-Szegő rearrangement inequalities in metric spaces and applications. https://arxiv.org/abs/2409.14182

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