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Fine Pólya-Szegő rearrangement inequalities in metric spaces and applications

Francesco Nobili, Ivan Yuri Violo

TL;DR

The paper develops a unified, fine-grained Pólya–Szegő rearrangement theory in general metric measure spaces using weighted decreasing rearrangements with $\omega=g\,dt$ and an isoperimetric profile $\mathcal{I}^{\flat}_g$. By coupling this framework with spaces carrying synthetic Ricci curvature lower bounds, it yields sharp energy inequalities for Sobolev and BV functions, with precise rigidity conclusions (e.g., radiality, spherical or Euclidean cones) and quantitative stability in both compact and noncompact settings. The authors derive a spectrum of geometric and functional inequalities under Ricci lower bounds, including diameter-enhanced Lichnerowicz-type estimates, Faber–Krahn-type results in Euclidean spaces with radial log-convex densities, and Sobolev/Neumann bounds in metric spaces, often accompanied by sharp equality cases and asymmetry enhancements in convex-cone geometries. These results extend classical rearrangement techniques to non-smooth spaces, offering new tools for isoperimetric, spectral, and rigidity questions in geometric analysis. The work thus provides a broad, robust bridge between rearrangement theory, isoperimetry, and functional inequalities in both abstract metric spaces and classical Euclidean settings with weighted densities.

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.

Fine Pólya-Szegő rearrangement inequalities in metric spaces and applications

TL;DR

The paper develops a unified, fine-grained Pólya–Szegő rearrangement theory in general metric measure spaces using weighted decreasing rearrangements with and an isoperimetric profile . By coupling this framework with spaces carrying synthetic Ricci curvature lower bounds, it yields sharp energy inequalities for Sobolev and BV functions, with precise rigidity conclusions (e.g., radiality, spherical or Euclidean cones) and quantitative stability in both compact and noncompact settings. The authors derive a spectrum of geometric and functional inequalities under Ricci lower bounds, including diameter-enhanced Lichnerowicz-type estimates, Faber–Krahn-type results in Euclidean spaces with radial log-convex densities, and Sobolev/Neumann bounds in metric spaces, often accompanied by sharp equality cases and asymmetry enhancements in convex-cone geometries. These results extend classical rearrangement techniques to non-smooth spaces, offering new tools for isoperimetric, spectral, and rigidity questions in geometric analysis. The work thus provides a broad, robust bridge between rearrangement theory, isoperimetry, and functional inequalities in both abstract metric spaces and classical Euclidean settings with weighted densities.

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.
Paper Structure (25 sections, 34 theorems, 241 equations)

This paper contains 25 sections, 34 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. A general Pólya-Szegő rearrangement theory
  3. The theory under Ricci lower bounds
  4. Applications to geometric and functional inequalities under Ricci lower bounds
  5. Applications to geometric and functional inequalities in the Euclidean space and beyond
  6. Preliminaries: calculus on metric measure spaces
  7. Decreasing rearrangement in metric spaces
  8. Definitions and notation
  9. Key assumptions
  10. Properties and regularization effects
  11. Pólya-Szegő inequalities: metric spaces
  12. Theory for Sobolev functions
  13. Theory for functions of bounded variation
  14. Pólya-Szegő inequalities: Ricci lower bounds
  15. Positive Ricci lower bound
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $({\rm X},{\sf d},\mathfrak m)$ be a metric measure space, $p>1$, let $\varnothing \neq \Omega\subset {\rm X}$ be open and let $u \in W^{1,p}_{loc}(\Omega)$ be such that $\mathfrak m(\{u>t\})<\infty$ for all $t > {\rm ess}\inf u$. Let $\omega=g{\rm d} t$ be a weight as above and let $u^*$ be the for some constants ${\sf C},\varepsilon>0$. Then, it holds and moreover, if the left-hand side is

Theorems & Definitions (73)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 63 more