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The Polya-Szego principle in the fractional setting: a glimpse on nonlocal functional inequalities

Alessandro Carbotti

TL;DR

The work addresses how the Polya–Szegő rearrangement principle extends to nonlocal, fractional settings to yield sharp geometric and functional inequalities. It develops a fractional Polya–Szegő framework in $W^{s,p}(\mathbb{R}^N)$, uses heat-kernel and extension representations, and derives strong results including fractional Sobolev and isoperimetric inequalities, Faber–Krahn-type bounds, and Talenti-type comparisons. It further explores stability (quantitative) forms, and generalizations to anisotropic and Gaussian contexts via Steiner/Ehrhard symmetrizations and related energies. These findings enrich nonlocal PDE analysis, spectral theory, and geometric measure theory by providing robust tools for extremal problems, with potential applications in probability, physics, and image processing.

Abstract

In this survey we present the fractional Polya Szego principle and its main consequences in the study of nonlocal functional inequalities. In particular, we show how symmetrization methods work also in the fractional setting and yield sharp results such as isoperimetric type inequalities. Further developments including stability issues and generalizations in the anisotropic and the Gaussian setting are also discussed.

The Polya-Szego principle in the fractional setting: a glimpse on nonlocal functional inequalities

TL;DR

The work addresses how the Polya–Szegő rearrangement principle extends to nonlocal, fractional settings to yield sharp geometric and functional inequalities. It develops a fractional Polya–Szegő framework in , uses heat-kernel and extension representations, and derives strong results including fractional Sobolev and isoperimetric inequalities, Faber–Krahn-type bounds, and Talenti-type comparisons. It further explores stability (quantitative) forms, and generalizations to anisotropic and Gaussian contexts via Steiner/Ehrhard symmetrizations and related energies. These findings enrich nonlocal PDE analysis, spectral theory, and geometric measure theory by providing robust tools for extremal problems, with potential applications in probability, physics, and image processing.

Abstract

In this survey we present the fractional Polya Szego principle and its main consequences in the study of nonlocal functional inequalities. In particular, we show how symmetrization methods work also in the fractional setting and yield sharp results such as isoperimetric type inequalities. Further developments including stability issues and generalizations in the anisotropic and the Gaussian setting are also discussed.

Paper Structure

This paper contains 15 sections, 8 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. The fractional laplacian
  4. The Pólya-Szegö principle for Gagliardo fractional seminorms
  5. Some applications
  6. Fractional Sobolev inequality
  7. Fractional Isoperimetric inequality
  8. Fractional Dirichlet eigenvalues and Faber-Krahn Inequality
  9. Talenti-type comparison for solutions to fractional equations
  10. Pólya-Szegö inequalities via partial symmetrization of extension functionals
  11. Stability of some Geometric Inequalities in the Fractional Setting
  12. Some generalizations
  13. The anisotropic case
  14. The Gaussian case
  15. Final comments and Open problems

Key Result

lemma 1

Let $1\le p\le\infty$. For every non-negative $u\in L^p(\mathbb{R}^N)$ it holds that $u^*\in L^p(\mathbb{R}^N)$ with

Figures (1)

  • Figure 1: Schwarz rearrangement of a 1D function.

Theorems & Definitions (25)

  • definition 1: Schwarz symmetrization
  • definition 2: Symmetric decreasing rearrangement of a function
  • remark 5
  • lemma 1: Rearrangement preserves $L^p$ norms
  • lemma 2: Riesz Rearrangement inequality
  • definition 3: Steiner symmetrization
  • definition 4: Steiner symmetrization of a function
  • remark 6
  • theorem 2: Fractional Pólya--Szegö Inequality
  • proof
  • ...and 15 more