# Some remarks on rearrangement for nonlocal functionals

## TL;DR

This paper investigates the properties of a nonlocal functional related to the Dirichlet p-norm, showing it does not decrease under rearrangement and establishing related properties like decay, compactness, and a fractional Polya-Szeg"o inequality.

## Contribution

It demonstrates the failure of rearrangement decrease for a specific nonlocal functional and establishes new properties including a fractional Polya-Szeg"o inequality.

## Key findings

- Nonlocal functional does not decrease under two-point rearrangement.
- Established decay and compactness properties for the functional.
- Proved a Polya-Szeg"o inequality for Riesz fractional gradients.

## Abstract

We prove that a nonlocal functional approximating the standard Dirichlet $p$-norm fails to decrease under two-point rearrangement. Furthermore, we get other properties related to this functional such as decay and compactness, and the Polya-Szeg\"o inequality for Riesz fractional gradients, a notion recently introduced in the literature.