# A P\'olya-Szeg\"o principle for general fractional Orlicz-Sobolev spaces

## TL;DR

This paper establishes Pólya-Szegö inequalities in fractional Orlicz-Sobolev spaces using polarization, providing a unified framework and deriving eigenvalue inequalities for nonlocal operators.

## Contribution

It introduces a general framework for fractional Orlicz-Sobolev spaces, proving inequalities and properties that unify various definitions in the literature.

## Key findings

- Proved Pólya-Szegö inequalities in fractional Orlicz-Sobolev spaces.
- Established density of smooth functions in these spaces.
- Derived Rayleigh-Faber-Krahn inequality for nonlocal eigenvalues.

## Abstract

In this article we prove modular and norm P\'olya-Szeg\"o inequalities in general fractional Orlicz-Sobolev spaces by using the polarization technique. We introduce a general framework which includes the different definitions of theses spaces in the literature, and we establish some of its basic properties such as the density of smooth functions. As a corollary we prove a Rayleigh-Faber-Krahn type inequality for Dirichlet eigenvalues under nonlocal nonstandard growth operators.