A note on P\'{o}lya-Szeg\"{o} inequality for fractional Orlicz-Sobolev seminorm in domains

TL;DR

This paper investigates how symmetric radial decreasing rearrangement affects the fractional Orlicz-Sobolev seminorm in domains, showing it can increase the seminorm and extending previous results to more general fractional settings.

Contribution

It extends the Pólya-Szegö inequality for fractional Orlicz-Sobolev seminorms to broader domains and more general functions beyond power-type behaviors.

Findings

Rearrangement can increase the fractional Orlicz-Sobolev seminorm.

Extension of Li-Wang's result to fractional seminorms in general domains.

Broader applicability to Orlicz functions beyond power functions.

Abstract

In this paper, we study the effect of symmetric radial decreasing rearrangement on fractional Orlicz-Sobolev seminorm in domains. Roughly speaking, we prove that symmetric radial decreasing rearrangement can increase the fractional Orlicz-Sobolev seminorm in domains. Our result extends that of Li-Wang [Commun. Contemp. Math. 21.07 (2019): 1850059.] to the setting of fractional seminorm in domains admitting behaviors more general than powers.