On fractional smoothness of modulus of functions

TL;DR

This paper investigates the fractional smoothness properties of the modulus and sign operators in Sobolev spaces, providing elementary proofs of their boundedness in certain fractional Sobolev spaces.

Contribution

It offers new elementary proofs demonstrating the boundedness of Nemytskii operators related to modulus and sign functions in fractional Sobolev spaces.

Findings

Boundedness of |u| and u^{ ext{±}} in H^s() for 0  s < 3/2

Elementary proofs of these boundedness results

Clarification of fractional smoothness properties of these operators

Abstract

We consider the Nemytskii operators $u\to |u|$ and $u\to u^{\pm}$ in a bounded domain $\Omega$ with $C^2$ boundary. We give elementary proofs of the boundedness in $H^s(\Omega)$ with $0\le s<3/2$.