Some remarks on a formula for Sobolev norms due to Brezis, Van Schaftingen and Yung

TL;DR

This paper investigates the behavior of Sobolev semi-norms at the critical case s=1, extending previous results to more general domains and non-smooth functions, clarifying questions raised in recent research.

Contribution

It provides new insights into the Gagliardo semi-norm at s=1 for weak L^q spaces, generalizing prior results to broader settings and less regular functions.

Findings

Extended formulas for Sobolev semi-norms at s=1

Generalized results to non-smooth functions

Addressed domain generalization issues

Abstract

We provide answers to some questions raised in a recent work by H. Brezis, J. Van Schaftingen and Po-Lam Yung concerning the Gagliardo semi-norm $|u|_{W^{s,q}}$ computed at $s = 1$, when the strong $L^q$ is replaced by weak $L^q$. In particular, we address generalization of their results for a general domain and non-smooth functions.