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

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.