On Two Questions by Brezis et al Concerning the Critical Difference Quotient Characterization of First-Order Sobolev Spaces

Let $N\in\mathbb N$ and $γ\in[-1,0)$. In [Anal. PDE 17 (2024)], Brezis, Seeger, Van~Schaftingen, and Yung asked how, in the exceptional range $γ\in[-1,0)$, $\dot{\mathrm{BV}}(γ)$ and $\dot W^{1,1}(γ)$ on ${\mathbb R}^N$ are related to other function spaces, especially to Hardy--Sobolev spaces, and w