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

RESUMO

Let $N\in\mathbb N$ and $\gamma\in[-1,0)$. In [Anal. PDE 17 (2024)], Brezis, Seeger, Van~Schaftingen, and Yung asked how, in the exceptional range $\gamma\in[-1,0)$, $\dot{\mathrm{BV}}(\gamma)$ and $\dot W^{1,1}(\gamma)$ on ${\mathbb R}^N$ are related to other function spaces, especially to Hardy--Sobolev spaces, and whether these spaces are normable. In this article, we prove that the homogeneous Hardy--Sobolev space is strictly embedded, respectively, into $\dot{\mathrm{BV}}(\gamma)$ and $\dot W^{1,1}(\gamma)$, and neither $\dot{W}^{1,1}(\gamma)/\mathbb R$ nor $\dot{BV}(\gamma)/\mathbb R$ is normable, which answers the above two questions.