On Two Questions by Brezis et al Concerning the Critical Difference Quotient Characterization of First-Order Sobolev Spaces
Year: 2026
Type: preprint
Abstract: 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 whether these spaces are normable. In this article, we prove that the homogeneous Hardy--Sobolev space is strictly embedded, respectively, into $\dot{\mathrm{BV}}(γ)$ and $\dot W^{1,1}(γ)$, and neither $\dot{W}^{1,1}(γ)/\mathbb R$ nor $\dot{BV}(γ)/\mathbb R$ is normable, which answers the above two questions.
Source: arXiv (Cornell University)
Institution: Beijing Normal University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G5710692465, G5884157002, G6737089312
OpenAlex ID: W7213526299