Brezis--Seeger--Van Schaftingen--Yung-Type Characterization of Homogeneous Ball Banach Sobolev Spaces and Its Applications

Let $\gamma \in \mathbb{R}\setminus \{0\}$ and $X({\mathbb{R}}^n)$ be a ball Banach function space satisfying some extra mild assumptions. Assume that $\mathrm{\Omega }={\mathbb{R}}^n$ or $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is an $(\epsilon ,\mathrm{\infty })$-domain for some $\epsilon \in (0,1]$. In this article, the authors prove that a function $f$ belongs to the homogeneous ball Banach Sobolev space ${\dot{W}}^{1,X}(\mathrm{\Omega })$ if and only if $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ and $$\underset{\lambda \in (0,\mathrm{\infty })}{\sup }\lambda {\parallel {\left[{\int }_{\{y\in \mathrm{\Omega }:\mathrm{\mid }f(\cdot )-f(y)\mathrm{\mid }>\lambda \mathrm{\mid }\cdot -y{\mathrm{\mid }}^{1+\frac{\gamma }{p}}\}}{\mid \cdot -y\mid }^{\gamma -n}dy\right]}^{\frac{1}{p}}\parallel }_{X(\mathrm{\Omega })}<\mathrm{\infty },$$ where $p\in [1,\mathrm{\infty })$ is related to $X({\mathbb{R}}^n)$. This result is of wide generality and can be applied to various specific Sobolev-type function spaces, including Morrey [Bourgain–Morrey-type, weighted (or mixed-norm or variable) Lebesgue, local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, which is new even in all these special cases; in particular, it coincides with the well-known result of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung when $X(\mathrm{\Omega }):=L^q({\mathbb{R}}^n)$ with $1<p=q<\mathrm{\infty }$, while it is still new even when $X(\mathrm{\Omega }):=L^q({\mathbb{R}}^n)$ with $1\le p<q<\mathrm{\infty }$.