  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
$\Omega=\mathbb{R}^n$ or $\Omega\subset\mathbb{R}^n$ is an
$(\varepsilon,\infty)$-domain for some $\varepsilon\in(0,1]$. In this article,
the authors prove that a function $f$ belongs to the homogeneous ball Banach
Sobolev space $\dot{W}^{1,X}(\Omega)$ if and only if $f\in
L_{\mathrm{loc}}^1(\Omega)$ and $$ \sup_{\lambda\in(0,\infty)}\lambda
\left\|\left[\int_{\{y\in\Omega:\
|f(\cdot)-f(y)|>\lambda|\cdot-y|^{1+\frac{\gamma}{p}}\}}
\left|\cdot-y\right|^{\gamma-n}\,dy
\right]^\frac{1}{p}\right\|_{X(\Omega)}<\infty, $$ where $p\in[1,\infty)$ 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 $X(\Omega):=L^q(\mathbb{R}^n)$ with
$1<p=q<\infty$, while it is still new even when $X(\Omega):=L^q(\mathbb{R}^n)$
with $1\leq p<q<\infty$.

Functional Analysis;Analysis of PDEs;Classical Analysis and ODEs