Bourgain–Brezis–Mironescu-Type Characterization of Inhomogeneous Ball Banach Sobolev Spaces on Extension Domains

Let \(\{\rho _\nu \}_{\nu \in (0,\nu _0)}\) ${\{{\rho }_{\nu }\}}_{\nu \in (0,{\nu }_0)}$ with \(\nu _0\in (0,\infty )\) ${\nu }_0\in (0,\infty )$ be a \(\nu _0\) ${\nu }_0$ -radial decreasing approximation of the identity on \(\mathbb {R}^n\) ${\mathbb{R}}^n$ , \(X(\mathbb {R}^n)\) $X({\mathbb{R}}^n)$ a ball Banach function space satisfying some extra mild assumptions, and \(\Omega \subset \mathbb {R}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ a \(W^{1,X}\) $W^{1,X}$ -extension domain. In this article, the authors prove that, for any f belonging to the inhomogeneous ball Banach Sobolev space \({W}^{1,X}(\Omega )\) $W^{1,X}(\mathrm{\Omega })$ , \(\begin{aligned} \lim _{\nu \rightarrow 0^+} \left\| \left[ \int _\Omega \frac{|f(\cdot )-f(y)|^p}{ |\cdot -y|^p}\rho _\nu (|\cdot -y|)\,\textrm{d}y \right] ^\frac{1}{p}\right\| _{X(\Omega )}^p =\frac{2\pi ^{\frac{n-1}{2}}\Gamma (\frac{p+1}{2})}{\Gamma (\frac{p+n}{2})} \left\| \,\left| \nabla f\right| \,\right\| _{X(\Omega )}^p, \end{aligned}\) $$\begin{array}{c}\hfill \underset{\nu \to 0^{+}}{lim}{∥{\left[{\int }_{\mathrm{\Omega }},\frac{{|f(·)-f(y)|}^p}{{|·-y|}^p},{\rho }_{\nu },(|,·,-,y|)\text{d}y\right]}^{\frac{1}{p}}∥}_{X(\mathrm{\Omega })}^p=\frac{2{\pi }^{\frac{n-1}{2}}\mathrm{\Gamma }(\frac{p+1}{2})}{\mathrm{\Gamma }(\frac{p+n}{2})}{∥,\left|\mathrm{\nabla },f\right|,∥}_{X(\mathrm{\Omega })}^p,\end{array}$$ where \(\Gamma \) $\mathrm{\Gamma }$ is the Gamma function and \(p\in [1,\infty )\) $p\in [1,\infty )$ is related to \(X(\mathbb {R}^n)\) $X({\mathbb{R}}^n)$ . Using this asymptotics, the authors further establish a characterization of \(W^{1,X}(\Omega )\) $W^{1,X}(\mathrm{\Omega })$ in terms of the above limit. To achieve these, the authors develop a machinery via using a method of the extrapolation and some recently found profound properties of \(W^{1,X}(\mathbb {R}^n)\) $W^{1,X}({\mathbb{R}}^n)$ to overcome those difficulties caused by that the norm of \(X(\mathbb {R}^n)\) $X({\mathbb{R}}^n)$ has no explicit expression and \(X(\mathbb {R}^n)\) $X({\mathbb{R}}^n)$ might not be translation invariant. This characterization has a wide range of generality and can be applied to various Sobolev-type spaces, such as Morrey [Bourgain–Morrey-type, weighted (or mixed-norm or variable), local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, which are all new. Particularly, when \(X(\Omega ):=L^p(\Omega )\) $X(\mathrm{\Omega }):=L^p(\mathrm{\Omega })$ with \(p\in (1,\infty )\) $p\in (1,\infty )$ , this characterization coincides with the celebrated results of J. Bourgain, H. Brezis, and P. Mironescu in 2001 and H. Brezis in 2002; moreover, this characterization is also new even when \(X(\Omega ):=L^q(\Omega )\) $X(\mathrm{\Omega }):=L^q(\mathrm{\Omega })$ with both \(q\in (1,\infty )\) $q\in (1,\infty )$ and \(p\in [1,q)\cup (q,\frac{n}{n-1}]\) $p\in [1,q)\cup (q,\frac{n}{n-1}]$ . In addition, the authors give several specific examples of \(W^{1,X}\) $W^{1,X}$ -extension domains as well as \(\dot{W}^{1,X}\) ${\dot{W}}^{1,X}$ -extension domains.