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

Let \(\{\rho _\nu \}_{\nu \in (0,\nu _0)}\) with \(\nu _0\in (0,\infty )\) be a \(\nu _0\)-radial decreasing approximation of the identity on \(\mathbb {R}^n\), \(X(\mathbb {R}^n)\) a ball Banach function space satisfying some extra mild assumptions, and \(\Omega \subset \mathbb {R}^n\) a \(W^{1,X}\)-extension domain. In this article, the authors prove that, for any f belonging to the inhomogeneous ball Banach Sobolev space \({W}^{1,X}(\Omega )\),

$$\begin{aligned} \lim _{\nu \rightarrow 0^+} \left\| \left[ \int _\Omega \frac{|f(\cdot )-f(y)|^p}{ |\cdot -y|^p}\rho _\nu (|\cdot -y|)\,\textrm{d}y \right] ^\frac{1}{p}\right\| _{X(\Omega )}^p =\frac{2\pi ^{\frac{n-1}{2}}\Gamma (\frac{p+1}{2})}{\Gamma (\frac{p+n}{2})} \left\| \,\left| \nabla f\right| \,\right\| _{X(\Omega )}^p, \end{aligned}$$

where \(\Gamma \) is the Gamma function and \(p\in [1,\infty )\) is related to \(X(\mathbb {R}^n)\). Using this asymptotics, the authors further establish a characterization of \(W^{1,X}(\Omega )\) 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 \(W^{1,X}(\mathbb {R}^n)\) to overcome those difficulties caused by that the norm of \(X(\mathbb {R}^n)\) has no explicit expression and \(X(\mathbb {R}^n)\) 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 \(X(\Omega ):=L^p(\Omega )\) with \(p\in (1,\infty )\), 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 \(X(\Omega ):=L^q(\Omega )\) with both \(q\in (1,\infty )\) and \(p\in [1,q)\cup (q,\frac{n}{n-1}]\). In addition, the authors give several specific examples of \(W^{1,X}\)-extension domains as well as \(\dot{W}^{1,X}\)-extension domains.