Extension Theorem and Bourgain--Brezis--Mironescu-Type Characterization of Ball Banach Sobolev Spaces on Domains

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded $(\epsilon ,\mathrm{\infty })$-domain with $\epsilon \in (0,1]$, $X({\mathbb{R}}^n)$ a ball Banach function space satisfying some extra mild assumptions, and $\{{\rho }_{\nu }{\}}_{\nu \in (0,{\nu }_0)}$ with ${\nu }_0\in (0,\mathrm{\infty })$ a ${\nu }_0$-radial decreasing approximation of the identity on ${\mathbb{R}}^n$. In this article, the authors establish two extension theorems, respectively, on the inhomogeneous ball Banach Sobolev space $W^{m,X}(\mathrm{\Omega })$ and the homogeneous ball Banach Sobolev space ${\dot{W}}^{m,X}(\mathrm{\Omega })$ for any $m\in \mathbb{N}$. On the other hand, the authors prove that, for any $f\in {\dot{W}}^{1,X}(\mathrm{\Omega })$, $$\underset{\nu \to 0^{+}}{\lim }{\parallel {[{\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }f(\cdot )-f(y){\mathrm{\mid }}^p}{\mathrm{\mid }\cdot -y{\mathrm{\mid }}^p}{\rho }_{\nu }(\mathrm{\mid }\cdot -y\mathrm{\mid })dy]}^{\frac{1}{p}}\parallel }_{X(\mathrm{\Omega })}^p=\frac{2{\pi }^{\frac{n-1}{2}}\mathrm{\Gamma }(\frac{p+1}{2})}{\mathrm{\Gamma }(\frac{p+n}{2})}{\parallel \mid \mathrm{\nabla }f\mid \parallel }_{X(\mathrm{\Omega })}^p,$$ where $\mathrm{\Gamma }$ is the Gamma function and $p\in [1,\mathrm{\infty })$ is related to $X({\mathbb{R}}^n)$. Using this asymptotics, the authors further establish a characterization of $W^{1,X}(\mathrm{\Omega })$ in terms of the above limit. To achieve these, the authors develop a machinery via using a method of the extrapolation, two extension theorems on weighted Sobolev spaces, 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 that $X({\mathbb{R}}^n)$ might be neither the reflection invariance nor the translation invariance. 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, all of which are new.