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

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a bounded $(\epsilon ,\infty )$-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 ${\nu }_0\in (0,\infty )$ 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 $W^{m,X}(\Omega )$ and the homogeneous ball Banach Sobolev space ${\dot{W}}^{m,X}(\Omega )$ for any $m\in \mathbb{N}$. On the other hand, the authors prove that, for any $f\in {\dot{W}}^{1,X}(\Omega )$, $$\underset{\nu \to 0^{+}}{\lim }{\Vert {\left[{\int }_{\Omega }\frac{|f(\cdot )-f(y){|}^p}{|\cdot -y{|}^p}{\rho }_{\nu }(|\cdot -y|)dy\right]}^{\frac{1}{p}}\Vert }_{X(\Omega )}^p=\frac{2{\pi }^{\frac{n-1}{2}}\Gamma (\frac{p+1}{2})}{\Gamma (\frac{p+n}{2})}{\Vert \left|\nabla f\right|\Vert }_{X(\Omega )}^p,$$ 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, 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.

58 pages, Submitted. arXiv admin note: substantial text overlap with arXiv:2307.10528, arXiv:2304.00949