Brezis--Seeger--Van Schaftingen--Yung-Type Characterization of Homogeneous Ball Banach Sobolev Spaces and Its Applications

Abstract

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 $(\epsilon ,\infty )$-domain for some $\epsilon \in (0,1]$. 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 $$\underset{\lambda \in (0,\infty )}{\sup }\lambda {\Vert {\left[{\int }_{\{y\in \Omega :|f(\cdot )-f(y)|>\lambda |\cdot -y{|}^{1+\frac{\gamma }{p}}\}}{\left|\cdot -y\right|}^{\gamma -n}dy\right]}^{\frac{1}{p}}\Vert }_{X(\Omega )}<\infty ,$$ 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 $1\le p<q<\infty$.

46 pages, Submitted. arXiv admin note: substantial text overlap with arXiv:2307.11392, arXiv:2304.00949