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

Abstract

Let $γ\in\mathbb{R}\setminus\{0\}$ and $X(\mathbb{R}^n)$ be a ball Banach function space satisfying some extra mild assumptions. Assume that $Ω=\mathbb{R}^n$ or $Ω\subset\mathbb{R}^n$ is an $(\varepsilon,\infty)$-domain for some $\varepsilon\in(0,1]$. In this article, the authors prove that a function $f$ belongs to the homogeneous ball Banach Sobolev space $\dot{W}^{1,X}(Ω)$ if and only if $f\in L_{\mathrm{loc}}^1(Ω)$ and $$ \sup_{λ\in(0,\infty)}λ\left\|\left[\int_{\{y\inΩ:\ |f(\cdot)-f(y)|>λ|\cdot-y|^{1+\fracγ{p}}\}} \left|\cdot-y\right|^{γ-n}\,dy \right]^\frac{1}{p}\right\|_{X(Ω)}<\infty, $$ 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 $X(Ω):=L^q(\mathbb{R}^n)$ with $1<p=q<\infty$, while it is still new even when $X(Ω):=L^q(\mathbb{R}^n)$ with $1\leq p<q<\infty$.