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

TL;DR

This paper characterizes homogeneous ball Banach Sobolev spaces using a new integral condition, extending classical results to a broad class of function spaces with diverse applications.

Contribution

It provides a general characterization of Sobolev spaces in terms of integral conditions for various ball Banach function spaces, including many special cases.

Findings

Unified integral criterion for Sobolev space membership.

Extension to Morrey, weighted, Lorentz, Orlicz, and other spaces.

New results even for classical Lebesgue spaces.

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 $(\varepsilon,\infty)$-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 $\dot{W}^{1,X}(\Omega)$ if and only if $f\in L_{\mathrm{loc}}^1(\Omega)$ and $$ \sup_{\lambda\in(0,\infty)}\lambda \left\|\left[\int_{\{y\in\Omega:\ |f(\cdot)-f(y)|>\lambda|\cdot-y|^{1+\frac{\gamma}{p}}\}} \left|\cdot-y\right|^{\gamma-n}\,dy \right]^\frac{1}{p}\right\|_{X(\Omega)}<\infty, $$ 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 $1\leq p<q<\infty$.