Brezis–Seeger–Van Schaftingen–Yung-type characterization of homogeneous ball Banach Sobolev spaces and its applications

Abstract

Let [Formula: see text] and [Formula: see text] be a ball Banach function space satisfying some extra mild assumptions. Assume that [Formula: see text] or [Formula: see text] is an [Formula: see text]-domain for some [Formula: see text]. In this paper, the authors prove that a function [Formula: see text] belongs to the homogeneous ball Banach Sobolev space [Formula: see text] if and only if [Formula: see text] and [Formula: see text] where [Formula: see text] is related to [Formula: see text]. 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 [Formula: see text] with [Formula: see text], while it is still new even when [Formula: see text] with [Formula: see text]. The novelty of this paper exists in that, to establish the characterization of [Formula: see text], the authors provide a machinery via using the generalized Brezis–Seeger–Van Schaftingen–Yung formula on [Formula: see text], the extension theorem on [Formula: see text], the Bourgain–Brezis–Mironescu-type characterization of the inhomogeneous ball Banach Sobolev space [Formula: see text], and the method of extrapolation to overcome those difficulties caused by that [Formula: see text] might be neither the rotation invariance nor the translation invariance and that the norm of [Formula: see text] has no explicit expression.