Let γ∈ℝ∖{0} and X(ℝn) be a ball Banach function space satisfying some extra mild assumptions. Assume that Ω=ℝn or Ω⊂ℝn is an (𝜀,∞)-domain for some 𝜀∈(0,1]. In this paper, the authors prove that a function f belongs to the homogeneous ball Banach Sobolev space Ẇ1,X(Ω) if and only if f∈L1loc(Ω) and
where
p∈[1,∞) is related to
X(ℝn). 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):=Lq(ℝn) with
1<p=q<∞, while it is still new even when
X(Ω):=Lq(ℝn) with
1≤p<q<∞. The novelty of this paper exists in that, to establish the characterization of
Ẇ1,X(Ω), the authors provide a machinery via using the generalized Brezis–Seeger–Van Schaftingen–Yung formula on
X(ℝn), the extension theorem on
Ẇ1,X(Ω), the Bourgain–Brezis–Mironescu-type characterization of the inhomogeneous ball Banach Sobolev space
W1,X(Ω), and the method of extrapolation to overcome those difficulties caused by that
X(ℝn) might be neither the rotation invariance nor the translation invariance and that the norm of
X(ℝn) has no explicit expression.
