New Characterizations of Musielak-Orlicz-Sobolev Spaces via Sharp Ball Averaging Functions

In this article, the authors establish a new characterization of the Musielak–Orlicz–Sobolev space on ${\mathbb{R}}^n$, which includes the classical Orlicz–Sobolev space, the weighted Sobolev space and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained by P. H"ast"o and A. M. Ribeiro [Commun. Contemp. Math. 19 (2017), 1650022, 13 pp] via weakening the assumption $f\in L^1({\mathbb{R}}^n)$ into $f\in L_{$\mathrm{l}\mathrm{o}\mathrm{c}$}^1(\mathbb R^n)$, which was conjectured to be true by H"ast"o and Ribeiro in the aforementioned same article.