New characterizations of Musielak-Orlicz-Sobolev spaces via sharp ball averaging functions

We establish a new characterization of the Musielak-Orlicz-Sobolev space on ?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?st? and A. M. Ribeiro [Commun. Contemp. Math., 2017, 19: 1650022] via weakening the assumption f ∈ L^1(R^n) into f ∈ L^1loc(R^n), which was conjectured to be true by Hosto and Ribeiro in the aforementioned same article.