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

Sibei Yang, Dachun Yang, Wen Yuan

Published 2018-09-02Version 1

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 $f\in L_{\mathrm{loc}}^1(\mathbb R^n)$, which was conjectured to be true by H\"ast\"o and Ribeiro in the aforementioned same article.