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

Abstract

We establish a new characterization of the Musielak–Orlicz–Sobolev space on {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. Hästö and A. M. Ribeiro [Commun. Contemp. Math., 2017, 19: 1650022] via weakening the assumption f\in L^{\mathrm{1}}({ℝ}^n) into f\in L_{}^{\mathrm{1}}({ℝ}^n), which was conjectured to be true by Hästö and Ribeiro in the aforementioned same article.

Key words： Musielak–Orlicz–Sobolev space; Orlicz–Sobolev space; variable exponent Sobolev space; sharp ball averaging function

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