We establish a new characterization of the Musielak-Orlicz-Sobolev space on ℝn, 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 ∈ L1(ℝn) into f ∈ L1loc(ℝn), which was conjectured to be true by Hästö and Ribeiro in the aforementioned same article.
Musielak-Orlicz-Sobolev space Orlicz-Sobolev space variable exponent Sobolev space sharp ball averaging function
46E35 42B35 42B25
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The authors would like to thank both referees for their very careful reading and several valuable comments which indeed improve the presentation of this article. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11871254, 11571289, 11571039, 11761131002, 11671185, 11871100) and the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2018-111).
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