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Frontiers of Mathematics in China

, Volume 14, Issue 1, pp 177–201 | Cite as

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

  • Sibei Yang
  • Dachun YangEmail author
  • Wen Yuan
Research Article
  • 21 Downloads

Abstract

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 fL1(ℝn) into fL1loc(ℝn), which was conjectured to be true by Hästö and Ribeiro in the aforementioned same article.

Keywords

Musielak-Orlicz-Sobolev space Orlicz-Sobolev space variable exponent Sobolev space sharp ball averaging function 

MSC

46E35 42B35 42B25 

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Notes

Acknowledgements

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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Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsGansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou UniversityLanzhouChina
  2. 2.Laboratory of Mathematics and Complex Systems (Ministry of Education of China)School of Mathematical Sciences, Beijing Normal UniversityBeijingChina

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