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

doi:10.1007/s11464-019-0744-1

Front. Math. China

2019, Vol. 14

Issue (1) : 177-201 https://doi.org/10.1007/s11464-019-0744-1

RESEARCH ARTICLE

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

Sibei YANG¹, Dachun YANG²(

), Wen YUAN²

¹. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China
². Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

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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 $f ∈ L 1 (ℝ n)$ into $f ∈ L 1 (ℝ n)$ , 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

Corresponding Author(s): Dachun YANG

Issue Date: 22 March 2019

Cite this article:

Sibei YANG,Dachun YANG,Wen YUAN. New characterizations of Musielak–Orlicz–Sobolev spaces via sharp ball averaging functions[J]. Front. Math. China, 2019, 14(1): 177-201.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-019-0744-1
https://academic.hep.com.cn/fmc/EN/Y2019/V14/I1/177

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