Commutators of Some Maximal Functions with Lipschitz Function on Orlicz Spaces

  • Pu ZhangEmail author
  • Jianglong Wu
  • Jie Sun


Let \(0\le \alpha <n\), \([b,M_{\alpha }]\) and \([b,M^{\sharp }]\) be the nonlinear commutators of the fractional maximal function \(M_{\alpha }\) and the sharp maximal function \(M^{\sharp }\) with a locally integrable function b. In this note, we give necessary and sufficient conditions for the boundedness of \([b,M_{\alpha }]\) and \([b,M^{\sharp }]\) on Orlicz spaces when the symbol b belongs to Lipschitz spaces, by which some new characterizations of non-negative Lipschitz functions (in terms of the Orlicz norm) are obtained.


Fractional maximal function sharp maximal function nonlinear commutator Lipschitz space Orlicz space 

Mathematics Subject Classification

42B25 42B20 46E30 47B47 26A16 



The authors would like to express their gratitude to the referee for valuable suggestions and comments.


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsMudanjiang Normal UniversityMudanjiangPeople’s Republic of China
  2. 2.Faculty of Information TechnologyMacau University of Science and TechnologyMacauPeople’s Republic of China

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