Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

Abstract

Abstract Let 0<α<n $0<\alpha<n$ and Mα $M_{\alpha}$ be the fractional maximal function. The nonlinear commutator of Mα $M_{\alpha}$ and a locally integrable function b is given by [b,Mα](f)=bMα(f)−Mα(bf) $[b,M_{\alpha}](f)=bM_{\alpha}(f)-M_{\alpha}(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of [b,Mα] $[b,M_{\alpha}]$ on variable Lebesgue spaces when b belongs to Lipschitz or BMO(Rn) $\mathit{BMO}({\mathbb{R}}^{n})$ spaces, by which some new characterizations for certain subclasses of Lipschitz and BMO(Rn) $\mathit{BMO}({\mathbb{R}}^{n})$ spaces are obtained.

Keywords

• Fractional maximal function
• Nonlinear commutator
• Variable Lebesgue space
• Lipschitz space
• BMO space