Characterizations of Commutators of Singular Integral Operators on Variable Exponent Spaces 
Received:September 19, 2019 Revised:March 17, 2020 
Key Word:
commutator Lipschitz space TriebelLizorkin space variable exponent singular integral operator

Fund ProjectL:Supported by the Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant Nos.2019D01C334; 2016D01C381) and the National Natural Science Foundation of China (Grant No.11661075). 

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Abstract: 
The main purpose of this paper is to characterize the Lipschitz space by the boundedness of commutators on Lebesgue spaces and TriebelLizorkin spaces with variable exponent. Based on this main purpose, we first characterize the TriebelLizorkin spaces with variable exponent by two families of operators. Immediately after, applying the characterizations of TriebelLizorkin space with variable exponent, we obtain that $b\in\dot{\Lambda}_{\beta}$ if and only if the commutator of Calder\'{o}nZygmund singular integral operator is bounded, respectively, from $L^{p(\cdot)}(\mathbb{R}^{n})$ to $\dot{F}^{\beta,\infty}_{p(\cdot)},$ from $L^{p(\cdot)}(\mathbb{R}^{n})$ to $L^{q(\cdot)}(\mathbb{R}^{n})$ with $1/p(\cdot)1/q(\cdot)=\beta/n.$ Moreover, we prove that the commutator of Riesz potential operator also has corresponding results. 
Citation: 
DOI:10.3770/j.issn:20952651.2020.05.008 
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