Characterizations of commutators of singular integral operators on variable exponent spaces 
Received:September 19, 2019 Revised:February 26, 2020 
Key Word:
commutator, Lipschitz space TriebelLizorkin space variable exponent singular integral operator

Fund ProjectL:The research was supported by National Natural Science Foundation of China (Grant No.11661075);~Natural Science Foundation of Xinjiang Uygur Autonomous Region (2016D01C381);~Natural Science Foundation of Xinjiang Uygur Autonomous Region (2019D01C334) 

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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, apply 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. 
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