Characterizations of Commutators of Singular Integral Operators on Variable Exponent Spaces

Abstract:

      The main purpose of this paper is to characterize the Lipschitz space by the boundedness of commutators on Lebesgue spaces and Triebel-Lizorkin spaces with variable exponent. Based on this main purpose, we first characterize the Triebel-Lizorkin spaces with variable exponent by two families of operators. Immediately after, applying the characterizations of Triebel-Lizorkin space with variable exponent, we obtain that $b\in\dot{\Lambda}_{\beta}$ if and only if the commutator of Calder\'{o}n-Zygmund 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.