奇异积分算子交换子在变指数空间上的特征

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 TriebelLizorkin space with variable exponent, we obtain that b ∈■β if and only if the commutator of Calderón-Zygmund singular integral operator is bounded, respectively, from■ to■,from■ to■ with■. Moreover, we prove that the commutator of Riesz potential operator also has corresponding results.

Characterizations of Commutators of Singular Integral Operators on Variable Exponent Spaces

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.