Characterizations of Commutators of Singular Integral Operators on Variable Exponent Spaces

DOI：10.3770/j.issn:2095-2651.2020.05.008

 作者 单位 房成龙 伊犁师范大学数学与统计学院, 新疆 伊宁 835000

这篇文章通过算子交换子在变指数Lebesgue空间, 变指数Triebel-Lizorkin空间的有界性去刻画了Lipschitz空间. 首先, 作者通过两个算子族刻画了Triebel-Lizorkin空间. 接着, 应用Triebel-Lizorkin空间等价刻画, 获得了b是Lipschitz函数的充要条件是奇异积分算子交换子从变指数Lebesgue空间到变指数Triebel-Lizorkin空间有界. 同时, 作者证明了Riesz位势算子交换子也有对应结果.

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.