关键词: Calderón commutator, Fourier transform, Littlewood-Paley theory, Calderón reproducing formula

Abstract: In this paper, we consider the boundedness on Triebel-Lizorkin spaces for the $d$-dimensional Calderón commutator defined by $T_{\Omega,a}f(x)={\rm p.\,v.}\int_{\mathbb{R}^d}\frac{\Omega(x-y)}{|x-y|^{d+1}}\big(a(x)-a(y)\big)f(y){\rm d}y,$ where $\Omega$ is homogeneous of degree zero, integrable on $S^{d-1}$ and has a vanishing moment of order one, and $a$ is a function on $\mathbb{R}^d$ such that $\nabla a\in L^{\infty}(\mathbb{R}^d)$. We prove that if $1<p,\,q<\infty$ and $\Omega\in L(\log L)^{2\tilde{q}}(S^{d-1})$ with $\tilde{q}=\max\{1/q,\,1/q'\}$, then $T_{\Omega,a}$ is bounded on Triebel-Lizorkin spaces $\dot{F}_{p}^{0,q}(\mathbb{R}^d)$.

Key words: Calderón commutator, Fourier transform, Littlewood-Paley theory, Calderón reproducing formula