数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1618-1632.doi: 10.1007/s10473-023-0411-1

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BOUNDEDNESS OF THE CALDERÓN COMMUTATOR WITH A ROUGH KERNEL ON TRIEBEL-LIZORKIN SPACES

Guoen HU, Jie LIU   

  1. Center of Fundamental Science, Huanghe Science and Technology University, Zhengzhou 450063, China
  • 收稿日期:2022-04-17 修回日期:2022-06-27 发布日期:2023-08-08
  • 通讯作者: †Guoen HU, E-mail: guoenxx@163.com
  • 作者简介:Jie LIU, E-mail: 57923206@qq.com
  • 基金资助:
    *The research was supported by the NNSF of China (11871108).

BOUNDEDNESS OF THE CALDERÓN COMMUTATOR WITH A ROUGH KERNEL ON TRIEBEL-LIZORKIN SPACES

Guoen HU, Jie LIU   

  1. Center of Fundamental Science, Huanghe Science and Technology University, Zhengzhou 450063, China
  • Received:2022-04-17 Revised:2022-06-27 Published:2023-08-08
  • Contact: †Guoen HU, E-mail: guoenxx@163.com
  • About author:Jie LIU, E-mail: 57923206@qq.com
  • Supported by:
    *The research was supported by the NNSF of China (11871108).

摘要: 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 $\dot{F}_{p}^{0,q}(\mathbb{R}^d)$.

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