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Boundedness on Triebel–Lizorkin spaces for the Calderón commutator with rough kernel

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  12 March 2025

Rong Liang  and
Xiangxing Tao Open the ORCID record for Xiangxing Tao [Opens in a new window]
Rong Liang
Affiliation:
Department of Mathematics, School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, People’s Republic of China e-mail: rliang0219@163.com
Xiangxing Tao*
Affiliation:
Department of Mathematics, School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, People’s Republic of China e-mail: rliang0219@163.com
*
e-mail: xxtao@zust.edu.cn
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Abstract

In this article, the authors consider the boundedness on Triebel–Lizorkin spaces for the d-dimensional Calderón commutator defined by

$$ \begin{align*}T_{\Omega,a}f(x)=\mathrm{p.\,v.}\int_{\mathbb{R}^d}\frac{\Omega(x-y)}{|x-y|^{d+1}}\big(a(x)-a(y)\big)f(y){d}y,\end{align*} $$
where $\Omega $ is homogeneous of degree zero, integrable on $S^{d-1}$ and has a vanishing moment of order one, a is a Lipschitz function on $\mathbb {R}^d$. The authors proved that if
$$ \begin{align*}\sup_{\zeta\in S^{d-1}}\int_{S^{d-1}}|\Omega(\theta)|\log ^{\beta} \big(\frac{1}{|\theta\cdot\zeta|}\big)d\theta<\infty\end{align*} $$
with $\beta \in (1,\,\infty )$, then $T_{\Omega ,a}$ is bounded on Triebel–Lizorkin spaces $\dot {F}_{p}^{0,q}(\mathbb {R}^d)$ for $1+\frac {1}{2\beta -1}<p,q<2\beta $.

Keywords

Calderón commutatorTriebel–Lizorkin spaceLittlewood–Paley theoryCalderón reproducing formula

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 13
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The research was supported by the NNSF of China under grants #12271483, #11971295, and #11771399.

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