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An estimate for the composition of rough singular integral operators

Part of: Harmonic analysis in several variables Special classes of linear operators

Published online by Cambridge University Press:  07 December 2020

Xiangxing Tao  and
Guoen Hu
Xiangxing Tao
Affiliation:
Department of Mathematics, School of Science, Zhejiang University of Science and Technology, Hangzhou310023, P.R. Chinae-mail:xxtao@zust.edu.cn
Guoen Hu*
Affiliation:
School of Applied Mathematics, Beijing Normal University, Zhuhai519087, P.R. China
*
e-mail: guoenxx@163.com
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Abstract

Let $\Omega $ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$ , $T_{\Omega }$ be the convolution singular integral operator with kernel $\frac {\Omega (x)}{|x|^d}$ . In this paper, we prove that if $\Omega \in L\log L(S^{d-1})$ , and U is an operator which is bounded on $L^2(\mathbb {R}^d)$ and satisfies the weak type endpoint estimate of $L(\log L)^{\beta }$ type, then the composition operator $UT_{\Omega }$ satisfies a weak type endpoint estimate of $L(\log L)^{\beta +1}$ type.

Keywords

Rough singular integral operatorcomposite operatorweak type endpoint estimate

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 47B33: Composition operators
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 911 - 922
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The research of X.T was supported by the NNSF of China under grant #11771399, and the research of G.H. (corresponding author) was supported by the NNSF of China under grant #11871108.

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