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Weighted estimates for the Calderón commutator

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  23 September 2019

Jiecheng Chen  and
Guoen Hu
Jiecheng Chen
Affiliation:
Department of Mathematics, Zhejiang Normal University, Jinhua321004, People's Republic of China (jcchen@zjnu.edu.cn)
Guoen Hu
Affiliation:
Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou450001, People's Republic of China (guoenxx@163.com)
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Abstract

In this paper the authors consider the weighted estimates for the Calderón commutator defined by

\mathcal{C}_{m+1, A}(a_1,\ldots,a_{m};f)(x)={\rm p. v.} \displaystyle\int_{\mathbb{R}}\displaystyle\frac{P_2(A; x, y)\prod\nolimits_{j=1}^m(A_j(x)-A_j(y))}{(x-y)^{m+2}}f(y){\rm d}y,
with P2(A;x, y) = A(x) − A(y) − A′(y)(x − y) and A′ ∈ BMO(ℝ). Dominating this operator by multi(sub)linear sparse operators, the authors establish the weighted bounds from $L^{p_1}(\mathbb {R},w_1) \times \cdots \times L^{p_{m+1}}(\mathbb {R},w_{m+1})$ to $L^{p}(\mathbb {R},\nu _{\vec {\kern 1pt w}})$, with p1, …, pm+1 ∈ (1, ∞), 1/p = 1/p1 + · · · + 1/pm+1, and $\vec {\kern 1pt w}=(w_1, \ldots , w_{m+1})\in A_{\vec {P}}(\mathbb {R}^{m+1})$. The authors also obtain the weighted weak type endpoint estimates for $\mathcal {C}_{m+1, A}$.

Keywords

Calderón commutatorweighted inequalitymultilinear singular integral operatorsparse operatormultiple weight

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 169 - 192
Copyright
Copyright © Edinburgh Mathematical Society 2019

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