Volume 35, Issue 3
A Note on Weak Type $(1,1)$ Estimate for the Higher Order Commutators of Christ-Journe Type

Anal. Theory Appl., 35 (2019), pp. 268-287.

Published online: 2019-04

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• Abstract

In this paper, a weak type $(1,1)$ estimate is established for the higher order commutator introduced by Christ and Journ\'e which is defined by

$$T[a_1,\cdots,a_l]f(x)=p.v. \int_{R^d} K(x-y)\Big(\prod_{i=1}^lm_{x,y}a_i\Big)\cdot f(y)dy,$$

where $K$ is the standard Calder\'on-Zygmund convolution kernel on $\mathbb{R}^d (d\geq2)$ and $m_{x,y}a_i=\int_0^1a_i(sx+(1-s)y)ds$.

• Keywords

Weak type $(1,1)$, higher order, commutator.

42B20, 42B25

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• TXT
@Article{ATA-35-268, author = {}, title = {A Note on Weak Type $(1,1)$ Estimate for the Higher Order Commutators of Christ-Journe Type}, journal = {Analysis in Theory and Applications}, year = {2019}, volume = {35}, number = {3}, pages = {268--287}, abstract = {

In this paper, a weak type $(1,1)$ estimate is established for the higher order commutator introduced by Christ and Journ\'e which is defined by

$$T[a_1,\cdots,a_l]f(x)=p.v. \int_{R^d} K(x-y)\Big(\prod_{i=1}^lm_{x,y}a_i\Big)\cdot f(y)dy,$$

where $K$ is the standard Calder\'on-Zygmund convolution kernel on $\mathbb{R}^d (d\geq2)$ and $m_{x,y}a_i=\int_0^1a_i(sx+(1-s)y)ds$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-0007}, url = {http://global-sci.org/intro/article_detail/ata/13116.html} }
TY - JOUR T1 - A Note on Weak Type $(1,1)$ Estimate for the Higher Order Commutators of Christ-Journe Type JO - Analysis in Theory and Applications VL - 3 SP - 268 EP - 287 PY - 2019 DA - 2019/04 SN - 35 DO - http://dor.org/10.4208/ata.OA-0007 UR - https://global-sci.org/intro/ata/13116.html KW - Weak type $(1,1)$, higher order, commutator. AB -

In this paper, a weak type $(1,1)$ estimate is established for the higher order commutator introduced by Christ and Journ\'e which is defined by

$$T[a_1,\cdots,a_l]f(x)=p.v. \int_{R^d} K(x-y)\Big(\prod_{i=1}^lm_{x,y}a_i\Big)\cdot f(y)dy,$$

where $K$ is the standard Calder\'on-Zygmund convolution kernel on $\mathbb{R}^d (d\geq2)$ and $m_{x,y}a_i=\int_0^1a_i(sx+(1-s)y)ds$.

Yong Ding & Xudong Lai. (1970). A Note on Weak Type $(1,1)$ Estimate for the Higher Order Commutators of Christ-Journe Type. Analysis in Theory and Applications. 35 (3). 268-287. doi:10.4208/ata.OA-0007
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