TY - JOUR
T1 - A Note on Weak Type $(1,1)$ Estimate for the Higher Order Commutators of Christ-Journé Type
AU - Yong Ding & Xudong Lai
JO - Analysis in Theory and Applications
VL - 3
SP - 268
EP - 287
PY - 2019
DA - 2019/04
SN - 35
DO - http://doi.org/10.4208/ata.OA-0007
UR - https://global-sci.org/intro/article_detail/ata/13116.html
KW - Weak type $(1,1)$, higher order, commutator.
AB - <p style="text-align: justify;">In this paper, a weak type $(1,1)$ estimate is established for the higher order commutator introduced by Christ and Journé which is defined by</p><p style="text-align: justify;">$$ T[a_1,\cdots,a_l]f(x)=p.v. \int_{R^d} K(x-y)\Big(\prod_{i=1}^lm_{x,y}a_i\Big)\cdot f(y)dy, $$</p><p style="text-align: justify;">where $K$ is the standard Calderόn-Zygmund convolution kernel on $\mathbb{R}^d (d\geq2)$ and $m_{x,y}a_i=\int_0^1a_i(sx+(1-s)y)ds$.</p>