A Note on Weak Type $(1,1)$ Estimate for the Higher Order Commutators of Christ-Journé Type

Abstract

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

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

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