On compactness of commutators of the Christ-Journé type operators

Abstract

This paper investigates the commutators $[b,T_a]$ associated with Christ-Journé type operator$T_a\left(f\right)\left(x\right)=\text{p.v.}\underset{R^n}{\int }K(x-y)m_{x,y}af\left(y\right)dy,$ where $m_{x,y}a={\int }_0^{1}a(tx+(1-t\left)y\right)dt$. We show that $[b,T_a]:=T_{a,b}$ is bounded on $L^p\left(R^n\right)$ for $1<p<\infty$ whenever $b\in \mathrm{BMO}\left(R^n\right)$, and compact on $L^p\left(R^n\right)$ whenever $b\in \mathrm{CMO}\left(R^n\right)$, where $\mathrm{CMO}\left(R^n\right)$ is the closure of $C_c^{\infty }\left(R^n\right)$ in the $\mathrm{BMO}\left(R^n\right)$ topology. Furthermore, we also study Christ-Journé type commutators with rough kernels $K\left(x\right)=\Omega \left(x\right)|x{|}^{-n}$, denoted by $T_{\left[a\right],b}$, and establish analogous results for their boundedness and compactness.