A charaterization of commutators for parabolic singular integrals

Abstract

In this paper, the authors give a characterization of the $L^p$-boundedness of the commutators for the parabolic singular integrals. More precisely, the authors prove that if $b\in \mathrm{BMO}_\varphi(\mathsf{R}^n,\rho)$, then the commutator $[b,T]$ is a bounded operator from $L^p(\mathsf{R}^n)$ to the Orlicz space $L_\psi(\mathsf{R}^n)$, where the kernel function $\Omega$ has no any smoothness on the unit sphere $S^{n-1}$. Conversely, if assuming on $\Omega$ a slight smoothness on $S^{n-1}$, then the boundedness of $[b,T]$ from $L^p(\mathsf{R}^n)$ to $L_\psi(\mathsf{R}^n)$ implies that $b\in \mathrm{BMO}_\varphi(\mathsf{R}^n,\rho)$. The results in this paper improve essentially and extend some known conclusions.