Necessary and sufficient conditions for boundedness of commutators of parametric Marcinkiewicz integrals with weighted Lipschitz functions

In this paper, we obtain the sharp maximal function estimate for the commutator $\mathcal{M}_{\Omega,b}^{\rho,m}$ generated by the parametric Marcinkiewicz integral $\mathcal{M}_{\Omega}^{\rho,m}$ and the locally integrable function $b$, where $\rho>0$, $m>1$ and $\Omega$ satisfies certain log-type regularity condition. Meanwhile, we prove the commutator $\mathcal{M}_{\Omega,b}^{\rho,m}$ is bounded from $L^p(\mu)$ to $L^q(\mu^{1-q})$ if and only if $b\in Lip_\beta(\mu)$, where $\mu\in A_1,0 \beta 1,1 p n/\beta$ and $1/q=1/p-\beta/n$.