The Commutators of Parameterized Littlewood–Paley Operators on Weighted Hardy Spaces

This paper is devoted to studying the behaviors of commutators for parameterized type Littlewood–Paley operators with \(b\in L^1_\textrm{loc}(\mathbb R^n)\) $b\in L_{\text{loc}}^1({\mathbb{R}}^n)$ on the weighted Hardy spaces. Under the assumption of that the homogeneous kernel \(\Omega \) $\mathrm{\Omega }$ satisfies certain mild regularities and \(b\in \mathcal {BMO}_{p}(\omega )\) $b\in {\mathcal{BMO}}_p(\omega )$ (a nontrivial subspaces of \(\mathrm BMO(\mathbb R^n)\) $\mathrm{B}MO({\mathbb{R}}^n)$ ) for certain \(\omega \in A_\infty \) $\omega \in A_{\infty }$ with \(0<p\le 1\) $0<p\le 1$ , we obtain the boundedness of commutators \(\mu ^\rho _{\Omega ,b}\) ${\mu }_{\mathrm{\Omega },b}^{\rho }$ generated by parameterized Marcinkiewicz operators \(\mu _\Omega ^\rho \) ${\mu }_{\mathrm{\Omega }}^{\rho }$ with b on the weighted Hardy spaces \(H^p(\omega )\) $H^p(\omega )$ . The corresponding results for the commutators of parameterized Littlewood–Paley area integrals \(\mu _{\Omega ,S}^\rho \) ${\mu }_{\mathrm{\Omega },S}^{\rho }$ and \(g_\lambda ^*\) $g_{\lambda }^{\ast }$ -functions \(\mu _{\Omega ,\lambda }^{\rho ,*}\) $\mu _{\mathrm{\Omega },\lambda }^{\rho ,\ast }$ are also given.