Journal of the
Korean Mathematical Society JKMS

The aim of this paper is to mainly establish the sufficient and necessary conditions for the boundedness of the commutator $\mathcal{M}^{\rho}_{\Omega, b}$ which is generated by the parameter Marcinkiwicz integral $\mathcal{M}^{\rho}_{\Omega}$ and the Lipschitz function $b$ on generalized Orlicz-Morrey space $L^{\Phi,\varphi}(\mathbb{R}^{d})$ in the sense of the Adams type result (or Spanne type result). Moreover, the necessary conditions for the parameter Marcinkiewizcz integral $\mathcal{M}^{\rho}_{\Omega}$ on the $L^{\Phi,\varphi}(\mathbb{R}^{d})$, and the commutator $[b,\mathcal{M}^{\rho}_{\Omega}]$ generated by the $\mathcal{M}^{\rho}_{\Omega}$ and the space $\mathrm{BMO}$ on the $L^{\Phi,\varphi}(\mathbb{R}^{d})$, are also obtained, respectively.

Keywords: Parameter Marcinkiewicz integral, commutator, Lipschits space, $\mathrm{BMO}$ space, generalized Orlicz-Morrey

MSC numbers: Primary 42B20, 42B35

Supported by: This work was financially supported by the Innovation Capacity Improvement Project for Colleges and Universities of Gansu Province (No. 2020A-010) and the Doctoral Scientific Research Foundation of Northwest Normal University (No. 6014/0002020203)