Commutators of log-Dini-type parametric Marcinkiewicz operators on non-homogeneous metric measure spaces

Abstract

Abstract Let ( X , d , μ ) $(\mathcal{X}, d, \mu )$ be a non-homogeneous metric measure space, which satisfies the geometrically doubling condition and the upper doubling condition. In this paper, the authors prove the boundedness in L p ( μ ) $L^{p} (\mu )$ of mth-order commutators M b , m ρ $\mathcal{M}^{\rho }_{b,m}$ generated by the Log-Dini-type parametric Marcinkiewicz integral operators with RBMO functions on ( X , d , μ ) $(\mathcal{X}, d, \mu )$ . In addition, the boundedness of the mth-order commutators M b , m ρ $\mathcal{M}^{\rho }_{b,m}$ on Morrey spaces M p q ( μ ) $M^{q}_{p}(\mu )$ , 1 < p ≤ q < ∞ $1< p \leq q< \infty $ , is also obtained for the parameter 0 < ρ < ∞ $0<\rho <\infty $ .

Keywords

• Log-Dini-type parametric Marcinkiewicz operator
• RBMO ( μ ) $\operatorname{RBMO}(\mu )$
• Commutator
• Non-homogeneous metric measure space