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

Abstract

Let $(\mathcal{X}, d, \mu )$ ( X , d , μ ) 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 $L^{p} (\mu )$ L p ( μ ) of mth-order commutators $\mathcal{M}^{\rho }_{b,m}$ M b , m ρ generated by the Log-Dini-type parametric Marcinkiewicz integral operators with RBMO functions on $(\mathcal{X}, d, \mu )$ ( X , d , μ ) . In addition, the boundedness of the mth-order commutators $\mathcal{M}^{\rho }_{b,m}$ M b , m ρ on Morrey spaces $M^{q}_{p}(\mu )$ M p q ( μ ) , $1< p \leq q< \infty $ 1 < p ≤ q < ∞ , is also obtained for the parameter $0<\rho <\infty $ 0 < ρ < ∞ .