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

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 m th-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 m th-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 $ .