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The main purpose of this paper is to prove that the boundedness of the commutator [...] Mκ,b∗ ${ℳ}_{\kappa ,b}^{*}$ generated by the Littlewood-Paley operator [...] Mκ∗ ${ℳ}_{\kappa }^{*}$ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of [...] Mκ∗ ${ℳ}_{\kappa }^{*}$ satisfies a certain Hörmander-type condition, the authors prove that [...] Mκ,b∗ ${ℳ}_{\kappa ,b}^{*}$ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from the space...