Commutators of Littlewood-Paley gκ∗ $g_{\kappa}^{*} $-functions on non-homogeneous metric measure spaces

The main purpose of this paper is to prove that the boundedness of the commutator $\mathcal{M}_{\kappa,b}^{*} $ generated by the Littlewood-Paley operator $\mathcal{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 $\mathcal{M}_{\kappa}^{*} $ satisfies a certain Hörmander-type condition, the authors prove that $\mathcal{M}_{\kappa,b}^{*} $ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from the space L log L(μ) to the weak Lebesgue space L1,∞(μ), and is bounded from the atomic Hardy spaces H1(μ) to the weak Lebesgue spaces L1,∞(μ).