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 generated by the Littlewood-Paley operator and RBMO ( ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of satisfies a certain Hörmander-type condition, the authors prove that is bounded on Lebesgue spaces ) for 1 < < ∞, bounded from the space log ) to the weak Lebesgue space ), and is bounded from the atomic Hardy spaces ) to the weak Lebesgue spaces ).