Commutators of Littlewood-Paley g*κ-functions on non-homogeneous metric measure spaces

The main purpose of this paper is to prove that the boundedness of the commutator M*(kappa,b) generated by the Littlewood-Paley operator M*(kappa) and RBMO(mu) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of M*(kappa) satisfies a certain Hormander-type condition, the authors prove that M*(kappa,b) is bounded on Lebesgue spaces L-p(mu) for 1 < p < infinity, bounded from the space L log L(mu) to the weak Lebesgue spaceL(1,infinity)(mu), and is bounded from the atomic Hardy spaces H-1(mu) to the weak Lebesgue spaces L-1,L-infinity(mu).