Littlewood-Paley g(lambda)(,mu)(*)-function and Its Commutator on Non-homogeneous Generalized Morrey Spaces

Let mu be a non-negative Radon measure on R-d which may be a non-doubling measure. In this paper, the authors prove that the Littlewood-Paley g(lambda,)(mu)(*)-function is bounded on the generalized Money space L-P,L-phi(mu), and also obtain that the commutator 4 .4 generated by the Littlewood-Paley function g(lambda,)(mu)(*)and the regular bounded mean oscillation space (=RBMO), which is due to X. Tolsa, is bounded on L-P,L-phi(mu). As a corollary, the authors prove that the commutator g(lambda,)(mu,b)(*) is bounded on the Money space Mqp(mu) defined by Sawano and Tanaka when we take phi(t) =t(1-p/q) with 1 < p < q < infinity.