Has A Non-doubling Measures Of Littlewood-Paley G Function And Its Boundedness Of Exchange

Let μ be a positive Radon measure on Rd which may be non-doubling. The only condition that μ must satisfy is μ(B(x, r))<Corn, for x∈Rd, r>0and some fixed constants Co>0and0<n≤d. In this paper, suppose that Littlewood-Paley g is bounded on L2(μ), by using the properties of Littlewood-Paley g and the Calderon-Zygmund decompsition for non-doubling measure, the boundedness of g from L1(μ) to L1:∞(μ) is obtaind. And then the boundedness of g from Hardy space H1(μ) to L(μ) is established. Moreover, for any function b∈RBMO(μ), we define higher-order commutator generated by Littlewood-Paley g and b. Based on the definiations and properties of RBMO(μ), the con-clusions of Sharp maximal operator, and the Calderon-Zygmund decompsition, we establish its boundedness on Lp(μ) for p∈(1,∞).