Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures

Let μ be a positive Radon measure on ℝd which may be nondoubling. The only condition that μ satisfies is μ(B(x,r))≤C0rn for all x∈ℝd, r>0, and some fixed constant C0. In this paper, we introduce the operator gλ,μ∗ related to such a measure and assume it is bounded on L2(μ). We then establish its boundedness, respectively, from the Lebesgue space L1(μ) to the weak Lebesgue space L1,∞(μ), from the Hardy space H1(μ) to L1(μ) and from the Lesesgue space L∞(μ) to the space RBLO(μ). As a corollary, we obtain the boundedness of gλ,μ∗ in the Lebesgue space Lp(μ) with p∈(1,∞)