Boundedness of Parametrized Littlewood-Paley Operators with Nondoubling Measures

Let μ be a nonnegative Radon measure on R which only satisfies the following growth condition that there exists a positive constant C such that μ B x, r ≤ Cr for all x ∈ R, r > 0 and some fixed n ∈ 0, d . In this paper, the authors prove that for suitable indexes ρ and λ, the parametrized g∗ λ function M∗,ρ λ is bounded on L μ for p ∈ 2,∞ with the assumption that the kernel of the operator M∗,ρ λ satisfies some Hörmander-type condition, and is bounded from L1 μ into weak L1 μ with the assumption that the kernel satisfies certain slightly stronger Hörmandertype condition. As a corollary,M∗,ρ λ with the kernel satisfying the above stronger Hörmander-type condition is bounded on L μ for p ∈ 1, 2 . Moreover, the authors prove that for suitable indexes ρ and λ,M∗,ρ λ is bounded from L∞ μ into RBLO μ the space of regular bounded lower oscillation functions if the kernel satisfies the Hörmander-type condition, and from the Hardy space H1 μ into L1 μ if the kernel satisfies the above stronger Hörmander-type condition. The corresponding properties for the parametrized area integral MρS are also established in this paper.