Estimates for parameter Littlewood-Paley g functions on nonhomogeneous metric measure spaces

Let (x,d,μ) be a metric measure space which satisfies the geometrically doubling measure and the upper doubling measure conditions. in this paper,the authors prove that,under the assumption that the kernel of m satisfies a certain hörmander-type condition,m,ρ is bounded from lebesgue spaces l p (μ) to lebesgue spaces l p (μ) for p ≥ 2 and is bounded from l 1 (μ) into l 1,∞ (μ). as a corollary,m,ρ is bounded on l p (μ) for 1 < p < 2. in addition,the authors also obtain that m,ρ is bounded from the atomic hardy space h 1 (μ) into the lebesgue space l 1 (μ). © 2016 guanghui lu and shuangping tao.