Estimates for Parameter Littlewood-Paley gκ⁎ Functions on Nonhomogeneous Metric Measure Spaces

Abstract

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 Mκ⁎ satisfies a certain Hörmander-type condition, Mκ⁎,ρ is bounded from Lebesgue spaces Lp(μ) to Lebesgue spaces Lp(μ) for p≥2 and is bounded from L1(μ) into L1,∞(μ). As a corollary, Mκ⁎,ρ is bounded on Lp(μ) for 1<p<2. In addition, the authors also obtain that Mκ⁎,ρ is bounded from the atomic Hardy space H1(μ) into the Lebesgue space L1(μ).