Commutators of Littlewood-Paley gκ∗$g_{\kappa}^{*} $-functions on non-homogeneous metric measure spaces

Abstract

The main purpose of this paper is to prove that the boundedness of the commutator Mκ,b∗$\mathcal{M}_{\kappa,b}^{*} $ generated by the Littlewood-Paley operator Mκ∗$\mathcal{M}_{\kappa}^{*} $ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of Mκ∗$\mathcal{M}_{\kappa}^{*} $ satisfies a certain Hörmander-type condition, the authors prove that Mκ,b∗$\mathcal{M}_{\kappa,b}^{*} $ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from the space L log L(μ) to the weak Lebesgue space L1,∞(μ), and is bounded from the atomic Hardy spaces H1(μ) to the weak Lebesgue spaces L1,∞(μ).

Keywords

• non-homogeneous metric measure space
• commutators
• gκ*-functions
• rbmo (μ)
• hardy space
• 42b25
• 42b35
• 30l99