On the Marcinkiewicz integral with variable kernels

In this paper we prove that the Marcinkiewicz integral μ_Ω with variable kernels is an operator of type (2, 2), where the kernel function Ω does not have any smoothness on the unit sphere in ℝⁿ. We prove further that, when the variable kernel Ω satisfies a class of integral Dini condition, μ_Ω is a bounded operator from the Hardy space H¹ (ℝⁿ) to L¹ (ℝⁿ) and from the weak Hardy space H ^1,∞ (ℝⁿ) to the weak L¹ space L ^1-∞(ℝⁿ), respectively. As corollaries of the above conclusions, we show that μ_Ω is also of the weak type (1, 1) and of type (p, p) for 1 < p < 2. Moreover, the L² boundedness of a class of the Littlewood-Paley type operators with variable kernels also are obtained, which are related to the Littlewood-Paley g*_λ-function and Lusin area integral, respectively.