On Marcinkiewicz integral with variable kernels

Abstract

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 R n . We prove further that, when the variable kernel Q satisfies a class of integral Dini condition, μ Ω is a bounded operator from the Hardy space H 1 (R n ) to L 1 (R n ) and from the weak Hardy space H 1, ∞(R n ) to the weak L 1 space L 1, ∞(R n ), 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 L 2 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.