On Weighted Inequalities for Parametric Marcinkiewicz Integrals

We establish a weighted Lp boundedness of a parametric Marcinkiewicz integral operator ρ Ω,h if Ω is allowed to be in the block space B (0,−1/2) q (Sn−1) for some q > 1 and h satisfies a mild integrability condition. We apply this conclusion to obtain the weighted Lp boundedness for a class of the parametric Marcinkiewicz integral operators ∗,ρ Ω,h,λ and ρ Ω,h,S related to the Littlewood-Paley g ∗ λ -function and the area integral S, respectively. It is known that the condition Ω ∈ B(0,−1/2) q (Sn−1) is optimal for the L2 boundedness of 1 Ω,1.