A rough Marcinkiewicz integral along smooth curves

We consider the boundedness of a kind of nonlinear integral operators on L^p spaces. Including the parametric Marcinkiewicz integrals with rough kernels along compound curves {Φ(ϕ(|y|))y ; y ∈ R^n} with Φ satisfying certain growth conditions and ϕ being differentiable function with monotonicity and some properties on the positive real line, we investigate the Lp bounds of these operators under the integral kernels given by the sphere functions Ω in H^1(Sn−1) or Ω in L(log+ L)1/2(Sn−1) and the radial function h ∈ ∆γ (R+). As applications, the corresponding results for parametric Marcinkiewicz integral operators related to area integrals and Littlewood-Paley g∗ λ - functions are presented.