On Rough Parametric Marcinkiewicz Integrals Along Certain Surfaces

In this paper, we study rough Marcinkiewicz integrals associated with surfaces defined by ΨP,ϕ={(˜P(w),ϕ(w)):w∈Rm}. We establish the Lp-boundedness of these integrals when the kernel functions lie in the Lq(Sm−1) space. Combining this result with Yano’s extrapolation technique, we further obtain the Lp-boundedness under weaker kernel conditions—specifically, when the kernels belong to either the block space Bq(0,−1/2)(Sm−1) or L(logL)1/2(Sm−1). Our results extend and refine several previously known results on Marcinkiewicz integrals, offering broader applicability and sharper conclusions.