On Rough Parametric Marcinkiewicz Integrals Along Certain Surfaces

In this paper, we study rough Marcinkiewicz integrals associated with surfaces defined by ${\mathsf{\Psi }}_{\mathcal{P},\varphi }=\{{}^{˜}(\mathcal{P}\left(w\right),\varphi (\left|w\right|\left)\right):w\in {\mathbb{R}}^m$}. We establish the $L^p$-boundedness of these integrals when the kernel functions lie in the $L^q\left({\mathbb{S}}^{m-1}\right)$ space. Combining this result with Yano’s extrapolation technique, we further obtain the $L^p$-boundedness under weaker kernel conditions—specifically, when the kernels belong to either the block space $B_q^{(0,-1/2)}\left({\mathbb{S}}^{m-1}\right)$ or $L{(logL)}^{1/2}\left({\mathbb{S}}^{m-1}\right)$. Our results extend and refine several previously known results on Marcinkiewicz integrals, offering broader applicability and sharper conclusions.