Generalized Marcinkiewicz Integral Operators Along Certain Surfaces

In this article, a class of rough generalized Marcinkiewicz operators is considered. Under the condition that the singular kernel belongs to the space $L^q\left({\mathbb{S}}^{n-1}\right)$, the boundedness of these operators is confirmed from the space of homogeneous Triebel–Lizorkin functions to the $L^p\left({\mathbb{R}}^{d+1}\right)$ space. Moreover, appropriate $L^p$ bounds are obtained which allow us to utilize Yano’s extrapolation procedure to prove the boundedness of the aforementioned operators under weaker assumptions on the kernels. In this work, several known past results are generalized, extended, and improved.