A class of rough generalized Marcinkiewicz integrals along twisted surfaces

In this paper, we investigated the mapping properties of generalized Marcinkiewicz integral operators associated with twisted surfaces. Under certain conditions on these surfaces, we established suitable $ L^p $ estimates for these operators, assuming the kernel functions belong to $ L^{q}\left(\mathbb{U}^{{{j}}-1}\times \mathbb{U} ^{{{k}}-1}\right) $. By combining these estimates with Yano's extrapolation technique, we further established the boundedness of these operators from the homogeneous Triebel-Lizorkin space $ \overset{.}{F}_{p}^{0, \tau}(\mathbb{R}^j\times\mathbb{R}^k) $ to the space $ L^p(\mathbb{R}^j\times\mathbb{R}^k) $ under significantly weaker assumptions on the kernels. Our results extended and improved many previously known results.