  In this article, we study maximal functions related to hypersurfaces with
vanishing Gaussian curvature in $\mathbb{R}^3$. Firstly, we characterize the
$L^p\rightarrow L^q$ boundedness of local maximal operators along homogeneous
hypersurfaces. Moreover, weighted $L^p$-estimates are obtained for the
corresponding global operators. Secondly, for a class of hypersurfaces that
lack a homogeneous structure and pass through the origin, we attempt to look
for other geometric properties instead of height of hypersurfaces to
characterize the optimal $L^p$-boundedness of the corresponding global maximal
operators.
