Boundedness for maximal operators over hypersurfaces in R³

In this article, we study maximal functions related to hypersurfaces with vanishing Gaussian curvature in R³. Firstly, we characterize the Lᵖ Lq boundedness of local maximal operators along homogeneous hypersurfaces. Moreover, weighted Lᵖ-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 Lᵖ-boundedness of the corresponding global maximal operators.