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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.
Li et al. (Mon,) studied this question.
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