Maximal functions related to homogeneous hypersurfaces in ${\mathbb{R}}^3$

We study maximal functions related to homogeneous polynomial hypersurfaces in ${\mathbb{R}}^3$. In a sense made precise in this paper, the region of $(p,q)$ for which we obtain $L^p\to L^q$ boundedness is optimal up to the endpoints for the corresponding local maximal operators. The boundedness exponents depend explicitly on both the height of the hypersurface and the type of the curve determined by the level set. As a corollary, we obtain $L^p$-estimates and weighted norm inequalities for the associated global maximal functions. Moreover, we also obtain optimal $L^p$-estimates for the global maximal operators associated with homogeneous polynomial hypersurfaces without transversality condition in ${\mathbb{R}}^3$.