Boundedness of maximal operators associated with convex hypersurfaces

In this talk, we discuss the problem of boundedness of maximal operators associated with hypersurfaces in the space of summable functions with degree $p$. We give a necessary condition for the boundedness of maximal operators in the space of square-summable functions in the case where one of the principal curvatures is nonzero. In the class of convex hypersurfaces, we obtain a criterion for the boundedness of the corresponding maximal operators in the space of square-summable functions.