On the critical boundedness index of maximal operators

The talk deals with boundedness problem for maximal operators associated to hypersurfaces in the space of integrable functions with degree $p$. A necessary condition for boundedness is given in the space of square-integrable functions in the case one nonvashing principal curvature. A criterion for the boundedness of the maximal operators in the space of square-integrable functions is obtained for a partial class of convex hypersurfaces.