Maximal Functions over Hypersurfaces with Flat Points

For n 4, let S be a flat hypersurface in n, and let d d , where C0 Ž n. and is the surface area measure on S. Then the maximal functions Mf associated to S and by MfŽx. supt 0 HS fŽx t .d Ž . , f SŽ n., are bounded on certain Orlicz spaces L Ž n..