  Using some resolution of singularities and oscillatory integral methods in
conjunction with appropriate damping and interpolation techniques, L^p
boundedness theorems for p > 2 are obtained for maximal operators over a wide
range of hypersurfaces. These estimates are sharp in many situations, including
the convex hypersurfaces of finite line type considered by Iosevich, Sawyer,
and others. As a corollary, we also give a generalization of the result of
Sogge and Stein that for some finite p the maximal operator corresponding to a
hypersurface whose Gaussian curvature does not vanish to infinite order is
bounded on L^p. Analogous estimates are proven for Fourier transforms of
surface measures, and these are sharp for the same hypersurfaces as the maximal
operators.
