Maximal averages over hypersurfaces and the Newton polyhedron

Using some resolution of singularities and oscillatory integral methods in conjunction with appropriate damping and interpolation techniques, Lp boundedness theorems for p >2 are obtained for maximal averages over hypersurfaces in Rn for n > 2. These estimates are sharp in various situations, including the convex hypersurfaces of finite line type considered by several authors. 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 Lp for some finite p. Analogous estimates are proven for Fourier transforms of surface measures, and these are sharp for the same hypersurfaces as the maximal operators. © 2011 Elsevier Inc. All rights reserved.