Maximal averages over hypersurfaces and the Newton polyhedron

Here φ(x) is a smooth cutoff function that localizes the surface S near some specific y ∈ S. The goal here is to determine the values of p for which M is bounded on L. The earliest work on this subject was done in the case where S is a sphere, when Stein [St1] showed M is bounded on L iff p > n+1 n for n > 1. This was later generalized by Greenleaf [Gr] to surfaces of nonvanishing Gaussian curvature, and the n = 1 case was later proven by Bourgain [Bo]. Since then, there have been a wide range of papers on this subject, which we will describe in more detail throughout this section. Although there are many interesting issues when p ≤ 2, for the purposes of this paper we always assume p > 2. Note that if M is bounded on some L, by interpolating with the L∞ case one has that M is bounded on L ′ for p′ > p. Hence our goal is to determine the optimal p0 ≥ 2 for which M is bounded on L for p > p0.