Estimates for maximal functions associated with hypersurfaces in R3 and related problems of harmonic analysis

We study the boundedness problem for maximal operators, associated with averages along smooth hypersurfaces S of finite type in 3-dimensional Euclidean space. For p > 2, we prove that if no affine tangent plane to S passes through the origin and S is analytic, then the associated maximal operator is bounded on if and only if p > h(S), where h(S) denotes the so-called height of the surface S (defined in terms of certain Newton diagrams). For non-analytic S we obtain the same statement with the exception of the exponent p = h(S). Our notion of height h(S) is closely related to A. N. Varchenko's notion of height h(φ) for functions φ such that S can be locally represented as the graph of φ after a rotation of coordinates. Several consequences of this result are discussed. In particular we verify a conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on S and the Lp-boundedness of the associated maximal operator, and a conjecture by Iosevich and Sawyer which relates the Lp-boundedness of, to an integrability condition on S for the distance to tangential hyperplanes, in dimension 3. In particular, we also give essentially sharp uniform estimates for the Fourier transform of the surface measure on S, thus extending a result by V. N. Karpushkin from the analytic to the smooth setting and implicitly verifying a conjecture by V. I. Arnold in our context. As an immediate application of this, we obtain an Fourier restriction theorem for S. © 2010 Institut Mittag-Leffler.