Introduction

Let S be a smooth hypersurface in R3 with Riemannian surface measure ds. We shall assume that S is of finite type, that is, that every tangent plane has finite order of contact with S. Consider the compactly supported measure dµ := ?ds on S, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The central problem that we shall investigate in this monograph is the determination of the range of exponents p for which a Fourier restriction estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

holds true.

This problem is a special case of the more general Fourier restriction problem, which asks for the exact range of exponents p and q for which an Lp-Lq Fourier restriction estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

holds true and which can be formulated for much wider classes of subvarieties S in arbitrary dimension n and suitable measures dµ supported on S. In fact, as observed by G. Mockenhaupt [M00] (see also the more recent work by I. Laba and M. Pramanik [LB09]), it makes sense in much wider settings, even for measures dµ supported on "thin" subsets S of Rn, such as Salem subsets of the real line.

The Fourier restriction problem presents one important instance of a wide circle of related problems, such as the boundedness properties of Bochner Riesz means, dimensional properties of Kakeya type sets, smoothing effects of averaging over time intervals for solutions to the wave equation (or more general dispersive equations), or the study of maximal averages along hypersurfaces. The common question underlying all these problems asks for the understanding of the interplay between the Fourier transform and properties of thin sets in Euclidean space, for instance geometric properties of subvarieties. Some of these aspects have been outlined in the survey article [M14], from which parts of this introduction have been taken.

The idea of Fourier restriction goes back to E. M. Stein, and a first instance of this concept is the determination of the sharp range of Lp-Lq Fourier restriction estimates for the circle in the plane through work by C. Fefferman and E. M. Stein [F70] and A. Zygmund [Z74], who obtained the endpoint estimates (see also L. Hörmander [H73] and L. Carleson and P. Sjölin [CS72] for estimates on more general related oscillatory integral operators). For subvarieties of higher dimension, the first fundamental result was obtained (in various steps) for Euclidean spheres Sn-1 by E. M. Stein and P. A. Tomas [To75], who proved that an Lp-L2 Fourier restriction estimate holds true for Snn-1, n = 3, if and only if p' = 2(2/(n - 1) + 1), where p' denotes the exponent conjugate to p, that is, 1/p + 1/p' = 1 (cf. [S93] for the history of this result). A crucial property of Euclidean spheres which is essential for this result is the non-vanishing of the Gaussian curvature on these spheres, and indeed an analogous result holds true for every smooth hypersurface Swith nonvanishing Gaussian curvature (see [Gl81]).

Fourier restriction estimates have turned out to have numerous applications to other fields. For instance, their great importance to the study of dispersive partial differential equations became evident through R. Strichartz' article [Str77], and in the PDE-literature dual versions which invoke also Plancherel's theorem are often called Strichartz estimates.

The question as to which Lp-Lq Fourier restriction estimates hold true for Euclidean spheres is still widely open. It is conjectured that estimate (1.2) holds true for S = Sn-1 if and only if p' > 2n/(n - 1) and p' = q(2/(n - 1) + 1), and there has been a lot of deep work on this and related problems by numerous mathematicians, including J. Bourgain, T. Wolff, A. Moyua, A. Vargas, L. Vega, and T. Tao (see, e.g., [Bou91], [Bou95], [W95], [MVV96], [TVV98], [W00], [TV00], [T03], and [T04] for a few of the relevant articles, but this list is far from being complete). There has been a lot of work also on conic hypersurfaces and some on even more general classes of hypersurfaces with vanishing Gaussian curvature, for instance in Barcelo [Ba85], [Ba86], in Tao, Vargas, and Vega [TVV98], in Wolff [W01], and in Tao and Vargas [TV00], and more recently by A. Vargas and S. Lee [LV10] and S. Buschenhenke [Bu12]. Again, these citations give only a sample of what has been published on this subject.

Recent work by J. Bourgain and L. Guth [BG11], making use also of multilinear estimates from work by J. Bennett, A. Carbery, and T. Tao [BCT06], has led to further important progress. Nevertheless, this and related problems continue to represent one of the major challenges in Euclidean harmonic analysis, bearing various deep connections with other important open problems, such as the Bochner-Riesz conjecture, the Kakeya conjecture and C. Sogge's local smoothing conjecture for solutions to the wave equation. We refer to Stein's book [S93] for more information on and additional references to these topics and their history until 1993, and to more recent related essays by Tao, for instance in [T04].

As explained before, we shall restrict ourselves to the study of the Stein-Tomastype estimates (1.1). For convex hypersurfaces of finite line type, a good understanding of this type of restriction estimates is available, even in arbitrary dimension (we refer to the article [I99] by A. Iosevich, which is based on work by J. Bruna, A. Nagel and S. Wainger [BNW88], providing sharp estimates for the Fourier transform of the surface measure on convex hypersurfaces). However, our emphasis will be to allow for very general classes of hypersurfaces S [subset]R3, not necessarily convex, whose Gaussian curvature may vanish on small, or even large subsets.

Given such a hypersurface S, one may ask in terms of which quantities one should describe the range of ps for which (1.1) holds true. It turns out that an extremely useful concept to answer this question is the notion of Newton polyhedron. The importance of this concept to various problems in analysis and related fields has been revealed by V.I. Arnol'd and his school, in particular through groundbreaking work by A. N. Varchenko [V76] and subsequent work by V. N. Karpushkin [K84] on estimates for oscillatory integrals, and came up again in the seminal article [PS97] by D. H. Phong and E. M. Stein on oscillatory integral operators.

Indeed, there is a close...