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This chapter presents three sets of problems and explains how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface S (here, a smooth, finite type hypersurface in R³ with Riemannian surface measure dσ‎). The first problem is a classical question about estimates for oscillatory integrals, and there exists a huge body of results on it, in particular for convex hypersurfaces. The other two problems had first been formulated by Stein: the study of maximal averages along hypersurfaces has been initiated in Stein's work on the spherical maximal function, and also the idea of Fourier restriction goes back to him.

This chapter discusses the Fourier restriction. 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. The chapter builds on previous discussions on Fourier restrictions, and presents a brief overview of the succeeding chapters.

This chapter showcases the remaining proofs of two results from the first chapter. It turns to the first of these results, based on a proposition made Chapter 1, on the characterization of linearly adapted coordinates. The chapter separately proves the two conditions discussed in the first proposition, before moving on to the next proposition. The second proposition obtained from Chapter 1 is about an invariant description of the notion of r-height. The chapter proves both parts of the proposition at the same time, which proves in the process an inequality from Chapter 9 before arriving at the proof of the proposition proper.

This chapter shows that one may reduce the desired Fourier restriction estimate to a piece Ssubscript Greek small letter psi of the surface S lying above a small, “horn-shaped” neighborhood Dsubscript Greek small letter psi of the principal root jet ψ‎, on which ∣x₂ − ψ‎(x₁)∣ ≤ ε‎xᵐ₁. Here, ε‎ > 0 can be chosen as small as one wishes. The proof then provides the opportunity to introduce some of the basic tools which will be applied frequently, such as dyadic domain decompositions, rescaling arguments based on the dilations associated to a given edge of the Newton polyhedron, in combination with Greenleaf's restriction and Littlewood–Paley theory, hence summing the estimates that have been obtained for the dyadic pieces.