 Fourier restriction for hypersurfaces in
Part II
Version 13.10.2014
Abstract.
This is the second in a serious of two articles, in which we prove a sharp  Fourier restriction theorem for a large class of smooth, finite type hypersurfaces in which includes in particular all realanalytic hypersurfaces.
Contents
 1 Introduction
 2 The case when : Reminder of the open cases
 3 Restriction estimates for the domain
 4 Spectral localization to frequency boxes where The case where not all ’s are comparable
 5 Interpolation arguments for the open cases where and or
 6 The case where
 7 Restriction estimates for the domains
 8 The remaining cases where and or Preliminaries
 9 The case where and
 10 The case where and
 11 The case where and What still needs to be done
 12 Proof of Proposition 11.3 (a) : Complex interpolation
 13 Proof of Proposition 11.3 (b) : Complex interpolation
 14 Appendix: Integral estimates of van der Corput type
1. Introduction
This is the second part of a pair of articles whose main goal is to prove our main result, Theorem 1.7, in [21] on  Fourier restriction estimates for smooth hypersurfaces of finite type in For the relevant statements, definitions and bibliographical references we therefore refer the reader to the introduction to that article. Under the assumption that our hypersurface is given as the graph of a smooth function defined near the origin and satisfying the conditions and we had covered in [21] all situations with the exception of the cases where the socalled linear height satisfies For this case, substantially more refined methods than the ones used in [21] are needed, since the use of Drury’s restriction estimate for nondegenerate curves turns out to be insufficient. In fact, the method that we shall develop in this second part will work whenever
Throughout this article, we shall make the following general
Assumption 1.1.
There is no linear coordinate system which is adapted to
Moreover, we may and shall assume that we are in linearly adapted coordinates, so that Recall also from [21] that this assumption implies that the principal face of the Newton polyhedron of is a compact edge which is intersected by the bisectrix
in an interior point, given by and that is contained in the principal line
and thus determines a weight so that also
Conventions: As in [21], we shall use the “variable constant” notation in this article, i.e., many constants appearing in the paper, often denoted by will typically have different values at different lines. Moreover, we shall use symbols such as or in order to avoid writing down constants. By we mean that there are constants such that and these constants will not depend on the relevant parameters arising in the context in which the quantities and appear. Similarly, by we mean that there is a (possibly large) constant such that and by we mean that there is a sufficiently small constant such that and again these constants do not depend on the relevant parameters.
By and we shall always denote smooth cutoff functions with compact support on where will be supported in a neighborhood of the origin, whereas will be support away from the origin in each of its coordinates i.e., for every These cutoff functions may also vary from line to line, and may in some instances, where several of such functions of different variables appear within the same formula, even designate different functions.
Also, if we speak of the slope of a line such as a supporting line to a Newton polyhedron, then we shall actually mean the modulus of the slope.
2. The case when : Reminder of the open cases
Recall from [21], Section 9, the following two Cases:

The principal face of the Newton polyhedron of is a compact edge, which lies on a line which we call the principal line of

is the vertex
What had remained open in [21] was the study of the piece of the surface corresponding to the domain containing the principal root jet in the cases (a) and (b), i.e.,
(2.1) 
when Indeed, we shall here develop an approach which will work whenever Our goal will thus be to prove the following extension of Proposition 12.1 in [21] to the case where
Proposition 2.1.
Assume that and that we are in Case (a) or (b). When is sufficiently small, and is sufficiently large in Case (a), then
whenever
In order to prove this proposition, we follow the domain decomposition algorithm for the domain developed in Section 12 of [21]. In Case (a), that algorithm led to a finite family of subdomains (socalled transition domains) and domains of the form
where the functions are of the form
with real coefficients , and where the exponents form a strictly increasing sequence
of rational numbers. Moreover, in the modified adapted coordinates given by
the function is given by
Notice that we can define these notions also for and then have and
Moreover, the domain is associated to an “edge” (which is indeed an edge, or can degenerate to a single point) of the Newton polyhedron of in the following way:
The edge with index will lie on a line
of slope (here, we shall always mean the modulus of the slope), where Introduce corresponding “dilations” by putting Then the domain
which represents the domain in the coordinates is invariant under these dilations, and the Newton diagram of the principal part of agrees with the edge
Recall also that the first edge agrees with the principal face of and lies on the principal line of the Newton polyhedron of and it intersects the bisectrix whereas for the edge will lie in the closed halfspace below the bisectrix.
Moreover, the Newton polyhedra of and of do agree in the closed halfspace above the bisectrix.
Now, in Section 8 of [21], setting we had distinguished between the cases where (Case 1), and (Case 2), and the case where (Case 3), and studied restriction estimates for the pieces of the surface corresponding to the domain
Only in Case 2 we had made use of the assumption so we can concentrate in the sequel on Case 2.
Notice also that our decomposition algorithm worked as well in Case (b), only that we had to skip the first step of the algorithm. We shall therefore first study the domain in Case (a), and in the last section describe the minor modifications needed to treat also the domains for which will then also cover Case (b) at the same time.
We can localize to the domain by means of a cutoff function
where Let us again fix a suitable smooth cutoff function on supported in an annulus such that the functions form a partition of unity. Here, denote the dilations associated to the weight In the original coordinates these correspond to the functions We then decompose the measure dyadically as
(2.2) 
where
Notice that by choosing the support of sufficiently small, we can choose as large as we need. It is also important to observe that this decomposition can essentially we achieved by means of a dyadic decomposition with respect to the variable which again allows to apply LittlewoodPaley theory (see [21]).
Moreover, changing to modified adapted coordinates in the integral defining and scaling by we find that
(2.3)  
where is given by
so that is a smooth function supported where (for some small ), whose derivatives are uniformly bounded and where
(2.4) 
with respect to the topology (and
In order to prove Proposition 2.1, we then still need to prove
Proposition 2.2.
Assume that that we are in Case 2, i.e., and and recall that When is sufficiently small and is sufficiently large, then for every
(2.5) 
where the constant is independent of
3. Restriction estimates for the domain
Let us assume that we are in Case (a), where the principal face is a compact edge. In the enumeration of edges of the Newton polyhedron associated to in Section 7 of [21], this edge corresponds to the index i.e.,
(3.1) 
The weight is here the principal weight from [21], and the line is the principal line of the Newton polyhedron of We then put
In particular, is the second coordinate of the point of intersection of the line
with the line and according to [21], display (1.11), is given by
(3.2) 
The domain that we have to study is then of the form
where Moreover,
(3.3) 
so that represents in the modified adapted coordinates
(3.4) 
compared to the adapted coordinates in which is represented by
Notice that the exponent may be noninteger (but rational), so that is in general only fractionally smooth, i.e., a smooth function of and some fractional power of only. The same applies to every with whereas is still smooth, i.e., when we express in our adapted coordinates, we still get a smooth function, whereas when we pass to modified adapted coordinates, we may only get fractionally smooth functions.
We shall write for the domain i.e.,
so that represents our domain in our modified adapted coordinates, in which is represented by
We assume that we are in Case 2, so that and
We choose minimal so that Since is homogeneous, the principal part of is then of the form (cf. (9.6) in [21])
(3.5) 
where is a homogeneous smooth function. Note that is rational, but not necessarily integer, since we are in modified adapted coordinates.
Observe also that this implies that we may write
(3.6) 
with smooth functions such that and
and smooth functions of which are either flat, or of finite type with smooth functions such that
For convenience, we shall also write when is flat, keeping in mind that in this case we may choose as large as we please (but ).
Notice that then, for consists of terms of degree strictly bigger than
Recall that the Newton diagram of the principal part is the line segment which must then contain a point with second coordinate given by It then follows easily that the following relations hold true:
Actually, since we even have
(3.7) 
As in [21], we define normalized measures corresponding to the by
where again is a smooth function with (for some small ) and is given by
for some Observe that
where
We write as by putting
(3.8) 
where is of the form
(3.9) 
with
(3.10) 
and is given by
(3.11) 
Recall that and that either and then is fixed (the type of the finite type function ), or and then we may assume that is as large as we please.
Observe that as that every is a power of
with positive exponents which are fixed rational numbers, except for those for which for which we may choose the exponents as large as we please.
Moreover, is a smooth function of all three arguments, and
(3.12) 
For sufficiently small, this implies in particular that when and
Assume we can prove that
(3.13) 
with independent of Then straightforward rescaling by means of the dilations leads to the estimate
(3.14) 
where (cf. 11.5) in [21]). So, our goal is to verify (3.13).
Observe also that the principal parts of and do agree.
Recall that and We shall often use the interpolation parameter Since, by definition, the second assumption implies
(3.15) 
We first derive some useful estimates from below for We put so that
(3.16) 
Note that is rational, but not necessarily entire. We next define
Let us also put and
and define accordingly, with replaced by
Lemma 3.1.

We have unless and In the latter case,

If and or and then
where the inequality is even strict unless and
Proof.
(a) The Newton polyhedron of is contained in the closed halfspace bounded from below by the principal line of which passes through the points and Moreover, it known that the principal line of is a supporting line to (this follows from Varchenko’s algorithm), and it has slope It is therefore parallel to the line passing through the points and and lies “above” (see Figure 1). Thus the second coordinate of the point of intersection of with is greater or equal to the second coordinate of the point of intersection of with so that
But, the point is determined by the equations and so that This shows that hence
Notice also that hence unless
So, assume that Then and the principal face of must be the edge (see Figure 1). Thus, if then clearly and And, if then we see that is the second coordinate of the point of intersection of with and thus
(cf. (3.2)).
(b) The inequality is equivalent to
so that the remaining statements are elementary to check. Q.E.D.
The following corollary is a straightforward consequence of the definition of and Lemma 3.1.
Corollary 3.2.

If and or and then unless and (where ).

If then unless where

If then unless and
Recall next that the complete phase corresponding to has the form
where
so that
4. Spectral localization to frequency boxes where
The case where not all ’s are comparable
Denote by the operator of convolution with where we recall that