A sharp bilinear cone restriction estimate

Thomas Wolff Department of Mathematics 253-37 Caltech Pasadena, Ca 91125 USA wolff@cco.caltech.edu

The purpose of this paper is to prove an essentially sharp L^{2} Fourier restriction estimate for light cones, of the type which is called bilinear in the recent literature.

Fix d ≥ 3, denote variables in ℝ^{d} by (\overline{x}, x_{d}) with \overline{x} ∈ ℝ^{d−1}, and let Γ = \{x : x_{d} = |\overline{x}| and 1 ≤ x_{d} ≤ 2\}. Let Γ_{1} and Γ_{2} be disjoint conical subsets, i.e.

Γ_{i} = \{x ∈ Γ : \frac{\overline{x}}{x_{d}} ∈ Ω_{i}\}

where Ω_{i} are disjoint closed subsets of the sphere S^{d−2}. Let f and g be two functions on Γ whose supports are contained in Γ_{1} and Γ_{2} respectively. We will prove the following estimate, where σ is surface measure on Γ, and f̂dσ is the ℝ^{d} Fourier transform:

\underline{Theorem 1} If p > 1 + \frac{2}{d} then

∥f̂dσĝdσ∥_{p} ≤ C_{p,Γ_{1},Γ_{2}}∥f∥_{2}∥g∥_{2}  (1)

Bilinear estimates of this general type have been used by several authors; see in particular [9]. The estimate (1) was formulated by Bourgain in [3], and it was proved in [3] when d = 3 and p > 2 − ϵ for some ϵ > 0, the case p = 2 being easier and implicit in [1]. Tao and Vargas [14] recently obtained the explicit range p > 2 − \frac{8}{121} when d = 3, and noted that one can also obtain a range p > 2 − ϵ_{0} in the four dimensional case. The range of p in Theorem 1 is known to be best possible when d = 3 except for the question of the endpoint - see [14], where the conjecture that (1) should hold for d = 3 and p ≥ \frac{5}{3} is attributed to Machedon and Klainerman - and is similarly best possible in higher dimensions; see the remarks in section 5 of the present paper.

Although Theorem 1 is sharp of its type in any dimension, it is more satisfactory in low dimensions, since when d is large the L^{2} norms on the right hand side of (1) are quite weak in comparison with other relevant norms and the exponent 1 + \frac{2}{d} is only a small improvement on the exponent 1 + \frac{2}{d−2} which follows from the Strichartz inequality. When d = 4, Theorem 1 implies (via a rescaling argument as in [15]) a statement analogous to a result of Barcelo [1] for the three dimensional case:

\underline{Corollary} When d = 4 the restriction of the Fourier transform to Γ defines a bounded operator from L^{p} to L^{p}(Γ) for any p < \frac{3}{2}.

The range of p here is again sharp. It should be pointed out that the geometric information needed for our results is simpler than what is likely to be needed either to solve the restriction problem for S^{2}, or to solve some of the other outstanding problems concerning the cone such as the multiplier problem and local smoothing, even in the 2+1dimensional case. On the other hand, there are very few hypersurfaces for which a sharp restriction theorem is known, and the approach below may be useful in connection with the sphere as well, insofar as it is possible to consider the sphere without first resolving the Kakeya problem.

As might be expected the proof of Theorem 1 uses Kakeya techniques related to Bourgain’s paper [2] and the now classical work of C. Fefferman and Cordoba. The necessary geometric information while not particularly deep is different from what has been used previously, and we prove what we need in section 1 below. In section 2 we discuss a lemma from [10], in section 3 we prove our main lemma (Lemma 3.5) and in section 4 we prove Theorem 1. In section 5 we prove the corollary and make some further related remarks. Finally, in an appendix we discuss the related question of mixed norm estimates for the restriction of the X-ray transform to the light rays. We prove an optimal local result (except for endpoint questions) in three and four dimensions and a partial result in higher dimensions. This is stated below as Theorem A.1.

We will use several ideas and lemmas from the previous work on the cone problem, e.g. from [3], [10] and [14]. Some aspects of the argument and also the fact that Theorem 1 should be an accessible result were suggested by the author’s recent paper [19].

List of notation Q(N): the cube in ℝ^{d} centered at the origin with side length N. |E|: measure or cardinality of the set E depending on the context.

χ_{E}: indicator function of E.

1. A property of light rays In this section we fix a suitable large constant B depending on the dimension d. A \underline{light ray} will mean a line in ℝ^{d} making a 45 degree angle with the plane x_{d} = 0.

We fix two disjoint conical sets Γ_{1} and Γ_{2} as described in the introduction and will say that a light ray is \underline{white} (resp. black) if its direction belongs to Γ_{1} (resp. Γ_{2}). Thus any white and black rays are transverse. We fix a small positive number ϵ.

Let δ > 0, and let W and B be sets respectively of white and black light rays with respective cardinalities m and n. For each white line W (or black line B) we associate to W (or B) the infinite cylinder whose axis is W (or B) and whose cross section radius is δ. We will denote these tubes by w and b. For each tube w (similarly b) we define

ϕ_{w}(x) = min(1, \frac{δ}{dist(x, w)})^{M}  (2)

where M is a large constant depending on ϵ. We assume that W (similarly B) is δseparated; by this we mean the following: if D is a disc in projective space with radius δ, then the tubes w whose axes belong to D have bounded overlap, i.e. no point belongs to more than B of them. We note this implies that the cardinality of lines in W which intersect a given compact set is bounded by a (negative) power of δ.

A μ-fold point is a point which belongs to at least μ white tubes, and a smooth μ-fold point is a point where the quantity

ΦW =^{def} ∑_{w∈W} ϕ_{w}

is at least equal to μ.

We fix a partition of Q(1) into pairwise disjoint δϵ-cubes; in this section we reserve the letter Q for these cubes (except for the standing notation Q(N) for cubes centered at the origin). In what follows we will be working with a relation ∼ between white or black tubes and the cubes Q. For any such relation we denote

nW(Q) = |{w ∈ W : w ∼ Q}|
nB(Q) = |{b ∈ B : b ∼ Q}|

If x is a point or E is a set contained in a cube Q then we will use the notation

w ∼ x (resp. w ∼ E)

to mean that w ∼ Q, where Q is the δϵ-cube containing x (resp. E), and we define

Φ̃W(x) = ∑_{w≁x^{w∈W}} ϕ_{w}(x)

Φ̃B(x) = ∑_{b≁x^{b∈B}} ϕ_{b}(x)

We also define (cf. [2]) a bush to be a set of tubes which are all the same color and which all pass through a common point p, and more generally an η-bush is a set of tubes which are all the same color and are all at distance < η from a common point p. We call any such point p a base point for the bush.

The purpose of this section is to prove the following lemma.

\underline{Lemma 1.1} Assume W and B are δ-separated. Then there is a relation ∼ between white or black tubes and δϵ-cubes Q so that the following hold, where C depends on d only; the implicit constants also depend on ϵ:

1. ∑_{Q} nW(Q) ≲ m(log \frac{1}{δ})^{5}.

2. ∑_{Q} nB(Q) ≲ n(log \frac{1}{δ})^{5}.

3. The δ-entropy of the set \{x ∈ Q(1) : Φ̃W(x) ≥ μ and ΦB(x) ≥ ν\} is ≲ δ^{−Cϵ} \frac{mn}{μ^{2}ν}.

4. The δ-entropy of the set \{x ∈ Q(1) : ΦW(x) ≥ μ and Φ̃B(x) ≥ ν\} is ≲ δ^{−Cϵ} \frac{mn}{μν^{2}} .

\underline{Remarks} 1. It is easy to see that the δ-entropy of the points which belong to μ white and ν black tubes can be as large as \frac{mn}{μν} - just take W and B to be bushes with a common basepoint and set μ = m, ν = n. Thus 3. gains a factor of μ over the “trivial” bound valid with Φ̃W replaced by ΦW. In the proof of Theorem 1, this factor will compensate for the factor appearing in Mockenhaupt’s estimate for the relevant square function, i.e. in Lemma 2.1 below. It is also important that the dependence on δ in 1. and 2. is only logarithmic, or more precisely that it does not involve the specific power δ^{−ϵ}. On the other hand the distinction between μ-fold points and smooth μ-fold points is purely technical - the functions ϕ_{b} are needed later on in order to estimate Schwartz tails.

2. It is natural to state Lemma 1.1 in the above manner, since only properties 1-4 of the relation ∼ will be used in the subsequent sections and not its exact definition. However, the relation will be constructed in an explicit and fairly simple way: roughly, arrange the white or black tubes into bushes, and define w ∼ Q if w belongs to a bush whose basepoint is in Q. This procedure together with the induction argument in section 4 below is a variant on the “two ends” argument in [17], [18].

Lemma 1.2 below is true because ϕ_{w} is essentially a rapidly decreasing sum of constants times characteristic functions of dilates of w; we leave the details to the reader. Lemma 1.3 is a geometrical fact; similar facts are used in various places in the literature, e.g. in [3] and [14].

\underline{Lemma 1.2} If x ∈ Q(1) is a smooth μ-fold point for the white tubes with μ ≥ δ^{B} then x is a basepoint for an η-bush (of white tubes) with cardinality ≳ (log \frac{1}{δ})^{−1}μ(\frac{η}{δ})^{M} for some η ≤ δ^{1−ϵ}. Conversely if C is a large fixed constant and x ∈ Q(1) is a basepoint for an η-bush with cardinality ≥ Cμ(\frac{η}{δ})^{M}, then x is a smooth μ-fold point.

\underline{Lemma 1.3} Let C ⊂ W be an η-bush with (say) η ≤ \sqrt{δ}, and let p be a basepoint for C. Define a set Ω by deleting from Q(1) the double of the δϵ-square Q containing p. Let b be any black tube. Then

∫_{Ω} ϕ_{b}ΦC ≲ δ^{−ϵ(d−2)}δ^{d}(\frac{η}{δ})^{2d−3}  (3)

\underline{Proof} First let b and w be a black and a white tube. For any λ ≤ 1 the set

{x ∈ Q(1) : ϕ_{b}(x) ≥ λ}

is contained in a tube with the same axis as b and with width about δλ^{− \frac{1}{M}}, and similarly with w. Since w and b are transverse we have the bound

|\{x ∈ Q(1) : min(ϕ_{b}(x), ϕ_{w}(x)) ≥ λ\}| ≲ (δλ^{− \frac{1}{M}})^{d}  (4)

Let Δ(b, w) be the quantity inf_{x∈Ω}(dist(x, b) + dist(x, w)) + δ. If λ is large compared with (\frac{δ}{Δ(b,w)})^{M} then the set in (4) does not intersect Ω. It follows therefore that

∫_{Ω} ϕ_{b}ϕ_{w} ≲ ∫_{Ω} min(ϕ_{b}, ϕ_{w})

≲ ∫_{0}^{( \frac{δ}{Δ(b,w)} )^{M}} (δλ^{− \frac{1}{M}})^{d}dλ
≲ δ^{d}(\frac{δ}{Δ(b, w)})^{M−d}  (5)

Now we prove the estimate (3) when η = δ. It is clear from (5) that the contribution to the left side from tubes w ∈ C such that Δ(b, w) ≳ δϵ is small. On the other hand let ρ be small compared with δϵ, and consider how many tubes w ∈ C there can be with Δ(b, w) ≤ ρ. The bush C is clearly contained in a Cδ-neighborhood of the portion of the light cone with origin at p which corresponds to the conical subset Γ_{1}. If b contains a certain point y which lies within ρ of Γ_{1} and is further than δϵ from p, then by transversality b must intersect Γ at a point within Cρ of y. Thus the number of tubes w with Δ(b, w) ≤ ρ is bounded by the δ-entropy of the set of lines in Γ which intersect a fixed Cρ-disc lying at distance further than δϵ from the vertex; equivalently, by the δ-entropy of a δ^{−ϵ}ρ-disc on

S^{d−2}, which is (\frac{δ^{−ϵ}ρ}{δ})^{d−2}. We conclude using (5) that there is a bound

∑(\frac{δ^{−ϵ}ρ}{δ})^{d−2}δ^{d}(\frac{δ}{ρ})^{M−d}

with the sum being over dyadic ρ ≥ δ. Thus we get the bound δ^{d−ϵ(d−2)} as claimed.

We now remove the restriction η = δ. If C is an η-bush then, for parameters ρ such that ρ ≥ η but ρ is small compared with ηϵ, the maximum number of η-separated lines

in C with Δ(b, w) ≤ ρ is bounded by (\frac{δ^{−ϵ}ρ}{η})^{d−2}; for this just apply the above argument replacing δ by η. The space of light rays is 2d − 3-dimensional, so any fixed light ray can be within η of at most (\frac{η}{δ})^{2d−3} δ-separated ones. It follows that for any ρ << ηϵ there are

≲ (\frac{η}{δ})^{2d−3}(\frac{δ^{−ϵ}(ρ+η)}{η})^{d−2} tubes w with Δ(b, w) ≤ ρ. We now apply (5) as above to bound the left side of (3) by

∑(\frac{η}{δ})^{2d−3}(\frac{δ^{−ϵ}(ρ + η)}{η})^{d−2}δ^{d}(\frac{δ}{ρ})^{M−d}

plus a negligible error, with the sum being over dyadic ρ ≥ δ. Estimate (3) follows from this. □

The following lemma is the main step in the argument. Essentially, it corresponds to Lemma 1.1 except that here we ignore the tails (they will be taken care of in the next lemma) and work with a fixed value of μ (hence the induction argument in the last part of the proof of Lemma 1.1 below).

\underline{Lemma 1.4} Given a value of μ_{0} we can partition W as

W = W_{g} ∪ W_{b}

where

1. W_{g} has no μ_{0}-fold points in Q(1), and

2. W_{b} = ∪_{i=1}^{R}C_{i} where each C_{i} is a bush with basepoint in Q(2) and R ≲ \frac{m}{μ_{0}} (log \frac{1}{δ})^{2}.

\underline{Proof} We fix a large enough constant C = C_{d} and then another large constant A. We will use a recursive argument. Accordingly, if W^{i} ⊂ W, then we let κ(W^{i}) be the maximum possible cardinality for a set of δ-separated μ_{0}- fold points for W^{i}. We have κ(W) ≲ δ^{−d} since all the tubes in W are contained in a fixed compact set.

Assume now that κ(W^{i}) = k. We will prove: W^{i} = W^{i+1} ∪ W_{b}^{i} where κ(W^{i+1}) ≤ \frac{k}{2}, and W_{b}^{i} is the union of ≲ A\frac{m}{μ_{0}} log \frac{1}{δ} δ-bushes.

Namely, let R_{i} be a set of δ-separated μ_{0}-fold points for W_{i} with maximum possible cardinality k. There are two cases.

(i) If k ≤ A\frac{m}{μ_{0}} log \frac{1}{δ} then we let W_{b}^{i} be all tubes w ∈ W^{i} such that dist(x, w) < δ for some x ∈ R_{i} and W^{i+1} = W^{i}\W_{b}^{i}. Evidently W_{b}^{i} is the union of ≲ A\frac{m}{μ_{0}} log \frac{1}{δ} δ-bushes; and κ(W^{i+1}) = 0 since any μ-fold point for W^{i} must lie within δ of some point of R_{i}.

(ii) If k > A\frac{m}{μ_{0}} log \frac{1}{δ} we choose A\frac{m}{μ_{0}} log \frac{1}{δ} points from R_{i} at random. We let W_{b}^{i} be the tubes w ∈ W^{i} such that dist(x, w) < δ for some x in the random sample, and W^{i+1} = W^{i}\W_{b}^{i}. Evidently W_{b}^{i} is the union of ≲ A\frac{m}{μ_{0}} log \frac{1}{δ} δ-bushes. We will show that with high probability κ(W^{i+1}) ≤ \frac{k}{2}.

For this, define for each w ∈ W^{i}

P (w) = k^{−1}|{x ∈ R_{i} : dist(w, x) < δ}|

Thus the probability that w is in W^{i+1} is at most

(1 − P (w))^{A \frac{m}{μ_{0}} log \frac{1}{δ}}

If P (w) ≥ C^{−1} \frac{μ_{0}}{m} it follows that the probability that w is in W^{i+1} is at most δ^{\frac{A}{C}}. If A is large enough then since the cardinality of the set of lines in W which intersect Q(2) is bounded by δ^{−B} it follows that with high probability no tubes with P (w) ≥ C^{−1} \frac{μ_{0}}{m} belong to W^{i+1}.

Now let R_{i+1} be a maximal set of δ-separated μ_{0}-fold points for W^{i+1}, and let R be a maximal 2δ-separated subset of R_{i+1}. Consider the quantity

∑_{w∈W^{i+1}} kP (w)  (6)

We have seen that with high probability (6) is less than \frac{k}{C} \frac{μ_{0}}{m} |W^{i+1}| ≤ \frac{k}{C} μ_{0}. On the other hand, we have

(6) = ∑_{x∈R_{i}} |{w ∈ W^{i+1} : dist(w, x) < δ}|
≥ μ_{0}|R|

The first line followed from the definition (6) by reversing the order of summation, and the second line then followed because every point in R is within δ of a point of R_{i} and no two points of R can be within δ of the same point of R_{i}. We conclude that with high probability |R| ≤ C^{−1}k. Since |R| and |R_{i}| are comparable it then follows that |R_{i+1}| ≤ \frac{k}{2}, as was to be shown.

We now proceed recursively. Let W^{0} = W and apply the preceding to express W^{0} = W_{b}^{0} ∪ W^{1}. Then apply the preceding to express W^{1} = W_{b}^{1} ∪ W^{2} and continue in this manner, stopping when we reach a situation where we are in case (i) above. Suppose we stop after T stages. Since κ(W^{i}) is initially ≲ δ^{−d} and decreases each time at least by a factor of 2, we then have T ≲ log \frac{1}{δ}. We now define W_{g} to be the set W^{i+1} defined at the last iteration. It satisfies κ(W_{g}) = 0 as required. On the other hand we define W_{b} = ∪_{i}W_{b}^{i}. This set is the union of the ≲ log \frac{1}{δ} sets W_{b}^{i}, each of which is the union of ≲ A\frac{m}{μ_{0}} log \frac{1}{δ} bushes. The lemma follows. □

The next lemma is a version of the preceding one incorporating Schwartz tails.

\underline{Lemma 1.5} Fix μ_{0} ≥ δ^{B}. Then W = W_{g} ∪ W_{b} where

1. Φ_{W_{g}} ≤ μ_{0} everywhere.

2. W_{b} = ∪_{k:2^{k}<δ^{−ϵ}}W_{b}^{k}, and for each k, W_{b}^{k} = ∪_{i=1}^{R_{k}}C_{i}, where C_{i} is a 2^{k}δ-bush with basepoint in Q(2) and R_{k} ≲ \frac{m}{2^{Mk}μ_{0}} (log \frac{1}{δ})^{3}.

\underline{Proof} Let w^{η} be the η-tube with the same axis as w. Notice that Lemma 1.3 is applicable also to the w^{η}’s (provided log η is comparable to log δ, which will be the case below), since we used δ-separation in the proof only to conclude that the cardinality of the white lines which intersect Q(2) was bounded by a negative power of δ.

We now define recursively a family of subsets W_{b}^{j}. Let W = W_{g}^{1} ∪ W_{b}^{1} be the decomposition from Lemma 1.3 for the given μ_{0}. If k ≥ 2 and if W_{b}^{j} have been defined for j < k then we let W_{g}^{k−1} = W\(∪_{j=1}^{k−1}W_{b}^{j}). The following inductive hypothesis will hold:

(∗) If j ≤ k − 1, then the family of tubes {w^{2^{j}δ} : w ∈ W_{g}^{k−1}} has no 2^{Mj}μ_{0}-fold points.

Let η = 2^{k}δ and apply Lemma 1.3 to the tubes {w^{η} : w ∈ W_{g}^{k−1}} replacing μ_{0} by 2^{Mk}μ_{0}. This decomposes W_{g}^{k−1} = W_{b}^{k} ∪ W_{g}^{k} where the tubes {w^{2^{k}δ} : w ∈ W_{g}^{k}} have no 2^{Mk}μ_{0}-fold points and W_{b}^{k} is the union of at most \frac{m}{2^{Mk}μ_{0}} (log \frac{1}{δ})^{2} 2^{k}δ-bushes. The inductive hypothesis (∗) is then satisfied for j ≤ k. We continue in this manner, stopping when 2^{Mk}μ_{0} becomes greater than m. This will occur at a stage k with 2^{k} < δ^{−ϵ}, since we have assumed μ_{0} ≥ δ^{B}. We define W_{g} to be the last W_{g}^{k}.

If Φ_{W_{g}}(x) ≥ (C log \frac{1}{δ})μ_{0} with C a large fixed constant then by Lemma 1.2 x must be

a 2^{Mk}μ_{0}-fold point for the tubes {w^{2^{k}δ} : w ∈ W_{g}} for some k, hence also a 2^{Mk}μ_{0}-fold point for the larger family {w^{2^{k}δ} : w ∈ W_{g}^{k}}, which is impossible by construction. The lemma now follows by replacing μ_{0} with (C log \frac{1}{δ})^{−1}μ_{0}. □

To prove Lemma 1.1 it suffices by symmetry to construct a relation between white tubes and δϵ-squares so that 1. and 3. hold. This will again be done recursively. A remark on terminology: in this argument, when we say that “C is a 2^{k}δ-bush” we mean that C is a 2^{k}δ-bush but not a 2^{k−1}δ-bush.

We apply Lemma 1.5 to W with μ_{0} = \frac{m}{2}, obtaining a set W_{g}^{1} with ΦW_{g}^{1} ≤ \frac{m}{2} and a

collection of stage 1 η-bushes C_{i}^{1} (thus each C_{i}^{1} is a 2^{k}δ-bush for some k with 2^{k}δ ≤ δ^{1−ϵ}). Then we apply Lemma 1.5 to W_{g}^{1} with μ_{0} = \frac{m}{4} obtaining W_{g}^{2} with ΦW_{g}^{2} ≤ \frac{m}{4} and stage

two η-bushes C_{i}^{2} and continue in this manner, taking μ_{0} = \frac{m}{2^{j}} at the jth stage. We stop the induction at stage R, where R by definition is the smallest integer such that \frac{m}{2^{R}} < δ^{B}.

Clearly R ≲ log \frac{1}{δ}. For each j_{0} ≤ R we now have a decomposition

W = W_{g}^{j_{0}} ∪ (∪_{j≤j_{0}} ∪_{i} C_{i}^{j})  (7)

where ΦW_{g}^{j_{0}} ≤ \frac{m}{2^{j_{0}}} , and (by part 2. of Lemma 1.5) we have the following:

For each j and k there are ≲ 2^{j}2^{−Mk}(log \frac{1}{δ})^{3} values of i such that C_{i}^{j} is a 2^{k}δ-bush.

For each C_{i}^{j} we fix a basepoint p_{i}^{j}. We now define the relation ∼:

\underline{Definition} A tube w and δϵ-square Q are related, w ∼ Q, if w belongs to an η-bush C_{i}^{j} such that p_{i}^{j} is in Q or one of its neighbors.

We show first that 1. holds. Suppose that C_{i}^{j} ⊂ W_{g}^{j−1} is a 2^{k}δ-bush. Then, using Lemma 1.2 and the fact that ΦW_{g}^{j−1} ≤ \frac{m}{2^{j−1}} , we get the following bound for the cardinality

of C_{i}^{j}:

|C_{i}^{j}| ≲ 2^{Mk}ΦC_{i}^{j}(p_{i}^{j}) ≲ 2^{Mk} \frac{m}{2^{j}}

By the preceding bound for the number of 2^{k}δ-bushes, we then have

∑_{i} |C_{i}^{j}| ≲ ∑_{k} 2^{j}2^{−Mk}(log \frac{1}{δ})^{3} · 2^{Mk} \frac{m}{2^{j}} ≲ m(log \frac{1}{δ})^{4}

Summing over j we get ∑_{i,j} |C_{i}^{j}| ≲ m(log \frac{1}{δ})^{5}. Thus, there are at most m(log \frac{1}{δ})^{5} pairs (w, C) where w is a white tube and C = C_{i}^{j} is an η-bush containing w. This obviously implies 1. It remains to prove 3.

Fix μ. If μ ≲ δ^{B} (and if B = B_{d} was chosen large enough) then 3. will clearly hold, since the right hand side will be greater than δ^{−d}. On the other hand, if μ is large compared with δ^{B} then we can choose j_{0} so that \frac{m}{2^{j_{0}}} is less than \frac{μ}{2} but greater than \frac{μ}{8}. We consider the decomposition (7) with this value of j_{0}. Thus ΦW_{g}^{j_{0}} ≤ \frac{μ}{2}, and, for each k we have

|\{(i, j) : j ≤ j_{0} and C_{i}^{j} is a 2^{k}δ − bush\}| ≲ 2^{−Mk}(log \frac{1}{δ})^{3} \frac{m}{μ}  (8)

Fix a black tube b, and fix also a choice of C_{i}^{j} with j ≤ j_{0}. Define Ω_{ij} by deleting from Q(1) the δϵ-square containing p_{i}^{j} and its neighbors. Lemma 1.3 implies that if C_{i}^{j} is a 2^{k}δ-bush then

∫_{Ω_{ij}} ϕ_{b}ΦC_{i}^{j} ≲ δ^{−(d−2)ϵ}2^{Ck}δ^{d}

where C depends on d.

Now sum over b, i and j ≤ j_{0} obtaining (provided M has been chosen large enough)

∑_{ij} ∫_{Ω_{ij}} ΦBΦC_{i}^{j} ≲ n ∑_{k} 2^{−Mk}(log \frac{1}{δ})^{3} \frac{m}{μ} · δ^{−(d−2)ϵ}2^{Ck}δ^{d} ≲ δ^{−(d−2)ϵ} \frac{nm}{μ}δ^{d}(log \frac{1}{δ})^{3}

where the first inequality followed from (8).

Suppose now that x is a point such that Φ̃W(x) ≥ μ. By the definition of the relation ∼ we have

Φ̃W(x) ≤ ΦW_{g}^{j_{0}}(x) + ∑_{x∈Ω_{ij}^{j≤j_{0}}} ΦC_{i}^{j}(x)

The first term on the right side is ≤ \frac{μ}{2}, so

Φ̃W(x) ≤ 2 ∑_{x∈Ω_{ij}^{j≤j_{0}}} ΦC_{i}^{j}(x)

whence

∫ ΦBΦ̃ ≤ 2 ∑_{ij} ∫_{Ω_{ij}} ΦBΦC_{i}^{j} ≲ δ^{−(d−2)ϵ} \frac{nm}{μ}δ^{d}(log \frac{1}{δ})^{3}

It follows that the measure of the set where ΦB ≥ ν and Φ̃ ≥ μ is

≲ δ^{−(d−2)ϵ} \frac{nm}{νμ^{2}} δ^{d}(log \frac{1}{δ})^{3}

Using that the functions ϕ_{w} are roughly constant on δ-discs it then follows that the δ-entropy is

≲ δ^{−(d−2)ϵ} \frac{nm}{νμ^{2}} (log \frac{1}{δ})^{3}

as claimed. □

What we actually use below is a slight variant on Lemma 1.1 where the infinite cylinders are replaced by finite ones. We introduce the following notation which will also be used in section 3.

\underline{Definition} 1. Suppose that g is a radial function in ℝ^{d} and R is a centered compact convex set. Then we use the notation g_{R} to mean g ∘ A, where A is an affine function mapping (the John ellipsoid for) R onto the unit ball.

2. ϕ will denote the function ϕ(x) = min(1, |x|^{−M}), where M is a sufficiently large constant.

Suppose now that we have collections B and W of cylinders of length 1 and cross section radius δ, which are δ-separated in the same sense as before; i.e. the ones whose direction belongs to a given δ-disc in projective space have bounded overlap, and furthermore the axis directions belong to Γ_{1} and Γ_{2} respectively. Let m = |W|, n = |B|. Fix (in addition to ϵ) another small positive η; the choice of M and the implicit constants below may now also depend on η. The quantities ΦW and Φ̃W are defined in the same way as before, except of course that we use the modified definiton of ϕ_{w} via the definition above.

\underline{Lemma 1.1′} With the above assumptions there is a relation ∼ between white or black tubes w ∈ W or b ∈ B and δϵ-cubes Q ⊂ Q(1) so that the following hold, where nW(Q) = |{w : w ∼ Q}|:

1. ∑_{Q} nW(Q) ≲ mδ^{−η}.

2. ∑_{Q} nB(Q) ≲ nδ^{−η}.

3. The δ-entropy of the set \{x ∈ Q(1) : Φ̃W(x) ≥ μ and ΦB(x) ≥ ν\} is ≲ δ^{−Cϵ} \frac{mn}{μ^{2}ν}.

4. The δ-entropy of the set \{x ∈ Q(1) : ΦW(x) ≥ μ and Φ̃B(x) ≥ ν\} is ≲ δ^{−Cϵ} \frac{mn}{μν^{2}} .

To prove this we define w ∼ Q if the infinite cylinder^{1} with the same axis as w is related to Q in the sense of Lemma 1.1 and if in addition the distance from w to the origin is less than δ^{− \frac{η}{2}} . Then 1. and 2. follow immediately from 1. and 2. of Lemma 1.1, and 3. and 4. follow from 3. and 4. of Lemma 1.1 using that the contribution to Φ from tubes further than δ^{− \frac{η}{2}} from the origin is negligibly small if M is large. □

2. A lemma of Mockenhaupt We cover the unit sphere S^{d−2} with a family of spherical caps c of radius N^{− \frac{1}{2}} with

bounded overlap; this gives also a covering of Γ by a family of “sectors” ρ = ρ_{c}, where

ρ_{c} = \{x ∈ Γ : \frac{\overline{x}}{x_{d}} ∈ c\}.

We will be using a variant on the square function estimate in [10]. To state it, let {ρ_{j}} be the sectors ρ which intersect Γ_{1} and let {ρ̃_{k}} be the sectors which intersect Γ_{2}. Let f and g be two functions on ℝ^{d} and assume that f = ∑_{j=1}^{μ} f_{j} and g = ∑_{k=1}^{ν} g_{k}, where suppf_{j} is contained in the N^{− \frac{1}{2}} -neighborhood of the sector ρ = ρ_{j}, and likewise suppg_{k} is contained in the N^{− \frac{1}{2}} -neighborhood of ρ̃_{k}. Let F = f̂, G = ĝ, and SF = (∑_{j} |f̂_{j}|^{2})^{\frac{1}{2}} SG = (∑_{k} |ĝ_{k}|^{2})^{\frac{1}{2}} .

1We allow the possibility that an infinite cylinder may contain several w’s. It is therefore easy to reduce to the case where the infinite cylinders are δ-separated.

\underline{Lemma 2.1} ∥F G∥_{2}^{2} ≲ min(μ, ν)∥(SF)(SG)∥_{2}^{2}.

\underline{Proof}[10] We claim that for a given point z ∈ ℝ^{d} there are ≲ min(μ, ν) pairs (j, k) such that z ∈ suppf_{j} + suppg_{k}.

We will use the following geometrically obvious fact (a consequence of the strict convexity of the sphere): let ϵ_{0} be a fixed positive constant and let ζ, ω_{1}, ω_{2} be points of S^{d−2} with |ω_{i} − ζ| ≥ ϵ_{0} for i = 1, 2. Let ℓ be a line in ℝ^{d−1} which passes through the point ζ and assume that both ω_{1} and ω_{2} are at distance at most δ from ℓ. Then |ω_{1} − ω_{2}| ≤ Cδ, where C depends on ϵ_{0}.

In order to prove the claim it suffices to show that for fixed j the set of k such that z ∈ suppf_{j} + suppg_{k} has bounded cardinality. To this end we fix ζ with (ζ, 1) ∈ ρ_{j}, and ω_{1} and ω_{2} such that (ω_{i}, 1) ∈ ρ̃_{k} and z ∈ suppf_{j} + suppg_{k_{i}} for i = 1, 2. If we let z = (w, t) then for suitable a, b ∈ [\frac{1}{2}, 2] we have

a + b = t + O(N^{− \frac{1}{2}} )
aω_{1} + bζ = w + O(N^{− \frac{1}{2}} )

and therefore

aω_{1} + (t − a)ζ = w + O(N^{− \frac{1}{2}} )

so that

ω_{1} − ζ = a^{−1}(w − tζ) + O(N^{− \frac{1}{2}} )  (9)

Estimate (9) says that the distance from ω_{1} to the line through ζ spanned by w − tζ is ≲ N^{− \frac{1}{2}} . Likewise the distance from ω_{2} to this line is ≲ N^{− \frac{1}{2}} . The disjoint conical support assumption implies that |ω_{i} − ζ| is bounded below for each i so we conclude that |ω_{1} − ω_{2}| ≤ CN^{− \frac{1}{2}} . This means that there are at most a bounded number of possible values for k, proving the claim.

The claim implies the lemma by a well-known calculation with the Plancherel theorem, which we omit. □

3. Main lemma It will be convenient to change the setup described in the introduction slightly in this

section. We fix a scale N, let Q(N) be the square centered at the origin with side N, and let Γ^{(N)} be the \frac{1}{N} -neighborhood of Γ_{1}; similarly Γ_{1}^{(N)} is the \frac{1}{N} -neighborhood of Γ_{1}, etc. Corresponding to the covering of Γ by sectors described in section 2 is a covering of Γ^{(N)} by \frac{1}{N} -neighborhoods of sectors, and in this section we use ρ to denote one of the latter.

Thus ρ is essentially a 1 × \overbrace{N^{− \frac{1}{2}} × … × N^{− \frac{1}{2}}}^{d−2 times} ×N^{−1}-rectangle. We fix disjoint sets E_{ρ} ⊂ ρ with ∪_{ρ}E_{ρ} = Γ^{(N)} and let ζ_{ρ} = χ_{E_{ρ}}.

Let f be a function supported on Γ_{1}^{(N)} with L^{2} norm 1, F = f̂, F_{ρ} = ζ̂_{ρ}f, and

SF (x) = (∑_{ρ} |F_{ρ}(x)|^{2})^{\frac{1}{2}}

Further let b be a fixed radial Schwartz function nonzero on Q(1) whose Fourier transform has compact support and whose ℤ^{d} translations form a partition of unity. For each

ρ we fix a tiling F^{ρ} of ℝ^{d} by rectangles σ with dimensions N × \overbrace{N^{\frac{1}{2}} × … × N^{\frac{1}{2}}}^{d−1 times}, the long direction being orthogonal to the light cone Γ at points of (the center line of) ρ, and we

let F = ∪_{ρ}F^{ρ}. We also let P^{ρ} be a tiling by N × \overbrace{N^{\frac{1}{2}} × … × N^{\frac{1}{2}}}^{d−2 times} ×1 rectangles dual to the sector ρ. For each ρ and each σ ∈ F^{ρ} we define F σ_{ρ} = bσF_{ρ}, where bσ (and also b_{π}, ϕσ, etc. in the subsequent argument) are as in the definition at the end of section 1; thus ∑σ F σ_{ρ} = F_{ρ}. For each (ρ, σ) we also further decompose F σ_{ρ} as ∑_{π∈P^{ρ}} F_{ρ}^{σ,π}, where F_{ρ}^{σ,π} = b_{π}F σ_{ρ}. The following fact (trivial to prove, since b has compact support) will be very important below:

\underline{Lemma 3.1} The inverse Fourier transforms of the functions F σ_{ρ} and F_{ρ}^{σ,π} are supported in a fixed dilate \overline{ρ} of ρ, and in particular are supported in the \frac{C}{N} -neighborhood of Γ.

The following fact is also clear from the Schwartz inequality since ∑_{σ∈F^{ρ}} ϕσ^{2} and ∑_{π∈P^{ρ}} ϕ_{π}^{2} are bounded for fixed ρ. Suppose that for each ρ a subset A^{ρ} ⊂ F^{ρ} is given. Then

∑_{ρ} | ∑_{σ∈A^{ρ}} F σ_{ρ}|^{2} ≲ ∑_{ρ} ∑_{σ∈A^{ρ}} |F σ_{ρ}|^{2}ϕσ^{−2} ≲ ∑_{π∈P^{ρ}^{σ∈A^{ρ}}^{ρ,σ,π}} |F_{ρ}^{σ,π}|^{2}ϕ_{π}^{−2}ϕσ^{−2}  (10)

The next two lemmas keep track of some relationships among the various decompositions of F which follow from orthogonality considerations and the uncertainty principle. We note the following: let π_{0} be a rectangle containing the origin, and let π be a translate of π_{0}. Then, the operator with kernel

K(x, y) = ϕ_{π}(x)^{−2}ϕ_{π_{0}}(x − y)^{100}ϕ_{π}(y)^{4}

maps L^{2} to L^{∞} with norm ≲ |π|^{\frac{1}{2}} , since one can easily show that ∫ |K(x, y)|^{2}dy ≲ |π| for fixed x.

\underline{Lemma 3.2} For fixed ρ and σ ∈ F^{ρ} we have ∑_{π} ∥ϕ_{π}^{−2}ϕσ^{−3}F_{ρ}^{σ,π}∥_{∞}^{2} ≲ N^{− \frac{d}{2}} ∥ϕσ^{−4}F σ_{ρ}∥_{2}^{2}.

\underline{Proof} Fix a Schwartz function κ whose Fourier transform is 1 on the unit ball and let \overline{κ} be the corresponding function whose Fourier transform is 1 on the set \overline{ρ} in Lemma 3.1, obtained from κ by composition with a linear map followed by multiplication by a character and by a scalar with magnitude about |ρ|. Then F_{ρ}^{σ,π} = \overline{κ} ∗ F_{ρ}^{σ,π}. Let σ_{0} and π_{0} be the rectangles in the tilings F^{ρ} and P^{ρ} which contain the origin. Then |\overline{κ}(z)| ≲ |ρ|ϕ_{π_{0}}(z)^{200} ≲ |ρ|ϕ_{π_{0}}(z)^{100}ϕ_{σ_{0}}(z)^{100}. We conclude that

|ϕ_{π}^{−2}(x)ϕσ^{−3}(x)F_{ρ}^{σ,π}(x)| ≲ ∫ K(x, y)|ϕ_{π}^{−4}(y)ϕσ^{−4}(y)F_{ρ}^{σ,π}(y)|dy

where

K(x, y) = |ρ|ϕ_{π}(x)^{−2}ϕσ(x)^{−3}ϕ_{π_{0}}(x − y)^{100}ϕ_{σ_{0}}(x − y)^{100}ϕ_{π}(y)^{4}ϕσ(y)^{4}
≲ |ρ|ϕ_{π}(x)^{−2}ϕ_{π_{0}}(x − y)^{100}ϕ_{π}(y)^{4}

We’ve seen that the norm of this kernel from L^{2} to L^{∞} is ≲ |π|^{\frac{1}{2}} |ρ| ≈ |ρ|^{\frac{1}{2}} ≈ N^{− \frac{d}{4}} . Accordingly

∑_{π} ∥ϕ_{π}^{−2}ϕσ^{−3}F_{ρ}^{σ,π}∥_{∞}^{2} ≲ N^{− \frac{d}{2}} ∑_{π} ∥ϕ_{π}^{−4}ϕσ^{−4}F_{ρ}^{σ,π}∥_{2}^{2}
= N^{− \frac{d}{2}} ∑_{π} ∥(ϕ_{π}^{−4}b_{π})ϕσ^{−4}F σ_{ρ}∥_{2}^{2}

and now we use that ∑_{π} |ϕ_{π}^{−4}b_{π}|^{2} ≲ 1 pointwise, obtaining the lemma. □

For each ρ and each σ ∈ F^{ρ} we define a parameter

h(σ) = (N^{− \frac{d+1}{2}} ∥ϕσ^{−4}F σ_{ρ}∥_{2}^{2})^{\frac{1}{2}}

We think of h(σ) as being essentially the L^{2} average of F_{ρ} on σ. We group the σ’s into families corresponding to the different possible dyadic values for h(σ); thus

F(h) = \{σ ∈ F : h(σ) ∈ [\frac{h}{2}, h]\}

and we define

F_{h} = ∑_{ρ} ∑_{σ∈F (h)∩F^{ρ}} F σ_{ρ}

\underline{Lemma 3.3} h^{2}|F(h)| ≲ N^{− \frac{d+1}{2}} .

\underline{Proof} Clearly

h^{2}|F(h)| ≲ N^{− \frac{d+1}{2}} ∑_{ρ} ∑_{σ∈F^{ρ}} ∥ϕσ^{−4}F σ_{ρ}∥_{2}^{2}

For fixed ρ we have ∑σ |bσϕσ^{−4}|^{2} ≲ 1 pointwise. So for fixed ρ we have ∑_{σ∈F^{ρ}} ∥ϕσ^{−4}F σ_{ρ}∥_{2}^{2} ≲ ∥F_{ρ}∥_{2}^{2}. If we sum over ρ and use orthogonality of the F_{ρ}’s the lemma follows. □

If g is a function supported on Γ_{2}^{(N)} with L^{2} norm 1 we will likewise denote ĝdσ by G, etc. Thus we obtain also functions G_{ρ}, Gσ_{ρ}, G_{ρ}^{σ,π}, G_{h}, and families of tubes G, G^{ρ}, G(h). The next lemma is a “local” estimate; it will then be combined with Lemma 1.1 to give the following Lemma 3.5 which is the main result of this section.

\underline{Lemma 3.4} Fix a square Q with side \sqrt{N}. Let F̃ and G̃ be subsets of F(h_{1}) and G(h_{2}) respectively and let μ and ν be the maximum values on the square Q of the functions ΦF^{}˜ and ΦG^{}˜ . Then

∫_{Q} |(∑_{ρ} ∑_{σ∈F^{}˜ ∩F^{ρ}} F σ_{ρ})(∑_{ρ_{2}} ∑_{σ_{2}∈G^{}˜ ∩G^{ρ_{2}}} Gσ_{2}_{ρ_{2}})|^{2} ≲ h_{1}^{2}h_{2}^{2}μν min(μ, ν)N^{\frac{d}{2}}  (11)

\underline{Proof} We subdivide F̃ and G̃ according to the possible dyadic values for ϕσ on Q. Thus we define

F̃(k) = {σ ∈ F̃ : min_{Q} ϕσ ∈ [2^{−(k+1)}, 2^{−k}]}

G̃(ℓ) = {σ ∈ G̃ : min_{Q} ϕσ ∈ [2^{−(ℓ+1)}, 2^{−ℓ}]}

We note that if σ ∈ F̃(k) then

∥ϕσϕ_{Q}∥_{∞} ≲ 2^{−k}  (12)

This follows from the rapid decay of ϕ and the fact that σ contains a translate of Q. Hence also ∥ϕσb_{Q}∥_{∞} ≲ 2^{−k}. Furthermore, from the definition of μ and ν, we have

|F̃(k)| ≲ 2^{k}μ and G̃(ℓ) ≲ 2^{ℓ}ν  (13)

The left side of (11) is ≲ ∑_{k=0}^{∞} ∑_{ℓ=0}^{∞} 2^{k+ℓ}A(k, ℓ), where

A(k, ℓ) = ∫ |(b_{Q}^{3} ∑_{ρ} ∑_{σ∈F^{}˜ (k)∩F^{ρ}} F σ_{ρ})(b_{Q}^{3} ∑_{ρ_{2}} ∑_{σ_{2}∈G^{}˜ (ℓ)∩G^{ρ_{2}}} Gσ_{2}_{ρ_{2}})|^{2}  (14)

Using Lemma 3.1 and that b̂ has compact support, one sees that the Fourier transform of the function b_{Q}^{3} ∑_{σ∈F^{}˜ (k)∩F^{ρ}} F σ_{ρ} is supported in the CN^{− \frac{1}{2}} -neighborhood of the sector ρ; and similarly with the second factor in (14). Lemma 2.1 is therefore applicable and implies that

A(k, ℓ) ≲ min(|F̃(k)|, |G̃(ℓ)|) ∫ ∑_{ρ} |b_{Q}^{3} ∑_{σ∈F^{}˜ (k)∩F^{ρ}} F σ_{ρ}|^{2} ∑_{ρ_{2}} |b_{Q}^{3} ∑_{σ_{2}∈G^{}˜ (ℓ)∩G^{ρ_{2}}} Gσ_{2}_{ρ_{2}}|^{2}

It follows by (10) that A(k, ℓ) is

≲ min(|F̃(k)|, |G̃(ℓ)|) ∫ b_{Q}^{12} ∑_{σ∈F (k)∩F^{ρ}^{ρ,σ,π}} ∑_{σ_{2}∈G(ℓ)∩G^{ρ_{2}}^{ρ_{2},σ_{2},π_{2}}} |F_{ρ}^{σ,π}|^{2}|G_{ρ_{2}}^{σ_{2},π_{2}}|^{2}ϕ_{π}^{−2}ϕσ^{−2}ϕ_{π_{2}}^{−2}ϕσ_{2}^{−2} (15)

We claim next that for each pair (π, π_{2}) we have

∫ b_{Q}^{4}ϕ_{π}^{2}ϕ_{π_{2}}^{2} ≲ N^{\frac{d−2}{2}}  (16)

Namely, π and π_{2} each have one “short” direction in which the width is 1, and these directions lie in Γ_{1} and Γ_{2} respectively, and are therefore transverse. It follows that π ∩π_{2} is contained within a bounded distance of a d − 2-plane, hence that

∫_{π∩π_{2}} b_{Q}^{4} ≲ N^{\frac{d−2}{2}}  (17)

Estimate (16) is just a version of (17) incorporating Schwartz tails, and is proved by estimating ϕ_{π}^{2} by an appropriate sum of constants times characteristic functions of translates of π (and similarly with ϕ_{π_{2}}^{2}) and then applying (17) to the terms in the resulting series.

We now consider the terms in the sum (15). For each pair (ρ, σ, π) and (ρ_{2}, σ_{2}, π_{2}) we have

∫ b_{Q}^{12}|F_{ρ}^{σ,π}|^{2}|G_{ρ_{2}}^{σ_{2},π_{2}}|^{2}ϕ_{π}^{−2}ϕσ^{−2}ϕ_{π_{2}}^{−2}ϕσ_{2}^{−2}

≲ ∥b_{Q}^{2}ϕ_{π}^{−2}ϕσ^{−1}F_{ρ}^{σ,π}∥_{∞}^{2}∥b_{Q}^{2}ϕ_{π_{2}}^{−2}ϕσ_{2}^{−1}G_{ρ_{2}}^{σ_{2},π_{2}}∥_{∞}^{2} ∫ b_{Q}^{4}ϕ_{π}^{2}ϕ_{π_{2}}^{2}
≲ N^{\frac{d−2}{2}} ∥b_{Q}^{2}ϕ_{π}^{−2}ϕσ^{−1}F_{ρ}^{σ,π}∥_{∞}^{2}∥b_{Q}^{2}ϕ_{π_{2}}^{−2}ϕσ_{2}^{−1}G_{ρ_{2}}^{σ_{2},π_{2}}∥_{∞}^{2}  (18)

by (16). It then follows that

∫ b_{Q}^{12}|F_{ρ}^{σ,π}|^{2}|G_{ρ_{2}}^{σ_{2},π_{2}}|^{2} ≲ N^{\frac{d−2}{2}} ∥ϕσ^{2}b_{Q}^{2}∥_{∞}^{2}∥ϕσ_{2}^{2}b_{Q}^{2}∥_{∞}^{2}∥ϕσ^{−3}ϕ_{π}^{−2}F_{ρ}^{σ,π}∥_{∞}^{2}∥ϕσ_{2}^{−3}ϕ_{π_{2}}^{−2}G_{ρ_{2}}^{σ_{2},π_{2}}∥_{∞}^{2}
≲ 2^{−4k−4ℓ}N^{\frac{d−2}{2}} ∥ϕσ^{−3}ϕ_{π}^{−2}F_{ρ}^{σ,π}∥_{∞}^{2}∥ϕσ_{2}^{−3}ϕ_{π_{2}}^{−2}G_{ρ_{2}}^{σ_{2},π_{2}}∥_{∞}^{2}  (19)

The first inequality followed from (18) by rearranging some factors, and the second inequality followed from (12).

Using (19) and Lemma 3.2 we may now bound (15) by

min(|F̃(k)|, |G̃(ℓ)|)2^{−4k−4ℓ}N^{− \frac{d+2}{2}} ∑_{ρ,ρ_{2}} ∑_{σ_{2}∈G^{}˜ (ℓ)∩G^{ρ_{2}}^{σ∈F^{}˜ (k)∩F^{ρ}}} ∥ϕσ^{−4}F σ_{ρ}∥_{2}^{2}∥ϕσ_{2}^{−4}Gσ_{2}_{ρ_{2}}∥_{2}^{2}

which by definition of h_{1} and h_{2} is

≲ min(|F̃(k)|, |G̃(ℓ)|)2^{−4k−4ℓ}N^{− \frac{d+2}{2}} · N^{d+1}|F̃(k)| |G̃(ℓ)|h_{1}^{2}h_{2}^{2}

We now use (13), and obtain a bound on (15) by

2^{−3k−3ℓ}μν min(2^{k}μ, 2^{ℓ}ν)N^{\frac{d}{2}} h_{1}^{2}h_{2}^{2}

Summing over k and ℓ gives the lemma. □

Fix ϵ > 0 and then η > 0 and partition Q(N) in nonoverlapping N^{1−ϵ}-squares; the letter R below will always denote one of these squares. We recall that f and g have L^{2} norm 1 and are supported on Γ_{1}^{(N)} and Γ_{2}^{(N)} respectively.

\underline{Lemma 3.5} On Q(N), for any h_{1} and h_{2} there are decompositions

F_{h_{1}} = F_{g} + F_{b} and F_{b} = ∑_{R} F_{b}^{R}
G_{h_{2}} = G_{g} + G_{b} and G_{b} = ∑_{R} G_{b}^{R}

where suppF_{b}^{R} ⊂ R, suppG_{b}^{R} ⊂ R, and the following estimates hold.

1. ∫_{Q(N)} |F_{g}G_{g}|^{2} + |F_{b}G_{g}|^{2} + |F_{g}G_{b}|^{2} ≲ N^{− \frac{d+2}{2} +Cϵ}

2. For each R we have F_{b}^{R} = α_{R}f̂_{R} and G_{b}^{R} = β_{R}ĝ_{R}, where α_{R} and β_{R} are supported on R and have L^{∞} norm ≤ 1, and f_{R} and g_{R} are supported on the N^{−(1−ϵ)}-neighborhoods of Γ_{1} and Γ_{2} respectively, and

∑_{R} ∥f_{R}∥_{2}^{2} + ∥g_{R}∥_{2}^{2} ≤ C_{η}N^{−ϵ+η}  (20)

\underline{Proof} Let W = F(h_{1}), B = G(h_{2}). We can assume that both h_{1} and h_{2} are greater than N^{−B_{1}} where B_{1} is a large dimension-dependent constant, since otherwise it is easy to check that the lemma is valid with F_{b} and G_{b} equal to zero. It follows that the cardinalities of W and B are bounded by N^{B_{2}}.

We apply Lemma 1.1′ after rescaling by N; thus δ in Lemma 1.1 is N^{− \frac{1}{2}} ; and we also set ϵ in Lemma 1.1′ equal to twice the present ϵ.

For each N^{1−ϵ}-square R we then define

F_{b}^{R} = \{ \begin{matrix}∑_{σ∼R^{σ∈W}} F σ_{ρ} & on R \\ 0 & elsewhere \end{matrix}

G_{b}^{R} = \{ \begin{matrix}∑_{σ∼R^{σ∈B}} Gσ_{ρ} & on R \\ 0 & elsewhere \end{matrix}

Define F_{b} to be equal to F_{b}^{R} on R for each R and similarly with G_{b}, and define F_{g} = F −F_{b}, G_{g} = G − G_{b}.

We will now show that

∫_{Q(N)} |F_{g}G|^{2} ≲ N^{− \frac{d+2}{2} +Cϵ}

Namely, fix a \sqrt{N}-square Q. Define μ to be the maximum on Q of ΦF^{}˜ , where F̃ is the tubes w ∈ F(h_{1}) such that w ≁ Q, and define ν to be the maximum on Q of Φ_{G(h_{2})}. By Lemma 3.4 we have

∫_{Q} |F_{g}G|^{2} ≲ h_{1}^{2}h_{2}^{2}μ^{2}νN^{\frac{d}{2}}

We now sum over Q and use 3. of Lemma 1.1. This gives

∫_{Q(N)} |F_{g}G|^{2} ≲ N^{Cϵ}h_{1}^{2}h_{2}^{2}N^{\frac{d}{2}} |F(h_{1})| |G(h_{2})|

which is ≲ N^{Cϵ} · N^{− \frac{d+2}{2}} by Lemma 3.3. We can clearly estimate ∫_{Q(N)} |F_{g}G|^{2} and ∫_{Q(N)} |F_{g}G_{g}|^{2} in the same way, and it follows that 1. holds.

We have the following almost orthogonality estimate:

∥ ∑_{σ∼R} F σ_{ρ}∥_{2}^{2} ≲ h_{1}^{2}N^{\frac{d+1}{2}} |\{σ : σ ∼ R\}|  (21)

Namely, for fixed ρ we have

∥ ∑_{σ∈F^{ρ}^{σ∼R}} F σ_{ρ}∥_{2}^{2} ≲ ∑_{σ∈F^{ρ}^{σ∼R}} ∥ϕσ^{−4}F σ_{ρ}∥_{2}^{2} ≲ h_{1}^{2}N^{\frac{d+1}{2}} |\{σ ∈ F^{ρ} : σ ∼ R\}|

where the first inequality follows from the Schwartz inequality since ∑σ ϕσ^{8} ≲ 1 pointwise and the second follows from the definition of h_{1}. Lemma 3.1 implies that the functions ∑_{σ∈F^{ρ}^{σ∼R}} F σ_{ρ} are essentially orthogonal for different ρ and (21) follows.

Using Lemma 3.1 again we see that, on each fixed N^{1−ϵ} square R, F_{b} = ∑_{σ∼R} F σ_{ρ} agrees with the Fourier transform of a function f_{R}^{0} supported on the \frac{C}{N} -neighborhood of Γ_{1}. We have

∑_{R} ∥f_{R}^{0}∥_{2}^{2} = ∑_{R} ∥ ∑_{σ∼R} F σ_{ρ}∥_{2}^{2} ≲ h_{1}^{2}N^{\frac{d+1}{2}} ∑_{R} |\{σ : σ ∼ R\}|
≲ h_{1}^{2}N^{\frac{d+1}{2}} |F(h_{1})|N^{η}
≲ N^{η}

The first inequality followed from (21), the second inequality followed from 1. of Lemma 1.1′ and the last inequality followed from Lemma 3.3. Now fix R and take a suitable Schwartz function κ supported in D(0, \frac{1}{2}) and whose Fourier transform is ≥ 1 on a large disc centered at the origin. Let κ_{R}(x) = e^{ik·x}N^{d(1−ϵ)}κ(N^{1−ϵ}x) for an appropriate k; if k is chosen correctly then κ̂_{R} ≥ 1 on R. Define α_{R} = \frac{1}{κ̂_{R}} on R and zero otherwise, and f_{R} = κ_{R} ∗ f_{R}^{0}. Then F_{b}^{R} = α_{R}f̂_{R}. To make the estimate (20) we will use the following fact, which follows from Schur’s test:

If s is a function supported in D(0, r) with ∥s∥_{∞} ≤ |D(0, r)|^{−1} and if suppf intersects every disc of radius r in measure ≤ γr, then ∥s ∗ f∥_{2} ≲ γ^{\frac{1}{2}} ∥f∥_{2}.

We apply this with s = κ_{R}, f = f_{R}^{0}, r ≈ N^{1−ϵ}, γ ≈ N^{−ϵ}, which is justified since f_{R}^{0} is supported on the \frac{C}{N} -neighborhood of Γ. It follows that ∥f_{R}∥_{2}^{2} ≲ N^{−ϵ}∥f_{R}^{0}∥_{2}^{2}, so we have the part of (20) which relates to f. We can of course treat g the same way, so the proof is complete. □

We note also that the L^{2} norms of F_{g}, F_{b}, G_{g} and G_{b} on Q(N) are all bounded by a constant; it suffices to prove this for F_{b} and G_{b}, and for them it follows from (20).

4. Proof of Theorem 1 We will use a lemma from the previous work:

\underline{Lemma 4.1} [3],[14] In order to prove Theorem 1 it suffices to prove that

∫_{Q(N)} |f̂dσĝdσ|^{p} ≲ N^{γ}∥f∥_{2}^{p}∥g∥_{2}^{p}

for fixed p > 1 + \frac{2}{d} and γ > 0.

This lemma originates in section 4 of [3], and the version stated above is a special case of Lemma 2.4 in part I of [14]. We also make a further reduction which follows by the uncertainty principle in the usual way: it suffices to prove that if f and g are functions with L^{2} norm 1 which are supported on the \frac{1}{N} -neighborhoods of Γ_{1} and Γ_{2} respectively, then

∫_{Q(N)} |f̂ĝ|^{p} ≲ N^{−p+γ}  (22)

if p > 1 + \frac{2}{d} and γ > 0.

The rest of this section is the proof of (22).

Fix p > 1 + \frac{2}{d} and let ϕ(N) be the supremum of the quantity

N^{p} ∫_{Q(N)} |f̂ĝ|^{p}  (23)

over functions f and g with L^{2} norm 1 which are supported in the \frac{1}{N} -neighborhoods of Γ_{1} and Γ_{2} respectively. Fix a sufficiently small ϵ and then a much smaller η; we will show that

ϕ(N) ≤ C(1 + N^{η}ϕ(N^{1−ϵ}))  (24)

for a suitable constant C.

Namely, choose f and g with L^{2} norm 1 so that the quantity (23) is essentially maximized. Then choose h_{1} and h_{2} using the pigeonhole principle so that

∫_{Q(N)} |F_{h_{1}}G_{h_{2}}|^{p} ≳ (log N)^{−2p}ϕ(N)

where F_{h_{1}} and G_{h_{2}} were defined in section 3. This is possible since it is easy to see that parameter values h which are less than a high negative power of N make a negligible contribution. Now apply Lemma 3.5 with this choice of h_{1} and h_{2}. With notation as in Lemma 3.5 we have (by the triangle inequality)

ϕ(N) ≲ (log N)^{2p} ∫_{Q(N)}(|F_{b}G_{g}|^{p} + |F_{g}G_{b}|^{p} + |F_{g}G_{g}|^{p}) + (log N)^{2p} ∑_{R} ∫_{R} |f̂_{R}ĝ_{R}|^{p}

In the first term, we estimate the L^{p} norm by the L^{1} and L^{2} norms using Holder’s inequality, and use that the L^{1} norms of F_{g}G_{b}, F_{b}G_{g} and F_{b}G_{b} are bounded by a constant by the remark at the end of section 3. In the second term, by definition of ϕ(N^{1−ϵ}), we can estimate the integral over a fixed R by

N^{−(1−ϵ)p}ϕ(N^{1−ϵ})∥f_{R}∥_{2}^{p}∥g_{R}∥_{2}^{p}

Making these estimates we conclude that

N^{−p}ϕ(N) ≲ (log N)^{2p} (∫_{Q(N)}(|F G_{g}|^{2} + |F_{g}G|^{2} + |F_{g}G_{g}|^{2}))^{p−1} +

+(log N)^{2p}N^{−(1−ϵ)p}ϕ(N^{1−ϵ}) ∑_{R} ∥f_{R}∥_{2}^{p}∥g_{R}∥_{2}^{p}

We now use Holder’s inequality on the sum over R and then insert the estimates in Lemma 3.5; this gives

N^{−p}ϕ(N) ≲ (log N)^{2p} (∫_{Q(N)}(|F G_{g}|^{2} + |F_{g}G|^{2} + |F_{g}G_{g}|^{2}))^{p−1}

+ (log N)^{2p}N^{−(1−ϵ)p}ϕ(N^{1−ϵ})(∑_{R} ∥f_{R}∥_{2}^{2})^{\frac{p}{2}} (∑_{R} ∥g_{R}∥_{2}^{2})^{\frac{p}{2}}

≲ (log N)^{2p}N^{(p−1)(Cϵ− \frac{d+2}{2} )} + (log N)^{2p}N^{−(1−ϵ)p}ϕ(N^{1−ϵ}) · N^{p(−ϵ+η)}

The assumption p > 1 + \frac{2}{d} implies that the exponent p − \frac{d+2}{2}(p − 1) is negative. We therefore obtain (24), since we can replace η by \frac{η}{p+1}, say.

If γ is given and if we take η sufficiently small then estimate (24) implies by an obvious induction that ϕ(N) ≲ N^{γ}; thus we have proved (22) and therefore Theorem 1. □

5. Further remarks We will now prove the corollary which was stated in the introduction. We first rephrase

it in a somewhat sharper form and in general dimensions. We will use mixed norms on Γ splitting the S^{d−2} and radial variables:

∥f∥_{L^{p}(L^{q})} =^{def} (∫_{S^{d−2}}(∫_{1}^{2} |f(tω)|^{q}dt)^{\frac{p}{q}} dω)^{\frac{1}{p}}

In the statement below, note that when d = 4 the condition on p reduces to p > 3; by duality we obtain a bound ∥f̂∥_{L^{p}(L^{2})} ≲ ∥f∥_{p} for any p < \frac{3}{2}, which clearly includes the result that was stated in the introduction. When d ≥ 5 the requirement that p be larger than 2 + \frac{4}{d} becomes significant so the statement becomes weaker.

\underline{Corollary 1} Assume that p > max(2 + \frac{4}{d}, 2 + \frac{2}{d−2}). Let f be a function on Γ. Then ∥f̂dσ∥_{p} ≤ C_{p}∥f∥_{L^{p}(L^{2})}.

\underline{Proof} This is the same as the proof of Theorem 2.2 in [15]; see also [14], where the rescaling maps for the cone employed below are used.

Fix a large number N and a spherical cap c ⊂ S^{d−2} centered at a point e ∈ S^{d−2} with

radius N^{−1}, i.e. c = \{ω ∈ S^{d−2} : |ω − e| < N^{−1}\}. Let Γ_{c} = \{x ∈ Γ : \frac{\overline{x}}{x_{d}} ∈ c\}. Define

T_{c} to be the linear map such that T (e, 1) = (e, 1), T (e, −1) = N^{2}(e, −1) and T y = Ny if y ∈ ℝ^{d} is orthogonal to (e, 1) and (e, −1). T_{c} maps light rays to light rays and has the following metric properties:

detT_{N} = N^{d}  (25)

and T_{c} expands the distance between any two light rays contained in c by a factor of roughly N, and roughly preserves distances on each individual such light ray.

Let c_{1} and c_{2} be two caps contained in c separated by an amount comparable to N^{−1} and let f and g be functions on Γ with L^{p}(L^{2}) norm 1 which are supported on Γ_{c_{1}} and Γ_{c_{2}} respectively. Define f̃dσ and g̃dσ to be the measures obtained by pushing forward fdσ and gdσ by the map T_{c}. Then f̃ and g̃ are functions on Γ whose conical supports are at least a constant distance apart, and their L^{p}(L^{2}) norms are comparable to N^{− \frac{d−2}{p′}}; hence their L^{2} norms are at most N^{− \frac{d−2}{p′}}. Furthermore we have the formulae

f̂dσ = ˆ f̃dσ ∘ T_{c}^{−1}
ĝdσ = ˆ g̃dσ ∘ T_{c}^{−1}

and therefore, by (25) and Theorem 1,

∫ |f̂dσĝdσ|^{\frac{p}{2}} = N^{d} ∫ |ˆf̃dσˆg̃dσ|^{\frac{p}{2}}
≲ N^{d−p \frac{d−2}{p′}} = N^{d−(p−1)(d−2)}

for any p > 2 + \frac{4}{d}. We now cover S^{d−2} with caps c_{j} of “width” N^{−1} as above and let f_{j} be functions on Γ with suppf_{j} ⊂ Γ_{c_{j}}. By applying the preceding estimate and summing over j we obtain

∑_{(j,k):dist(c_{j},c_{k})≈N^{−1}} ∫ |{ \atop χ_{j}fdσ}{ \atop χ_{k}fdσ}|^{\frac{p}{2}} ≲ N^{d−(p−1)(d−2)} ∑_{j} ∥f_{j}∥_{p}^{p}

The exponent of N is negative if p > 2 + \frac{2}{d−2}. The result now follows exactly as in [15], since the supports of the Fourier transforms of the functions { \atop χ_{j}fdσ}{ \atop χ_{k}fdσ} have finite overlap if N is fixed and dist(c_{j}, c_{k}) ≈ N^{−1}. □

We now consider the Mockenhaupt square function

SF (x) = (∑_{ρ} |F_{ρ}|^{2})^{\frac{1}{2}}

where F = f̂dσ with f supported on Γ, f = ∑_{ρ} f_{ρ} with f_{ρ} supported in the sector ρ of width about N^{− \frac{1}{2}} and F_{ρ} = { \atop f_{ρ}dσ}. The following simple result appears natural in higher dimensions where the expected critical exponent is 2 + \frac{2}{d−2}; we do not consider the question of L^{4}(ℝ^{3}) estimates except to note that Theorem 1 can of course be substituted into the numerology in [14].

In addition, we consider a “weaker” square function S̃ defined as follows: let F = f̂dσ be as above, let Δ run through a covering of Γ by discs of radius N^{− \frac{1}{2}} , suppose that f_{Δ} is supported in Δ and f = ∑_{Δ} f_{Δ}, F_{Δ} = { \atop f_{Δ}dσ} and

S̃F = (∑_{Δ} |F_{Δ}|^{2})^{\frac{1}{2}}

\underline{Corollary 2} If 2 ≤ p ≤ 2 + \frac{4}{d} then there is an estimate

∥F ∥_{p} ≲ N^{( \frac{1}{2} − \frac{1}{p}) \frac{d−2}{4} +ϵ}∥SF ∥_{p}  (26)

for any ϵ > 0.

\underline{Proof} We consider first the “bilinear” version; in this version, one can prove a stronger result where S̃F replaces SF. Thus we let f and g as in Theorem 1 and F = f̂dσ, G = ĝdσ, and will show that

∥F G∥_{\frac{p}{2}} ≲ N^{( \frac{1}{2} − \frac{1}{p}) \frac{d−2}{2} +ϵ}(∥S̃F ∥_{p}^{2} + ∥S̃G∥_{p}^{2})  (27)

Namely, we have

∥F G∥_{L^{\frac{p}{2}} (Q)} ≲ N^{− \frac{1}{2} +ϵ}∥b_{Q}F ∥_{2}∥b_{Q}G∥_{2}

when p > 2+ \frac{4}{d}. This follows by applying (22) (with N replaced by N^{\frac{1}{2}} and Q(N) replaced by Q) to the functions b_{Q}F and b_{Q}G. By interpolation with L^{2} there is also an estimate

∥F G∥_{L^{\frac{p}{2}} (Q)} ≲ N^{− \frac{d+2}{2} ( \frac{1}{2} − \frac{1}{p})+ϵ}∥b_{Q}F ∥_{2}∥b_{Q}G∥_{2}

when 2 ≤ p ≤ 2 + \frac{4}{d}. The b_{Q}F_{Δ}’s are essentially orthogonal (their Fourier supports are essentially disjoint) so we can estimate ∥b_{Q}F ∥_{2} by ∥b_{Q}S̃F ∥_{2}; using this and then Holder’s inequality we obtain

∥F G∥_{L^{\frac{p}{2}} (Q)} ≲ N^{− \frac{d+2}{2} ( \frac{1}{2} − \frac{1}{p})+ϵ}∥b_{Q}S̃F ∥_{2}∥b_{Q}S̃G∥_{2}
≲ N^{− \frac{d+2}{2} ( \frac{1}{2} − \frac{1}{p})+ϵ} · N^{\frac{d}{2} (1− \frac{2}{p} +ϵ)}∥b_{Q}S̃F ∥_{p}∥b_{Q}S̃G∥_{p}
≤ N^{( \frac{1}{2} − \frac{1}{p}) \frac{d−2}{2} +ϵ}(∥b_{Q}S̃F ∥_{p}^{2} + ∥b_{Q}S̃G∥_{p}^{2})

Now take an ℓ^{\frac{p}{2}} sum over Q. Using the rapid decay of b we obtain (27).

The same argument clearly applies to S, so we also have

∥F G∥_{\frac{p}{2}} ≲ N^{( \frac{1}{2} − \frac{1}{p}) \frac{d−2}{2} +ϵ}(∥SF ∥_{p}^{2} + ∥SG∥_{p}^{2})  (28)

In the case of S, since the maps T_{c} essentially take sectors contained in Γ_{c} to sectors one can pass from the estimate (28) to the “linear” one (i.e. (26)) by rescaling, just as in [14] or in the proof of Corollary 1. □

\underline{Further remarks} 1. It will be clear to the experts that one could also obtain a partial result on the (higher dimensional) cone multiplier/local smoothing problem using the estimate (27) together with the usual technology as discussed for example in [10] and an estimate for a Nikodym type light ray maximal function, followed by another rescaling argument to pass from the bilinear to the linear estimate. We do not present this here because the estimate we have at present for the maximal function is rather crude.

2. Let p_{d} = 2 + \frac{2}{d−2}. It is natural to ask the following question: is there an estimate

∥f̂∥_{L^{p′}(L^{p})} ≲ ∥f∥_{p′}  (29)

provided p > p_{d}. One could also weaken this by asking instead for the estimate (p ≥ p_{d})

∀ϵ∃Cϵ : ∥f̂∥_{L^{p′}(L^{p})} ≤ Cϵλϵ∥f∥_{p′}  (30)

if suppf ⊂ D(0, λ), λ ≥ 1.

This statement would easily imply the restriction conjecture for the sphere S^{d−2}. Namely, suppose that f ∈ L^{p′}(ℝ^{d−1}) with p′ as above and that f is supported in Q(N), and apply (29) to the function f(\overline{x})e^{2πix_{d}}ϕ(\frac{x_{d}}{N}) where ϕ is a suitable bump function. (if one assumes instead (30) then this argument still works using Tao’s ϵ-removal lemma, see [14] for example.) Of course (29) would also solve the cone restriction problem, so it appears to be a natural common generalization.

The statements (29) or (30) are also related to several other conjectures in the literature. For example, (30) may be seen to be weaker than the “Radon transform” conjecture in [13], and is therefore also weaker than the so-called local smoothing conjecture [11]. We sketch the argument as follows: let Rf be the Radon transform of f restricted to the planes orthogonal to light rays as discussed in [13]; we will use the notation of that paper. Observe that the partial Fourier transform of Rf in the s variable can be identified with the restriction of f̂ to the cone. Because of this, a rescaling argument followed by an application of the Hausdorff-Young theorem in the s variable shows, assuming [13], formula (33) (and that p′ ≤ 2!), that if suppf ⊂ D(0, λ) ⊂ ℝ^{d} then

∥f̂∥_{L^{p′}(L^{p})} ≲ λ^{\frac{d−1}{p} +α}∥f∥_{p′}

Thus if [13], (33) were true for all α > −\frac{d−1}{p} as is conjectured in [13] then it would follow that (30) holds.

In the four dimensional case, estimate (29) is superficially similar to Corollary 1, the difference being that the radial dependence is now L^{p} instead of L^{2}, but since it would imply the restriction conjecture for S^{2} it should not be accessible using only “soft” Kakeya information like our Lemma 1.1.

3. We indicate why the exponent 1 + \frac{2}{d} in Theorem 1 is sharp. When d = 3 this is pointed out in [14] and the idea is the same in general dimensions - namely, to use a surface which has two families of rulings, with each ruling being a light ray. It suffices to take the doubly ruled surface to be a plane Π.

Let ρ_{i}, i = 1, 2 be sectors of Γ of width N^{− \frac{1}{2}} as above. Let ℓ_{i} be the center line of ρ_{i} and ν_{i} the normal to Γ along ℓ_{i}. Then ν_{1} and ν_{2} span a certain 2-plane Π through the origin. Let R be the intersection of the cube Q(N) with the \sqrt{N}-neighborhood of Π. One can think of R as the union of rectangles dual to the \frac{1}{N} -neighborhood of ρ_{i} for either value i = 1 or 2. For either i one can therefore construct in a standard way a function f_{i} supported in ρ_{i} with L^{2} norm 1, such that |f̂_{i}dσ|^{2} ≳ N · |R|^{−1} on, say, the dilation of R by a small factor ϵ_{0}. Here |R| ≈ N^{\frac{d+2}{2}} . It follows that for any p one has

∫ |{ \atop f_{1}dσ}{ \atop f_{2}dσ}|^{p} ≳ N^{− \frac{d}{2} p}N^{\frac{d+2}{2}}

This approaches ∞ as N → ∞ if p < 1 + \frac{2}{d}, which shows that (1) cannot hold when p < 1 + \frac{2}{d}.

Appendix: Estimates for the restricted X-ray transform

The motivation for this appendix was to clarify the relationship between Lemma 1.1 and other approaches that have been taken to the restriction of the X-ray transform to the light rays - see for example [3], [6], [7], [8], [14] and [16]. This leads to a family of mixed norm estimates which we formulate as Theorem A.1 below.

Let L be the space of light rays with the integral defined by

∫_{L} f(ℓ)dℓ = ∫_{S^{d−2}} ∫_{Y (ω)} f(ℓ(y, ω))dydω

Here ℓ(y, ω) is the line through y with direction (ω, 1), and Y (ω) is the hyperplane perpendicular to ℓ(0, ω). We define mixed norms on G by

∥f∥_{L^{q}(L^{r})} = (∫_{S^{d−2}}(∫_{Y (ω)} |f(ℓ(y, ω))|^{r}dy)^{\frac{q}{r}} dω)^{\frac{1}{q}}

We define the X-ray transform as an operator from functions on ℝ^{d} to functions on L via

Xf(ℓ) = ∫_{ℓ} f

and will be interested in estimates for X from L^{p} to L^{q}(L^{r}).

We first discuss necessary conditions in order to formulate a plausible conjecture; we omit details here. Suppose that X is bounded from L^{p} to L^{q}(L^{r}). Then dilations give the condition

\frac{d}{p} − \frac{d − 1}{r} = 1  (31)

See e.g. [4] and [7]. Furthermore, the maps T_{c} used in section 5 give the condition

\frac{d − 2}{q} ≥ \frac{d}{p} − \frac{d}{r}  (32)

Again see [7]. Another condition can be obtained by considering the example f = χ_{E} where E is the δ-neighborhood of the cone segment Γ. This takes the form

\frac{1}{p} ≤ \frac{d}{2r}  (33)

It is natural to expect that (31), (32), (33) are essentially also sufficient for boundedness. We will not consider endpoint questions and will therefore work locally. Index juggling leads to the following

\underline{Plausible conjecture}: Let p = q = \frac{d^{2}−2d+2}{d} and r = \frac{d^{2}−2d+2}{2}. Then X is bounded from the Sobolev space W^{p,ϵ}(Q(1)) to L^{q}(L^{r}) for any ϵ > 0.

By W^{p,ϵ}(Q(1)) we mean functions supported in Q(1) with ∥f∥_{p,ϵ} =^{def} ∥(1 − Δ)\frac{ϵ}{4} f∥_{p} < ∞.

There is an obvious bound on L^{1}, namely, by Fubini’s theorem

∥Xf∥_{L^{∞}(L^{1})} = ∥f∥_{1}  (34)

Interpolating (34) with the preceding conjecture we obtain the following conjectural bound on L^{p}.

\underline{Plausible conjecture_{p}}: Assume that 1 ≤ p ≤ \frac{d^{2}−2d+2}{d}. Define r via \frac{d}{p} − \frac{d−1}{r} = 1 and q via \frac{d−2}{q} = \frac{d}{p} − \frac{d}{r}. Then X is bounded from W^{p,ϵ}(Q(1)) to L^{q}(L^{r}) for any ϵ > 0.

This would imply all local W^{p,ϵ} → L^{q}(L^{r}) estimates with the given p which are not ruled out by (31) (in the local form where ≤ replaces =), (32) and (33).

We will prove the following:

\underline{Theorem A.1} If d = 3 or d = 4 then the above conjectures are true. If d ≥ 5 then the second conjecture is true on L^{p} provided p ≤ \frac{d+1}{2}.

\underline{Remarks} 1. We note that q and r coincide when p = \frac{d}{2}, q = r = d − 1, and that this case is covered by our result. This is new except when d = 3 (see below); it is analogous to the result of Drury [5] (see also [12] and [4]) for the full X-ray transform.

2. Consider the case d = 3. In this case, the angular parameter ω runs over a one dimensional space and the restricted X-ray transform as defined here is a special case of the restricted X-ray transform associated to a “rigid line complex” [6], [7]. If d = 3 and q = r, then the estimate in Theorem A.1 is an estimate from W^{\frac{3}{2} ,ϵ} to L^{2}. The latter estimate is known, actually in the sharper form where ϵ = 0 - cf. [16] (I thank Allan Greenleaf for this reference) and [6] - and a dual formulation of this same estimate is used in [14]. However, Theorem A.1 is new also in the three dimensional case if p > \frac{3}{2}.

3. It may be possible to obtain a scale invariant result (i.e. ϵ = 0) by modifying the argument below, at least if one assumes strict inequality in (32) and (33) and ignores the three dimensional case, but we do not attempt that here because the formulation of Lemma 1.1 in the body of the paper is unsuitable for that purpose. We note though that our estimate on W^{pϵ} can immediately be “upgraded” to a (local, of course) estimate on L^{p} provided one assumes strict inequality in (31), (32), (33). This is because one can interpolate with the known fact that X is bounded from a negative order L^{2} Sobolev space to L^{2}. We leave details to the reader.

4. A proof of the above conjectures for the full range of p in general dimensions has to be hard, since this would include a version of the Kakeya conjecture. Namely, if the first conjecture is true in ℝ^{d}, then a Kakeya set in ℝ^{d−1} must have Minkowski dimension at least d + \frac{4}{d} − 3, as may be seen by applying the restricted X-ray bound to the indicator function of a cylinder over the δ-neighborhood of the Kakeya set. From this and known arguments (namely the subadditivity of the minimal possible Minkowski dimension for a Kakeya set in ℝ^{n} as a function of n) follows that the first conjecture if true in all dimensions would imply that Kakeya sets have full Minkowski dimension.

We will need the following numerical inequalities (trivial in principle, but we give proofs for the reader’s convenience). Here θ ∈ [\frac{1}{2}, 1] (we emphasize that θ ≥ \frac{1}{2}) and the variables x, y, a, b, a_{j}, b_{k} are nonnegative real numbers.

min(ax, by)^{θ} max(ax, by)^{1−θ} ≤ min(x, y)^{θ} max(x, y)^{1−θ} max(a, b)^{θ} min(a, b)^{1−θ} (35)
min(∑_{j} a_{j}, ∑_{k} b_{k})^{θ} max(∑_{j} a_{j}, ∑_{k} b_{k})^{1−θ} ≤ ∑_{j,k} min(a_{j}, b_{k})^{θ} max(a_{j}, b_{k})^{1−θ}  (36)

\underline{Proofs} For (35) we may assume that x ≤ y. If also ax ≤ by, then

min(ax, by)^{θ} max(ax, by)^{1−θ} = a^{θ}b^{1−θ} min(x, y)^{θ} max(x, y)^{1−θ}

and (35) follows. If ax ≥ by then

min(ax, by)^{θ} max(ax, by)^{1−θ} = (\frac{by}{ax})^{2θ−1}a^{θ}x^{θ}b^{1−θ}y^{1−θ}
≤ a^{θ}x^{θ}b^{1−θ}y^{1−θ}
= min(x, y)^{θ} max(x, y)^{1−θ} max(a, b)^{θ} min(a, b)^{1−θ}

since a ≥ b.

For (36) we can assume ∑_{j} a_{j} ≤ ∑_{k} b_{k} = 1. In fact, we can assume in addition that ∑_{j} a_{j} = 1. This follows from (35): let t = ∑_{j} a_{j} and consider the effect of replacing a_{j} by t^{−1}a_{j}. The left side of (36) increases by a factor of t^{−θ}, and (35) implies the right side increases by at most this much.

The right side of (36) is smallest if θ = 1 so we are reduced to proving that ∑_{j} a_{j} = ∑_{k} b_{k} = 1 implies ∑_{j,k} min(a_{j}, b_{k}) ≥ 1. But

∑_{j} ∑_{k} min(a_{j}, b_{k}) ≥ ∑_{j} min(a_{j}, ∑_{k} b_{k}) ≥ min(∑_{j} a_{j}, ∑_{k} b_{k})

so we are done. □

We start the proof of Theorem A.1 by giving a convenient restatement of Lemma 1.1; this differs from Lemma 1.1 only in that the Schwartz tails have been discarded and entropy replaced by measure, and is therefore an immediate corollary of Lemma 1.1.

Let W and B be δ-separated sets of white and black δ-tubes (thus they satisfy the transversality assumptions); assume each tube intersects the unit square. We let ∼ be the relation in Lemma 1.1 and will use the notation w ∼ x and nW(Q) defined there. Let

ΦW(x) = ∑_{w∈W} χ_{w}(x), ΦB(x) = ∑_{b∈B} χ_{b}(x)

ΦW^{∗}(x) = ∑_{w∼x^{w∈W}} χ_{w}(x), ΦB^{∗}(x) = ∑_{b∼x^{b∈B}} χ_{b}(x)

Φ̃W = ΦW − ΦW^{∗}, Φ̃B(x) = ΦB − ΦB^{∗}

\underline{Lemma A.1} The following hold, where C depends on d only; the implicit constants also depend on ϵ, and Q runs over a partition of Q(1) into δϵ-squares:

1. ∑_{Q} nW(Q) ≲ |W|(log \frac{1}{δ})^{5}.

2. ∑_{Q} nB(Q) ≲ |B|(log \frac{1}{δ})^{5}.

3. |\{x ∈ Q(1) : Φ̃W(x) ≥ μ and ΦB(x) ≥ ν\}| ≲ δ^{−Cϵ} \frac{|W| |B|}{μ^{2}ν}δ^{d}.

4. |\{x ∈ Q(1) : Φ̃B(x) ≥ ν and ΦW(x) ≥ μ\}| ≲ δ^{−Cϵ} \frac{|W| |B|}{μν^{2}}δ^{d}. □

The rough idea now is to regard 3. and 4. of Lemma A.1 as a “virtual” L^{\frac{3}{2}} to L^{3} estimate and to interpolate between this and an L^{1} to L^{1} estimate, namely the following:

\underline{Lemma A.2} |\{x ∈ Q(1) : ΦW(x) ≥ μ and ΦB(x) ≥ ν\}| ≲ δ^{d−1} min(\frac{|W|}{μ}, \frac{|B|}{ν}).

\underline{Proof} It is clear that ∥ ∑_{w∈W} χ_{w}∥_{L^{1}(Q(1))} ≲ |W|δ^{d−1}, hence the measure of the μ-fold points is ≲ \frac{|W|}{μ}δ^{δ−1}, which implies the lemma. □

Fix θ ∈ [\frac{1}{2}, 1] and define Ψ_{θ} = min(ΦB, ΦW)^{θ} max(ΦB, ΦW)^{1−θ}

S_{θ} = min(ΦB^{∗}, ΦW^{∗})^{θ} max(ΦB^{∗}, ΦW^{∗})^{1−θ}

T_{θ} = (Φ̃B)^{θ}ΦW^{1−θ} + (Φ̃W)^{θ}ΦB^{1−θ}

We will use below that

Ψ_{θ} ≲ S_{θ} + T_{θ}  (37)

This is a consequence of the numerical inequality min(a+b, c+d)^{θ} max(a+b, c+d)^{1−θ} ≲ a^{θ}(c+d)^{1−θ} +c^{θ}(a+b)^{1−θ} +min(b, d)^{θ} max(b, d)^{1−θ} which follows for example from (36).

We now estimate T_{θ} for appropriate θ by interpolation between Lemmas A.1 and A.2. \underline{Lemma A.3} Let p and q satisfy 1 ≤ q ≤ 3 and \frac{1}{q} ≥ \frac{2}{p} − 1. Let θ = \frac{1}{4}(3 − \frac{1}{q}). Then

∥δ^{d−2}T_{θ}∥_{L^{q}(Q(1))}^{q} ≲ δ^{−Cϵ}(δ^{2d−3}|B| · δ^{2d−3}|W|)^{\frac{q}{2p}}  (38)

\underline{Proof} It suffices to consider the case where \frac{1}{q} = \frac{2}{p} − 1 since the δ-separation implies that the quantity (δ^{2d−3}|B| · δ^{2d−3}|W|) is ≲ 1.

Define Y (μ, ν) to be the set where Φ̃W ≥ μ and ΦB ≥ ν. Lemmas A.1 and A.2 give

|Y (μ, ν)| ≲ δ^{−Cϵ} min(\frac{|W| |B|}{μ^{2}ν}δ^{d}, \frac{|W|}{μ}δ^{d−1}, \frac{|B|}{ν}δ^{d−1})

and therefore also

|Y (μ, ν)| ≲ δ^{−Cϵ}(\frac{|W| |B|}{μ^{2}ν}δ^{d})^{\frac{q}{p} −1}(\frac{|W|}{μ}δ^{d−1})^{1− \frac{q}{2p}} (\frac{|B|}{ν}δ^{d−1})^{1− \frac{q}{2p}}
= δ^{−Cϵ} \frac{(|B| |W|)^{\frac{q}{2p}} δ^{d−2+ \frac{q}{p}}}{μ^{θq}ν^{(1−θ)q}}

where we used the value of θ to obtain the last line. Summing over dyadic levels for μ and ν between 1 and a negative power of δ gives

∥ΦB^{1−θ}(Φ̃W)^{θ}∥_{q}^{q} ≲ δ^{−Cϵ}(|B| |W|)^{\frac{q}{2p}} δ^{d−2+ \frac{q}{p}}

This and the analogous estimate with the roles of B and W reversed imply

∥T_{θ}∥_{q}^{q} ≲ δ^{−Cϵ}(|B| |W|)^{\frac{q}{2p}} δ^{d−2+ \frac{q}{p}}

which is equivalent to (38) when \frac{1}{q} = \frac{2}{p} − 1. □

We will now pass to a similar estimate for Ψ_{θ}. We will use a rescaling argument and induction on δ like the final argument in [17] or [18]. The rescaling argument requires another relation between the exponents, which is essentially the dual relation to (31). We remark at this point that the quantity which we need to estimate in order to prove Theorem A.1 is min(ΦB, ΦW) and not the slightly larger Ψ_{θ}. It is possible that the slightly stronger result obtained by considering Ψ_{θ} could prove useful, but the main reason we use Ψ_{θ} is that the rescaling argument in the proof is difficult to carry out with min(ΦB, ΦW).

\underline{Lemma A.4} Assume that q ≤ 3, \frac{1}{q} ≥ \frac{2}{p} − 1, and 1 ≤ \frac{q}{p} ≤ \frac{d}{d−1}. Then for any ϵ > 0 there is a constant Aϵ making the following estimate valid; here θ = \frac{1}{4}(3 − \frac{1}{q}):

∥δ^{d−2}Ψ_{θ}∥_{L^{q}(Q(1))}^{q} ≤ Aϵδ^{−Cϵ}(δ^{2d−3}|B| · δ^{2d−3}|W|)^{\frac{q}{2p}}  (39)

\underline{Proof} We start with the following observation concerning rescaling.

\underline{Claim} Suppose that δ is small enough and that (39) has been proved with δ replaced by δ^{1−ϵ}. Let Q be a δϵ-cube, and let B and W be δ-separated sets of tubes. Then

∥Ψ_{θ}∥_{L^{q}(Q)}^{q} ≤ δ^{\frac{Cϵ^{2}}{2}} · Aϵδ^{−Cϵ}(δ^{2d−3}|W| · δ^{2d−3}|B|)^{\frac{q}{2p}}

Namely, for each w ∈ W let k(w) be the cardinality of the set of tubes w_{1} ∈ W such that w_{1} ∩ Q is contained in the double of w; similarly for each b ∈ B let k(b) be the cardinality of the set of tubes b_{1} ∈ B such that b_{1}∩Q is contained in the double of b. Notice that k(w) and k(b) are between 1 and δ^{−(d−2)ϵ}. Let W(μ) = {w ∈ W : k(w) ∈ [μ, 2μ]}, B(ν) = {b ∈ B : k(b) ∈ [ν, 2ν]}, and (analogously to the earlier definitions) let

ΦW^{μ} = ∑_{w∈W_{μ}} χ_{w}, ΦB^{ν} = ∑_{b∈B_{ν}} χ_{b}

Ψ_{θ}^{μν} = min(ΦB^{ν}, ΦW^{μ})^{θ} max(ΦB^{ν}, ΦW^{μ})^{1−θ}

Then

Ψ_{θ} ≤ ∑_{μ,ν} Ψ_{θ}^{μν}  (40)

where the sum is over dyadic values of μ and ν. This follows from (36).

By (40) and pigeonholing, there are values of μ and ν such that

∥Ψ_{θ}^{μν}∥_{L^{q}(Q)} ≳ (log \frac{1}{δ})^{−2}∥Ψ_{θ}∥_{L^{q}(Q)}

We assume without loss of generality that μ ≥ ν. Now let \overline{B} (resp. \overline{W}) be subsets of B(ν) (resp. W(μ)) which are maximal with respect to the following property:

(∗): If b_{1}, b_{2} ∈ \overline{B} (resp \overline{W}), then b_{1} ∩ Q is not contained in the double of b_{2}.

Let \overline{Φ}B (resp. \overline{Φ}W) be the sums of the characteristic functions of the tubes of width C_{0}δ coaxial with the tubes in \overline{B} (resp. \overline{W}), and

\overline{Ψ}_{θ} = min(\overline{Φ}B, \overline{Φ}W)^{θ} max(\overline{Φ}B, \overline{Φ}W)^{1−θ}

Then ΦW^{μ} ≲ μ\overline{Φ}W and ΦB^{ν} ≲ ν\overline{Φ}B, pointwise on Q; this follows from maximality of \overline{W} and \overline{B} provided C_{0} is large enough. Hence also

Ψ_{θ}^{μν} ≲ μ^{θ}ν^{1−θ}\overline{Ψ}_{θ}

by (35). Taking L^{q} norms we conclude that

∥Ψ_{θ}^{μν}∥_{L^{q}(Q)}^{q} ≲ μ^{θq}ν^{(1−θ)q}∥\overline{Ψ}_{θ}∥_{L^{q}(Q)}^{q}  (41)

Furthermore property (∗) implies

|\overline{B}| ≲ ν^{−1}|B(ν)|, |\overline{W}| ≲ μ^{−1}|W(μ)|  (42)

We now dilate the situation by a factor δ^{−ϵ}. This maps Q to a cube Q′ of side 1, and maps \overline{B} and \overline{W} to δ^{1−ϵ}-separated families of C_{0}δ^{1−ϵ}-tubes. Accordingly we can apply the hypothesis that (39) holds at scale δ^{1−ϵ}. We conclude that

∥δ^{(1−ϵ)(d−2)}\overline{Ψ}_{θ}(δϵx)∥_{L^{q}(Q′,dx)}^{q} ≲ Aϵδ^{−Cϵ(1−ϵ)}(δ^{(2d−3)(1−ϵ)}|\overline{W}| · δ^{(2d−3)(1−ϵ)}|\overline{B}|)^{\frac{q}{2p}}

Making the change of variables x → δϵx and factoring out the powers of δϵ we get

δ^{−dϵ}δ^{−q(d−2)ϵ}∥δ^{d−2}\overline{Ψ}_{θ}∥_{L^{q}(Q)}^{q} ≲ Aϵδ^{−Cϵ(1−ϵ)}δ^{− \frac{q}{p} (2d−3)ϵ}(δ^{2d−3}|\overline{W}| · δ^{2d−3}|\overline{B}|)^{\frac{q}{2p}}

We now substitute in the estimates (41) and (42), obtaining

μ^{−θq}ν^{−(1−θ)q}δ^{−dϵ}δ^{−q(d−2)ϵ}∥δ^{d−2}Ψ_{θ}^{μν}∥_{L^{q}(Q)}^{q} ≲ (μν)^{− \frac{q}{2p}} · δ^{− \frac{q}{p} (2d−3)ϵ}
·Aϵδ^{−Cϵ(1−ϵ)}(δ^{2d−3}|W_{μ}| · δ^{2d−3}|B_{ν}|)^{\frac{q}{2p}}

or equivalently

∥δ^{d−2}Ψ_{θ}^{μν}∥_{L^{q}(Q)}^{q} ≲ μ^{q(θ− \frac{1}{2p})}ν^{q(1−θ− \frac{1}{2p})} · δ^{− \frac{q}{p} (2d−3)ϵ+dϵ+q(d−2)ϵ}
·Aϵδ^{−Cϵ(1−ϵ)}(δ^{2d−3}|W_{μ}| · δ^{2d−3}|B_{ν}|)^{\frac{q}{2p}}

But ν ≤ μ ≲ δ^{−(d−2)ϵ}, and the exponents q(θ − \frac{1}{2p}) and q(1−θ − \frac{1}{2p}) are both nonnegative. Since \frac{q}{p} ≤ \frac{d}{d−1}, a little juggling of indices shows that therefore

μ^{q(θ− \frac{1}{2p})}ν^{q(1−θ− \frac{1}{2p})}δ^{− \frac{q}{p} (2d−3)ϵ+dϵ+q(d−2)ϵ} ≲ 1

It follows that

∥δ^{d−2}Ψ_{θ}^{μν}∥_{L^{q}(Q)}^{q} ≲ δ^{Cϵ^{2}} · Aϵδ^{−Cϵ}(δ^{2d−3}|W_{μ}| · δ^{2d−3}|B_{ν}|)^{\frac{q}{2p}}

and therefore

∥δ^{d−2}Ψ_{θ}∥_{L^{q}(Q)}^{q} ≲ (log \frac{1}{δ})^{2q}δ^{\frac{Cϵ^{2}}{2}} · δ^{\frac{Cϵ^{2}}{2}} Aϵδ^{−Cϵ}(δ^{2d−3}|W| · δ^{2d−3}|B|)^{\frac{q}{2p}}

The factor (log \frac{1}{δ})^{2q}δ^{\frac{Cϵ^{2}}{2}} is evidently small for small δ, so the proof of the claim is complete.

We assume now that (39) has been proved for parameter values δ > δ_{0} for a certain δ_{0} (the case where δ is large is easy if Aϵ has been chosen appropriately) and will prove it when δ^{1−ϵ} > δ_{0}. This will evidently establish the lemma.

We use (37), and observe that a bound like (39) with Ψ_{θ} replaced by T_{θ} follows from Lemma A.3; the implicit constant in Lemma A.3 is small compared with Aϵ if Aϵ has been chosen appropriately. To estimate S_{θ}, subdivide Q(1) in δϵ-cubes Q. On each fixed Q we can apply the claim to the restriction of S_{θ} to Q, replacing W by {w ∈ W : w ∼ Q} and similarly with B.

We obtain for each Q

∥δ^{d−2}S_{θ}∥_{L^{q}(Q)}^{q} ≤ δ^{\frac{Cϵ^{2}}{2}} · Aϵ(δ^{2d−3}nW(Q) · δ^{2d−3}nB(Q))^{\frac{q}{2p}}  (43)

We now sum over Q concluding that

∥δ^{d−2}S_{θ}∥_{L^{q}(Q(1))}^{q} ≤ δ^{\frac{Cϵ^{2}}{2}} · Aϵ ∑_{Q}(δ^{2d−3}nW(Q) · δ^{2d−3}nB(Q))^{\frac{q}{2p}}
≤ δ^{\frac{Cϵ^{2}}{2}} · Aϵ(∑_{Q} δ^{2d−3}nW(Q))^{\frac{q}{2p}} (∑_{Q} δ^{2d−3}nB(Q))^{\frac{q}{2p}}

≲ δ^{\frac{Cϵ^{2}}{2}} · Aϵ(C log \frac{1}{δ})^{5 \frac{q}{p}} (δ^{2d−3}|W|)^{\frac{q}{2p}} (δ^{2d−3}|B|)^{\frac{q}{2p}}

The three inequalities followed respectively from (43), from Holder’s inequality (recall

that q ≥ p) and from 1. and 2. of Lemma A.1. The factor δ^{\frac{Cϵ^{2}}{2}} · (C log \frac{1}{δ})^{5 \frac{q}{p}} is small for small δ; the result now follows by combining the last inequality with the preceding bound for ∥T_{θ}∥_{L^{q}(Q)}^{q}. □

Lemma A.4 is our main estimate and the rest of the argument is basically just another rescaling argument. This is fairly routine, so we will omit some details. In order to carry out the argument efficiently we first make some further definitions and remarks.

We define a map X^{∗} from functions on L to functions of ℝ^{d} via

X^{∗}f(x) = ∫_{S^{d−2}} f(ℓ(x, ω))dω

This is easily seen to be the adjoint map to X. If c is a spherical cap on S^{d−2}, then define L_{c} to be the set of light rays ℓ ∈ L whose direction is (ω, 1) for some ω ∈ c. For given p and r and δ, and a set Y ⊂ L, define

Eδ^{p,r}(Y) = ∥χ_{Yδ}∥_{L^{p}(L^{r})}

where Yδ is the δ-neighborhood of Y (with respect to a smooth metric on L).

Next, fix a cap c centered at a point e ∈ S^{d−2} with radius σ. The map T_{c} in section 5 takes light rays to light rays, so there is an action T_{c} : L → L, which has the following metric properties:

(a): If Y ⊂ L_{c} then ∥χ_{T_{c}Y} ∥_{L^{p}(L^{r})} ≈ σ^{− \frac{d−2}{p} − \frac{d}{r}} ∥χ_{Y} ∥_{p,r}

(b): If Y ⊂ L_{c} then Eδ^{p,r}(T_{c}Y) ≲ σ^{− \frac{d−2}{p} − \frac{d}{r}} Eσδ^{p,r}(Y)

(a) is proved as follows: within c, T_{c} expands distances along S^{d−2} by a factor σ^{−1} (hence volumes by σ^{−(d−2)}), and if ω ∈ c then the action on the fiber {x ∈ ℝ^{d} : x⊥(ω, 1)}

expands volumes by roughly \overbrace{σ^{−1} × … × σ^{−1}}^{d−2 times} ×σ^{−2}, i.e. by σ^{−d}. Thus L^{p}(L^{r}) norms expand by σ^{− \frac{d−2}{p} − \frac{d}{r}}. Also (b) follows from (a) by observing that T_{c} maps the Cσδneighborhood of Y ⊂ L_{c} onto a set which includes the δ-neighborhood of T_{c}Y.

Further, if Y ⊂ L_{c} then

X^{∗}χ_{Y} (x) ≈ σ^{d−2}X^{∗}χ_{T_{c}Y} (T_{c}x)  (44)

This follows from the definition of X^{∗} and the formula for volume expansion along S^{d−2}.

We will now rephrase Lemma A.4 using some of the preceding notation and at the same time will replace it by a somewhat weaker result with a less cumbersome statement.

\underline{Lemma A.5} Let Z ⊂ L, let C be a large constant and let S be a set of points in ℝ^{d} with the following properties:

1. The intersection of S with the δ-neighborhood of any given ray ℓ ∈ Z is contained in a cube of side 1.

2. If x ∈ S, then there are two spherical caps c_{1} and c_{2} on S^{d−2} with width C^{−1} and whose distance apart is at least C^{−1}, such that min(X^{∗}(χ_{L_{c_{1}}∩Z})(x), X^{∗}(χ_{L_{c_{2}}∩Z})(x)) ≥ μ.

Then

|S| ≲ δ^{−ϵ}μ^{−q}Eδ^{p,r}(Z)^{q}

for any fixed ϵ > 0, provided q ≤ 3, q ≥ p ≥ r, and \frac{1}{q} ≥ \frac{2}{r} − 1, \frac{q}{r} ≤ \frac{d}{d−1}.

\underline{Proof} We first make a couple of reductions. First, it suffices to prove the lemma with the assumption 1. replaced by the stronger assumption that A ⊂ Q(1). This follows in a standard way using that q ≥ p ≥ r: if the result is proved for A contained in a square of side 1, then one can tile by such squares, take an L^{q} sum over the squares and use the hypothesis 1. It then also suffices to prove Lemma A.5 when p = r, since Eδ^{p,r}(Z) increases with p when Z is contained in a fixed compact subset. In addition, it suffices by a simple covering argument to prove the lemma assuming that the caps c_{1} and c_{2} in 2. are independent of x.

Now define Z_{i} = Z ∩ L_{c_{i}}, let W and B be maximal δ-separated subsets of Z_{1} and Z_{2} respectively and (for each w ∈ W) let D_{w} the δ-disc in L centered at w. Then X^{∗}(χ_{Z_{1}∩D(w)}) ≲ δ^{d−2}χ_{w}, where on the right side χ_{w} is the characteristic function of the δ-neighborhood of the line w. So X^{∗}(χ_{Z_{1}}) ≲ ∑_{w} δ^{d−2}χ_{w} = ΦW, where ΦW is as in Lemma A.1. Accordingly min(X^{∗}(χ_{L_{c_{1}}∩Z})(x), X^{∗}(χ_{L_{c_{2}}∩Z})(x)) ≲ min(ΦB, ΦW) ≤ Ψ_{θ}. The result now follows from Lemma A.4 using Tchebyshev’s inequality and that

(δ^{2d−3}|B|δ^{2d−3}|W|)^{\frac{1}{2r}} ≲ Eδ^{r,r}(Z). □

The point will now be that for appropriate values of the exponents the statement of Lemma A.5 is essentially invariant under the rescaling maps T_{c}.

\underline{Lemma A.6} Assume that q ≤ 3, q ≥ p ≥ r, \frac{1}{q} ≥ \frac{2}{r} − 1, \frac{q}{r} ≤ \frac{d}{d−1}, and

\frac{d − 2}{p} + \frac{d}{r} ≤ d − 2 + \frac{d}{q}  (45)

Let Y ⊂ L. Then

∥X^{∗}χ_{Y} ∥_{L^{q}(Q(1))} ≲ δ^{−ϵ}Eδ^{p,r}(Y)

\underline{Proof} A standard argument shows that it will suffice to prove the corresponding distributional estimate

|{x ∈ Q(1) : X^{∗}χ_{Y} (x) ≥ λ}| ≲ δ^{−ϵ}λ^{−q}Eδ^{p,r}(Y)^{q}  (46)

in the case where λ is bounded below by a high power of δ, say

λ ≥ δ^{\frac{B}{2} (d−2)}

where B is a large constant depending on d. This is because of the δ^{−ϵ} factors and the fact that very small values of λ clearly make a negligible contribution.

To prove (46), let A = {x ∈ Q(1) : X^{∗}χ_{Y} (x) ≥ λ} and define Aσ to be all points x with the property that there are two σ-caps c_{1} and c_{2} on S^{d−2} whose distance apart is between σ and Cσ and such that X^{∗}(χ_{Y ∩L_{c_{i}}}) ≥ C^{−1}δϵλ for i = 1, 2. We claim that ∪σAσ ⊃ A; the union here is over dyadic σ ≥ δ^{B}.

Namely, if x ∈ A, then take the smallest σ such that X^{∗}(χ_{Y ∩L_{c}})(x) ≥ (Cσ)\frac{ϵ}{B} λ for some cap c of width Cσ. (The lower bound on λ implies that then σ ≥ δ^{B}) Consider a covering of σ by caps c_{i} of width σ. The minimality of σ implies that X^{∗}(χ_{Y ∩L_{c_{i}}})(x) is small compared with X^{∗}(χ_{Y ∩L_{c}})(x) for each fixed i. It follows that the contribution from a fixed finite number of the c_{i}’s is similarly small, and therefore there must be two c_{i}’s, call them c_{1} and c_{2}, which are at distance ≥ σ apart such that X^{∗}(χ_{Y ∩L_{c_{i}}})(x) ≳ σ \frac{ϵ}{B} λ for i = 1 and 2. This implies the claim.

By pigeonholing we may now choose σ so that

|Aσ| ≥ δϵ|A|  (47)

Cover S^{d−2} with a family of Cσ-caps c_{i} with bounded overlap. This gives a further decompostion

Aσ = ∪_{i}Aσ^{c_{i}}

where Aσ^{c_{i}} is the set of x for which the two σ-caps c_{1} and c_{2} in the definition of Aσ may be taken to be contained in c_{i}.

We now fix one of the c_{i}’s and apply Lemma A.5 to the sets Z = T_{c_{i}}(L_{c_{i}} ∩ Y) and S = T_{c_{i}}(Aσ^{c_{i}}). Formula (44) shows that the hypothesis 2. is satisfied with μ ≈ σ^{−(d−2)}λ, and since A ⊂ Q(1) and T_{c} preserves lengths in the (e, 1) direction, one can easily see that hypothesis 1. is also satisfied. It follows that

|T_{c_{i}}(Aσ^{c_{i}})| ≲ δ^{−ϵ}(σ^{−(d−2)}λ)^{−q}Eδ^{p,r}(T_{c_{i}}(L_{c_{i}} ∩ Y))
≲ (σ^{−(d−2)}λ)^{−q}σ^{−q( \frac{d−2}{p} + \frac{d}{r})}Eσδ^{p,r}(L_{c_{i}} ∩ Y)^{q}

by property (b) above. Thus, using also (25)

|Aσ| ≲ δ^{−ϵ}σ^{d}(σ^{−(d−2)}λ)^{−q}σ^{−q( \frac{d−2}{p} + \frac{d}{r})}Eσδ^{p,r}(L_{c_{i}} ∩ Y)^{q}

which implies that

|Aσ| ≲ δ^{−ϵ}Eσδ^{p,r}(L_{c_{i}} ∩ Y)^{q}

by the assumption (45).

Now observe that the σδ-neighborhoods of the sets L_{c_{i}} ∩ Y are essentially disjoint (no point y ∈ L belongs to more than a bounded number). Accordingly we can sum over c_{i} to obtain

|Aσ| ≲ δ^{−ϵ}Eσδ^{p,r}(Y)^{q}

We now use (47) and the fact that Eϵ^{p,r} increases with ϵ. The result follows. □

\underline{Proof of Theorem A.1} Let p, q, r be as in Theorem A.1. Because the statement is obtained by interpolation with (34) we can assume that p has its largest possible value, namely \frac{5}{3} if d = 3 and \frac{d+1}{2} if d ≥ 4. The following relations on the dual exponents will hold:

\frac{p′}{r′} ≤ \frac{d}{d − 1}

\frac{d − 2}{q′} + \frac{d}{r′} ≤ d − 2 + \frac{d}{p′}

\frac{1}{p′} ≥ \frac{2}{r′} − 1

r′ ≤ q′ ≤ p′ ≤ 3

Namely the first two are dual to (31) and (32) respectively. The third follows since p ≤ \frac{d+1}{2} and r is defined by (31), while the last is most easily checked by using the explicit values of p, q and r. Thus Lemma A.6 is applicable and shows that

∥X^{∗}χ_{Y} ∥_{L^{p′}(Q(1))} ≲ δ^{−ϵ}Eδ^{q′,r′}(Y)  (48)

We now pass to the dual estimate. If f is supported in Q(1) then we define Xδf(ℓ) = δ^{−(d−1)} ∫_{ℓδ} f, where ℓδ is the tube of width δ with axis ℓ.

Fix a nonnegative function f supported in Q(1) with ∥f∥_{p} = 1 and consider the quantity ∥Xδf∥_{L^{q}(L^{r})}. By duality there is a function g : L → ℝ such that ∥g∥_{L^{q′}(L^{r′})} = 1 and

∫_{L} gXδf ≳ ∥Xδf∥_{L^{q}(L^{r})}

Since Xδf is roughly constant on δ-discs and since values of Xδf which are less than a high power of δ make a negligible contribution to the norm, we can then conclude that there is a function g : L → ℝ with ∥g∥_{L^{q′}(L^{r′})} = 1, with

∫_{L} gXδχ_{E} ≳ δϵ∥Xδf∥_{L^{q}(L^{r})}

and such that g has the special form

g = μχ_{Y}  (49)

where μ is a scalar, and the set Y is a union of δ-discs. Note that this implies μEδ^{q′,r′}(Y) ≲ 1. We also let Ỹ be the corresponding union of 2δ-discs.

Letting g be as in (49), we have

∥Xδf∥_{L^{q}(L^{r})} ≲ δ^{−ϵ} ∫_{L} μχ_{Y} Xδf
≲ δ^{−ϵ} ∫_{L} μχỸ Xf
= δ^{−ϵ} ∫_{ℝ_{d}} X^{∗}(μχỸ)f

Now apply (48) to χỸ and use Holder’s inequality, obtaining

∥Xδf∥_{L^{q}(L^{r})} ≲ δ^{−ϵ}∥f∥_{p}  (50)

since μEδ^{q′,r′}(Ỹ) ≲ 1.

It remains to trade ϵ derivatives for the δ^{−ϵ} factor, which is done in the usual way. Suppose that f has W^{pϵ}-norm 1 and has support in Q(1). If ϕ is an appropriately chosen C_{0}^{∞} function and ϕ_{j}(x) = 2^{dj}ϕ(2^{j}x) then we can express f = g + ∑_{j} ϕ_{j} ∗ f_{j}, where ĝ has compact support, and where ∑_{j} 2^{ηj}∥f_{j}∥_{p} ≲ ∥f∥_{pϵ} for small η. It follows using the smoothing effect of ϕ_{j} that

Xf ≲ 1 + ∑_{j} X_{2^{−j}}|f_{j}|

and now the theorem follows by applying (50) with a small enough value of ϵ to the terms in the series. □

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