Local smoothing type estimates on L^{p} for large p

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

This paper is a contribution to the theory of Fourier transform estimates related to the light cone Γ = \{ξ ∈ ℝ^{n+1} : ξ_{n} = \sqrt{∑_{j=1}^{n−1} |ξ_{j}|^{2}}\}. There has been quite a bit of recent work on such estimates (see for example [1], [8], [10], [11], [2], [18], [6], [17], [20], [16]) but a number of apparently deeper questions remain open.

The best known are the so-called cone multiplier and local smoothing conjectures, both of which would follow (cf. [10] and [11]) from a sharp estimate by the associated square function,

∀ϵ ∃Cϵ : ∥f∥_{p} ≤ CϵNϵ∥Sf∥_{p}  (1)

Here p = 4 in the case of the 2 + 1 dimensional problem, which is the case that we will consider in this paper. Also f is a function with suppf̂ ⊂ Γ_{N}(1), where Γ_{N}(1) is the points which are at distance about N from the origin and at distance ≤ 1 from Γ. Sf corresponds to a decomposition of Γ into sectors of angular width N^{− \frac{1}{2}} :

(Sf(x)^{2} = ∑_{Θ} |f_{Θ}(x)|^{2}

where f_{Θ} = Ξ_{Θ} ∗ f, with Ξ_{Θ} having Fourier support near the part of Γ_{N}(1) corresponding to an arc of the circle Θ of angular length N^{− \frac{1}{2}} and the sum runs over a covering by such arcs.

We do not know how to prove (1). However, we can prove a related estimate with a high value of p, which has some of the same consequences that (1) would have.

Let us define

∥f∥_{p,mic} = (∑_{Θ} ∥f_{Θ}∥_{p}^{p})^{\frac{1}{p}}

with the obvious modification when p = ∞. Here p ≥ 2; it is then clear that ∥f∥_{p,mic} ≤ ∥Sf∥_{p}.

\underline{Theorem 1} The following estimate is valid if p > 74 and f̂ is supported in Γ_{N}(1).

∀ϵ ∃Cϵ : ∥f∥_{p} ≤ CϵNϵN^{\frac{1}{2} − \frac{2}{p}} ∥f∥_{p,mic}  (2)

We remark that for the indicated range of p this estimate is clearly sharp except for issues connected with the Nϵ factor. Further, that the main point is to obtain some value of p < ∞ for which (2) holds. We have tried to arrange the argument in a reasonably efficient way, but improvements in the value of p should certainly be possible.

It is easy to see that (2) fails when p is below the Strichartz exponent 6; for this, consider a function f such that |f_{Θ}| is approximately equal to 1 on a proportional subset of the unit cube for each Θ. It is therefore natural to conjecture that p ≥ 6 is the correct range.

Like other estimates on high L^{p} spaces, Theorem 1 means that we understand the “large” values of f, but only them. Thus we record the following (immediate) corollary:

\underline{Corollary 1} Assume that f ∈ L^{2}, f̂ is supported in Γ_{N}(1) and ∥f∥_{∞,mic} ≤ 1. Then

|\{|f| ≥ N^{\frac{1}{2} −η}\}| ≤ C_{η}N^{−2+δ(η)}∥f∥_{2}^{2}  (3)

where δ(η) → 0 as η → 0.

In the limit as η → 0 this estimate may easily be seen to be sharp and it is considerably stronger than what can be obtained from any of the various results in the literature (cf. [2], [18], [17], [6], [20]). If one weakens (3) a bit by replacing ∥f∥_{2}^{2} with ∥Sf∥_{4}^{4}, then the resulting estimate would be a consequence of the conjectured (1), but the known partial results on (1) lose a factor of N^{\frac{1}{2}} [8] or slightly less than that ([2], [17], [20]). We remark here that the assumption ∥f∥_{∞,mic} ≤ 1 is just a normalization and scales out if one replaces ∥f∥_{2}^{2} with ∥Sf∥_{4}^{4}.

Theorem 1 easily implies the optimal L^{74} local smoothing bound, and a corresponding result for the cone multipliers. Let ∥f∥_{p,α} be the inhomogeneous Sobolev norm with α derivatives in L^{p}.

\underline{Corollary 2} (i) If u is a solution of □u = 0, u(·, 0) = f, \frac{du}{dt}(·, 0) = g in 2 + 1 dimensions then

∥u∥_{L^{p}(ℝ^{2}×[1,2])} ≤ C_{pα} (∥f∥_{p,α} + ∥g∥_{p,α−1})

if p > 74 and α > \frac{1}{2} − \frac{2}{p}.

(ii) Let ρ_{1} be a C_{0}^{∞} function of one variable supported in the interval (1, 2), and let ρ_{2} be a C_{0}^{∞} function of two variables. Then the cone multiplier operators T_{α} defined via T̂_{α}f = m_{α}f̂, where m_{α}(x) = |x_{3} − \sqrt{x_{1}^{2} + x_{2}^{2}}|^{α}ρ_{1}(x_{3})ρ_{2}(x_{1}, x_{2}), are bounded on L^{p} if p > 74 and α > \frac{1}{2} − \frac{2}{p}.

In connection with the cone multiplier statement, we note that the usual argument ([5] and [15], p. 422) which proves the optimal Bochner-Riesz estimate on L^{p} assuming an L^{2} restriction theorem does not work as easily for the cone. Nevertheless it seems possible that bounds like Corollary 2(ii) could be proved without going through Theorem 1.

Theorem 1 also implies the following result, answering a question which the author heard of some time ago from P. Mattila.

\underline{Corollary 3} Let E be a set in ℝ^{2} × ℝ with Hausdorff dimension greater than 1, and let F be a set in ℝ^{2} with the property that if (x, t) ∈ E then the one dimensional Lebesgue outer measure of C(x, t) ∩ E is nonzero. Then F cannot be of two dimensional Lebesgue measure zero.

Part of the point here is that no set theoretic restrictions are needed, in contrast to many related situations where one has to be more careful. The reason for this is that Theorem 1 is a measure estimate and makes no reference to any notion of dimension. On the other hand, the special case of Corollary 3

(∗) If F is a compact set in ℝ^{2} and there is a compact set E ⊂ ℝ^{2} with Hausdorff dimension greater than 1 such that F contains some circle centered at each point of E, then F has positive measure

is new also, although it appears to us that (∗) could be proved without going through the entire proof of Theorem 1. Assertion (∗) with “positive measure” replaced by “Hausdorff dimension two” follows from [18], but the combinatorial techniques from [3] used there are known to be hard to improve. On the other hand, it has been known for a while (Schlag [13]) that (∗) would be a consequence of (1). The argument can be set up so that it only needs the large values estimate of Corollary 1, and it proves Corollary 3 as well. In the case where E has Hausdorff dimension greater than \frac{3}{2} − ϵ for suitably small ϵ, (∗) is a consequence of the abovementioned partial results on (1), and when dimE > \frac{3}{2} a geometric proof was given by Mitsis [9].

In addition, we give a further corollary (see Proposition 6.1 and the subsequent discussion) which is essentially a refinement of Corollary 3 to an L^{p} estimate.

We now give a brief sketch of the proof of Theorem 1. It involves a combination of incidence geometry and L^{2} arguments.

In section 1 we consider a certain “continuum incidence problem” involving approximately tangent pairs of circles and prove a bound in this problem which generalizes a result in [18]. The proof given here is also a lot simpler than the proof in [18], although the same techniques are used (roughly a combination of those in [8] and [3]), so this may be of some independent interest.

To motivate the subsequent sections we recall a certain general principle applicable to many problems in this area. Namely, one often wants to use an induction argument of the following type: let X(N) be an estimate making reference to functions living on Fourier scale N, for present purposes the estimate of Theorem 1. Suppose it has been proved on scale N^{\frac{1}{2}} . If f lives on Fourier scale N then apply X(N^{\frac{1}{2}} ) to the part of f whose Fourier transform lives on a given sector of the cone of angular width N^{− \frac{1}{4}} , which can be rescaled using Lorentz transformations so that it becomes a portion of Γ_{N^{\frac{1}{2}}} . The resulting question of controlling interactions between different such sectors can be localized in x-space to N^{− \frac{1}{2}} -discs, so after dilating by N^{\frac{1}{2}} one can apply X(N^{\frac{1}{2}} ) again.

In practice this logic does not work very well, essentially because solutions of the differential inequality |q̇| ≤ q^{2} can blow up in finite time so that applying the inductive hypothesis twice leads to problems. However it would be a different matter it instead one could apply the inductive hypothesis once on scale N^{\frac{1}{2}} and once on a significantly smaller scale, let us say N^{\frac{1}{2} (1−ϵ_{0})}. As it turns out, in our context it is possible to do this.

Namely, in section 2 we put the result proved in section 1 into a dual form where it becomes a property of light rays instead of circles. In section 3 we obtain from this a certain “localization” property of functions with Fourier support near the light cone (related to the “two ends” condition used in several recent related papers) which allows us to change scales from N^{\frac{1}{2}} to N^{\frac{1}{2} (1−ϵ_{0})} under certain circumstances. Once we have this property, it is possible to set up an induction argument which proves Theorem 1. This is done in section 5; section 4 is the proofs of some lemmas. Other than the localization property, we need only uncertainty principle type L^{2} techniques. In particular, we use neither stationary phase nor L^{4} orthogonality.

Finally, section 6 is the proofs of the corollaries.

A few remarks concerning generalizations to higher dimensions. The analogue of Theorem 1 in n + 1 dimensions is

∥f∥_{p} ≤ CϵNϵN^{\frac{n−1}{2} − \frac{n}{p}} ∥f∥_{p,mic}  (4)

with the natural definition of ∥f∥_{p,mic}, and the possible range of validity is p ≥ \frac{2n+2}{n−1}. Estimate (4) is substantially easier to prove in higher dimensions, and much better values of p (of the form 2 + O(\frac{1}{n}) as n → ∞) can be obtained; this will be done in a forthcoming paper by H. Farag and the author [4]. At present it is unclear to us whether or not one should be able to obtain a sharp result.

One could also consider the analogous question for spheres instead of cones. However, it is possible here to “go down” from estimate (4) for the cone in ℝ^{n+1} to the corresponding estimate for the sphere in ℝ^{n}, so the case of spheres may be of less interest.

List of notation We give here only notation which will be used throughout the paper.

A t-cube is a cube in ℝ^{3} of side t whose vertices belong to tℤ^{3}; thus any two are identical or disjoint. We will always tacitly assume that t is dyadic; thus the grids may be taken to be nested.

χ_{E}: indicator function of the set E.

|E|: Lebesgue measure or cardinality of E depending on the context.

Eδ(X): δ-entropy of X, i.e. maximum possible cardinality for a δ-separated subset of X.

We fix once and for all a small constant ϵ_{0} > 0 and then a large constant M.

ϕ(x): the function (1 + |x|^{2})^{− \frac{M}{2}} .

a_{R}: an affine map taking the unit cube centered at 0 to the rectangle R.

ϕ_{R}: the composition ϕ ∘ a_{R}^{−1} ψ(x): A function mapping ℝ^{3} → ℝ such that

1. ψ = η^{2}, where η̂ is supported in a small disc centered at 0.

2. ψ ≠ 0 on a large cube centered at the origin.

3. The ℤ^{3} translations of ψ form a partition of unity.

ψ_{R}: the composition ψ ∘ a_{R}^{−1}.

Γ: the forward light cone \{ξ ∈ ℝ^{3} : ξ_{3} = \sqrt{ξ_{1}^{2} + ξ_{2}^{2}}\}

Γ_{N}: the cone segment Γ ∩ \{|ξ| ∈ [\frac{N}{2} , 2N]\}.

Γ_{N}(1): the 1-neighborhood of the cone segment Γ_{N}. Similarly we let Γ_{N}(C) be the C-neighborhood of the cone segment \{ξ : |ξ| ∈ [\frac{N}{2^{C}} , 2^{C}N]\}.

For fixed N, we take a partition of unity subordinate to a covering of the circle by arcs Θ of length about N^{− \frac{1}{2}} , and use this to form a partition of unity y_{Θ} on Γ_{N}(C) in the natural way. We let Ξ_{Θ} be a function whose Fourier transform coincides with y_{Θ} when |ξ| ≈ N. If the support of f̂ is contained in Γ_{N}(C) we define

∥f∥_{p,mic} = (∑_{Θ} ∥Ξ_{Θ} ∗ f∥_{p}^{p})^{\frac{1}{p}}

for p < ∞ and

∥f∥_{∞,mic} = sup_{Θ} ∥Ξ_{Θ} ∗ f∥_{∞}

\underline{1. A bound for circle tangencies} As was mentioned in the introduction, this section of the paper is essentially a generalization and simplification of the result of [18]. One can obtain the result of [18] by specializing Lemma 1.4 below to the case of families of circles with δ-separated radii and then using fairly standard arguments.

In this section we use the notation c(x, r) for the circle with center x and radius r, and we will always assume that x is in the disc centered at the origin with radius \frac{1}{4} and that \frac{1}{2} ≤ r ≤ 2. If c_{i} = c(x_{i}, r_{i}) then we define d(c_{1}, c_{2}) = |x_{1} − x_{2}| + |r_{1} − r_{2}|, Δ(c_{1}, c_{2}) = ||x_{1} − x_{2}| − |r_{1} − r_{2}||. We fix large constants C_{1}, C_{2}, C_{3}, C_{0}, which will be specified below, and are chosen in the indicated order. We also fix numbers δ and t with δ small compared with t. A \underline{δ,t-rectangle} is by definition the δ-neighborhood of an arc of

length \sqrt{\frac{δ}{t}} on some circle. We say that a circle c is \underline{tangent} to a δ, t-rectangle R if the C_{1}δ-neighborhood of c contains R. Two δ, t-rectangles are \underline{close} if there is a 2δ, t-rectangle which contains them both and \underline{comparable} if there is a C_{0}δ, t-rectangle which contains them both; if they are not close they are \underline{nonclose} and if they are not comparable they are \underline{incomparable}.

\underline{Lemma 1.1} If R_{1} and R_{2} are incomparable (δ, t)-rectangles both tangent to a common

circle c, and if p_{i} ∈ R_{i}, then |p_{1} − p_{2}| ≥ C_{3}\sqrt{\frac{δ}{t}}.

\underline{Proof} Both R_{1} and R_{2} are contained in the C_{1}δ-neighborhood of c. If they con-

tain points with |p_{1} − p_{2}| < C_{3}\sqrt{\frac{δ}{t}} then it follows that they are contained in the C_{1}δ-

neighborhood of an arc of c of length < (C_{3} + 2)\sqrt{\frac{δ}{t}}, so we can take C_{0} = max(C_{1}, (C_{3} + 2)^{2}). □

The definitions of “close” and “comparable” are almost the same, but for technical reasons we need both of them. The distinction between the two is insignificant because of the following

\underline{Lemma 1.2} For any constant A there is a constant C(A) such that if R is any (Aδ, t) rectangle then the cardinality of any set of pairwise nonclose (δ, t) rectangles contained in R is at most C.

\underline{Proof} This is left to the reader. □

With the given choice of C_{0} we therefore have the following fact.

\underline{Lemma 1.3} Any set of pairwise non-close (δ, t)-rectangles has a pairwise incomparable subset with comparable cardinality.

\underline{Proof} This follows from Lemma 1.2 and standard counting arguments. □

We now fix two finite sets of circles, W and B, which we call white and black respectively, and let m = |W| and n = |B|. We make the following assumptions:

W and B are each δ-separated.  (5)
If c_{1} and c_{2} are the same color then d(c_{1}, c_{2}) ≤ t.  (6)
If c_{1} and c_{2} are opposite colors then d(c_{1}, c_{2}) ≥ t.  (7)
If c_{1} and c_{2} are opposite colors then d(c_{1}, c_{2}) ≤ 100t.  (8)

Define w ∈ W and b ∈ B to be \underline{incident} if Δ(b, w) < δ, and define I(W, B) to be the set of pairs (w, b), w ∈ W, b ∈ B, such that w and b are incident. In principle, one wants to make a nontrivial estimate of the cardinality |I(W, B)|, but the “clamshell” configuration where all the white and black circles are tangent at a point shows that this is not possible, so it is necessary to set things up a bit differently. We make the following further definitions.

A \underline{rectangle of type (≥ μ, ≥ ν)} is a (δ, t)-rectangle which is tangent to at least μ white circles and to at least ν black circles. Additionally, a \underline{rectangle of type (μ, ν)} is a (δ, t)- rectangle which is tangent to between μ and 2μ white circles and to between ν and 2ν black circles, a \underline{rectangle of type (μ, ≥ ν)} is a (δ, t)-rectangle which is tangent to between μ and 2μ white circles and to at least ν black circles, and a \underline{rectangle of type (≥ μ, ν)} is defined analogously. The main result of this section is

\underline{Lemma 1.4} If ϵ > 0 then there is a constant Cϵ such that the cardinality of any set of pairwise incomparable (δ, t)-rectangles of type (≥ μ, ≥ ν) is bounded by

Cϵ(mn)ϵ ((\frac{mn}{μν})^{\frac{3}{4}} + \frac{m}{μ} + \frac{n}{ν})

The remainder of the section is the proof of Lemma 1.4 and its corollary, Lemma 1.17 below. We first recall some geometric and combinatorial facts; these are largely elementary, and most of them have been used before in similar contexts (e.g. [8], [12], [18]). However, our present notation is rather different, so we discuss the proofs.

\underline{Lemma 1.5} If w and b are incident, then there is a (δ, t)-rectangle R such that both w and b are tangent to any (δ, t)-rectangle close to R. Conversely, if w and b are tangent to a common (δ, t)-rectangle, then Δ(b, w) ≤ C_{2}δ. If they are tangent to comparable (δ, t)-rectangles then Δ(b, w) ≲ δ where the implicit constant should be chosen after C_{0}.

\underline{Proof} Let w = c(x_{1}, r_{1}), b = c(x_{2}, r_{2}); we can suppose that r_{1} > r_{2}. Then ||x_{1} − x_{2}| − |r_{1} − r_{2}|| < δ. Take the ray emanating from x_{1} and passing through x_{2}. It intersects w

at a point p. Let R be the δ-neighborhood of the arc of w centered at p with length \sqrt{\frac{δ}{t}},

and let S be the 100δ-neighborhood of the arc of w centered at p with length 10\sqrt{\frac{δ}{t}}. It is not difficult to verify the following two facts provided C_{1} has been chosen large enough:

1. S is contained in the C_{1}δ-neighborhoods of both w and b.

2. Any rectangle which is close to R in the sense defined above is contained in S.

The first statement of the lemma follows. For the converse statement, we use that the area of the intersection of the C_{1}δ-neighborhoods of w and of b is ≲ δ^{2}_{\sqrt{tΔ(w,b)}} (see [19],

Lemma 3.1(a)). If w and b are tangent to a common δ, t-rectangle, then the intersection of their C_{1}δ-neighborhoods contains this rectangle, so we get

δ · \sqrt{\frac{δ}{t}} ≲ \begin{matrix}δ & 2 \\ √ & \overline{tΔ(w, b)} \end{matrix}

which gives Δ(b, w) ≤ C_{2}δ. The final statement of the lemma follows in a similar way. □

\underline{Lemma 1.6} For any w and b the cardinality of a set of pairwise incomparable (δ, t)- rectangles tangent to both w and b cannot exceed a fixed constant C.

\underline{Proof} If any such rectangle exists then Δ(w, b) ≤ C_{2}δ by Lemma 1.5. The intersection of the C_{1}δ-neighborhoods of w and b is then contained in the Cδ-neighborhood of an arc

of length C\sqrt{\frac{δ}{t}} ([19], Lemma 3.1b), and by Lemma 1.2 the latter set contains only a bounded number of pairwise incomparable (δ, t)-rectangles. □

We define Ĩ(B, W) similarly to I(B, W) replacing δ with C_{2}δ.

\underline{Lemma 1.7} (a) If R is any set of pairwise nonclose (δ, t)-rectangles then

|Ĩ(B, W)| ≳ |{(R, b, w) : R ∈ R, and b and w are both tangent to R}|

(b) There is a set R of pairwise incomparable (δ, t)-rectangles such that

|I(B, W)| ≲ |{(R, b, w) : R ∈ R, and b and w are both tangent to R}|

\underline{Proof} Consider a pair (b, w) ∈ B × W. If (b, w) ∉ Ĩ(B, W) then no rectangle R as in the lemma can exist, by Lemma 1.5, and if (b, w) ∈ Ĩ(B, W) then just a bounded number of pairwise nonclose such R’s can exist, by Lemma 1.6. This proves (a). For (b), use Lemma 1.5 to choose for each (b, w) ∈ I(B, W) a rectangle R such that any rectangle close to R is tangent to w and b. Take a maximal pairwise nonclose subset of these R’s, and then use Lemma 1.3. □

\underline{Lemma 1.8} Let c_{1}, c_{2}, c_{3} be three circles. Let R be a set of pairwise incomparable (δ, t)-rectangles with the following property:

(∗) For each R ∈ R there is a circle c such that d(c, c_{i}) ≥ t for each i, such that c and c_{1} are tangent to R, and furthermore such that there are two (δ, t)-rectangles R_{2} and R_{3} such that c and c_{i} are tangent to R_{i} for i = 2, 3, and such that R, R_{2} and R_{3} are pairwise incomparable.

Then the cardinality of R cannot exceed a fixed constant.

\underline{Proof} We use the following fact, which is included in the so-called Marstrand three circle lemma.

(M): Let C be all circles c such that d(c, c_{i}) ≥ t and Δ(c, c_{i}) ≤ C_{2}δ, and such that the C_{1}δ-neighborhood of c intersects the C_{1}δ-neighborhood of c_{i} (i = 1, 2, 3) at

points p_{i} satisfying |p_{i} − p_{j}| ≥ C_{3}\sqrt{\frac{δ}{t}}. Then the set ∪_{c∈C}(C_{1}δ-neighborhood of c) ∩ (C_{1}δ-neighborhood of c_{1}) is contained in the union of the C_{1}δ-neighborhoods of two arcs

of c_{1} each of length less than ≲ \sqrt{\frac{δ}{t}}.

This fact is implicit in [8] and is stated in [18], Lemma 1.8. A detailed proof is contained in the proof of Lemma 3.2 in [19].

To pass from this to Lemma 1.8, let c be a circle as in (∗). Then Δ(c, c_{i}) ≤ C_{2}δ for each i by Lemma 1.5. Let p_{i} be a point of R_{i}, i = 2, 3, and let p_{1} be a point of R. All three rectangles R, R_{2} and R_{3} are tangent to c, so by Lemma 1.1 we must have

|p_{i}−p_{j}| ≥ C_{3}\sqrt{\frac{δ}{t}}. The definition of tangent implies that R is contained in the intersection of the C_{1}δ-neighborhoods of c and of c_{1}. Accordingly, by (M), R is contained in the union

of the C_{1}δ-neighborhoods of two arcs of c_{1} each of length ≲ \sqrt{\frac{δ}{t}}. Now Lemma 1.2 implies that there are at most a bounded number of pairwise incomparable R’s. □

\underline{Lemma 1.9} Let c_{1} = c(x_{1}, r_{1}) and c_{2} = c(x_{2}, r_{2}) be circles with d(c_{1}, c_{2}) ≥ t, and assume that r_{1} ≥ r_{2}. Let R_{1} and R_{2} be comparable (δ, t)-rectangles and assume that c_{i} is tangent to R_{i} for i = 1, 2. Then

(i) c_{2} is contained in the Cδ-neighborhood of the interior of c_{1}.

(ii) For any constant A there is a constant C = C(A) such that the cardinality of a set of pairwise incomparable (δ, t) rectangles each of which is tangent to c_{1} and intersects the Aδ-neighborhood of the interior of c_{2} does not exceed C.

\underline{Proof} (i) Let c_{i} = C(x_{i}, r_{i}). Lemma 1.5 implies Δ(c_{1}, c_{2}) ≤ Cδ. Then z ∈ c_{2} implies |z −x_{1}| ≤ |z −x_{2}|+|x_{2} −x_{1}| ≤ |z −x_{2}|+r_{1} −r_{2} +Cδ ≤ r_{2} +δ +r_{1} −r_{2} +Cδ = r_{1} +(C +1)δ, as claimed.

(ii) Δ(c_{1}, c_{2}) ≲ δ by Lemma 1.5. Using that r_{1} ≥ r_{2} it follows similarly to the proof of (i) that the intersection of the δ-neighborhood of c_{1} with the Aδ-neighborhood of the interior of c_{2} is contained in the Cδ-neighborhood of c_{2} for suitable C. The lemma now follows using that the intersections of the Cδ-neighborhoods of c_{1} and c_{2} has diameter

≲ \sqrt{\frac{δ}{t}} by [19], Lemma 3.1, and using also Lemma 1.2. □

In the proof of Lemma 1.4 we employ the “Canham bound plus divide and conquer” strategy of (for example) [3]. The next few lemmas provide the necessary combinatorial ingredients.

\underline{Lemma 1.10} (Canham Lemma) The cardinality of any set of pairwise incomparable (δ, t) rectangles of type (μ, ν) is ≲ \frac{mn^{\frac{2}{3}}}{μν^{\frac{2}{3}}} + \frac{n}{ν}. The cardinality of any set of pairwise

incomparable (δ, t) rectangles of type ≥ μ, ≥ ν is ≲ \frac{mn^{\frac{2}{3}}}{μν^{\frac{2}{3}}} + \frac{n}{ν} log \frac{m}{μ}.

\underline{Proof} The second statement follows readily from the first by summing over the different dyadic levels for μ and ν.

To prove the first statement, let R be a set of pairwise incomparable rectangles of type (μ, ν). Observe that for each w ∈ W and b ∈ B there is at most a bounded number of rectangles R ∈ R which are tangent to both w and b, by Lemma 1.5; this fact will be used several times below.

The letter b always denotes an element of B, and w an element of W. We define the following sets:

P: all pairs (w, b) such that there is R ∈ R with both w and b tangent to R.

P(b): all pairs (w, b) ∈ P with given second member b.

Q: all quadruples (w_{1}, w_{2}, w_{3}, b) such that the following holds: there are distinct rectangles R_{i} ∈ R such that, for each i, w_{i} and b are both tangent to R_{i}.

Q(b): all quadruples (w_{1}, w_{2}, w_{3}, b) ∈ Q with given last member b.

Lemma 1.8 implies that, for given w_{1}, w_{2}, w_{3}, there are at most C rectangles R_{1} ∈ R for which some b ∈ B may be found such that (w_{1}, w_{2}, w_{3}, b) ∈ Q and such that the choice of R_{1} in the definition is the given one. For each choice of R_{1}, there are at most 2ν ways to choose b, since R_{1} must have type (μ, ν). Accordingly

|Q| ≤ m^{3} · C · 2ν  (9)

On the other hand, for each b ∈ B there are exactly |P(b)|^{3} triples (w_{1}, w_{2}, w_{3}) such that, for some R_{1}, R_{2}, R_{3} ∈ R and each i ∈ {1, 2, 3}, w_{i} and b are both tangent to R_{i}. Similarly, there are exactly |P(b)|^{2} pairs (w_{1}, w_{2}) such that, for some R_{1}, R_{2} ∈ R and each i ∈ {1, 2}, w_{i} and b are both tangent to R_{i}. For each of these, there are at most 4μ choices of w_{3} which are tangent to either R_{1} or R_{2}. It follows that

|Q(b)| ≥ |P(b)|^{3} − 12μ|P(b)|^{2}  (10)

Therefore

|P| = ∑_{b} |P(b)|
≤ ∑_{b:|P(b)|<24μ} |P(b)| + ∑_{b:|P(b)|≥24μ} |P(b)|

≤ 24μn + ∑_{b:|P(b)|≥24μ}(2|Q(b)|)^{\frac{1}{3}}

≤ 24μn + n^{\frac{2}{3}} (2 ∑_{b:|P(b)|≥24μ} |Q(b)|)^{\frac{1}{3}}
≲ μn + ν^{\frac{1}{3}} n^{\frac{2}{3}} m

where the second inequality follows from (10) and the last inequality follows from (9). If R ∈ R then there are at least μν pairs (w, b) ∈ W ×B such that w and b are both tangent to R. It follows that |P| ≲ \frac{n}{ν} + \frac{n^{\frac{2}{3}} m}{ν^{\frac{2}{3}} μ} as claimed. □

\underline{Lemma 1.11} Assume W and B have no (δ, t) rectangles of type (≥ 1, ≥ ν_{0}) nor of type (≥ μ_{0}, ≥ 1). Then

|I(W, B)| ≲ μ0^{\frac{1}{3}} nm^{\frac{2}{3}} log ν_{0} + ν_{0}m log μ_{0}

\underline{Proof} Using Lemma 1.7 there is a pairwise incomparable set of (δ, t)-rectangles R with (sums over μ and ν run through dyadic values)

|I(W, B)| ≲ ∑_{R∈R} |{(b, w) : b and w are both tangent to R}|
≲ ∑_{μ,ν} μν · #(members of R which are of type μ, ν)

≲ ∑_{ν≤ν_{0}^{μ≤μ_{0}}} μν(\frac{nm^{\frac{2}{3}}}{νμ^{\frac{2}{3}}} + \frac{m}{μ})

≲ μ0^{\frac{1}{3}} nm^{\frac{2}{3}} log ν_{0} + ν_{0}m log μ_{0}

At the next to last line, we used Lemma 1.10 with the roles of white and black reversed. □

\underline{Lemma 1.12} (Cell decomposition lemma) Suppose we randomly choose r black circles. Then with high probability there is a decomposition of the white circles

W = W^{∗} ∪ (∪_{i=1}^{R}W_{i}) (disjoint decomposition)

with

(i) R ≤ r^{3}a(r), where a(r) grows extremely slowly - in particular slower than any power of log r.

(ii) For each i, there are ≲ \frac{n log n}{r} black circles b such that Δ(b, w) < C_{2}δ for some w ∈ W_{i}.

(iii) Each circle in W^{∗} satisfies Δ(b, w) < C_{2}δ for some circle b in the random sample.

\underline{Proof} This lemma is due essentially to workers in computational geometry (see for example [3], [14]), and was adapted for analysis purposes in [18]. The version stated here differs from Proposition 2.1 in [18] in that there we had W = B, but the present bipartite version follows in exactly the same way. □

Notice that the same parameter (namely C_{2}δ) appears on the right hand side in (ii) and (iii). It would be more convenient to be able to use C_{2}δ in (ii) and δ in (iii), but this is not possible without additional work; see the proof of the claim in [18], p. 997. We must therefore phrase the next lemma in terms of Ĩ(B, W).

\underline{Lemma 1.13} With high probability the cardinality of W^{∗} is ≲ \frac{r|I^{}˜ (B,W)|}{n}.

\underline{Proof} Part (iii) of Lemma 1.12 implies that the probability of a given circle w ∈ W to belong to W^{∗} is bounded by

\frac{r}{n}|\{b ∈ B : Δ(b, w) < C_{2}δ\}|

Accordingly the expectation of the cardinality of W^{∗} is bounded by \frac{r|I^{}˜ (B,W)|}{n}, which suffices. □

We define a \underline{cluster of white circles} to be a subset C ⊂ W, with the property that there is a (δ, t) rectangle R such that every circle in C is tangent to a (δ, t) rectangle comparable to R. In a similar manner we define a cluster of black circles. The next two lemmas are related to considerations in section 1 of [20].

\underline{Lemma 1.14} Let C be a cluster of white circles and let b be a black circle. Then the cardinality of any set of pairwise incomparable )δ, t)-rectangles each of which is tangent to some circle in C and to b is bounded by a constant.

\underline{Proof} Consider the circles in the cluster with smaller radius than b. Choose a circle in the cluster, w_{1}, whose radius is as large as possible subject to this constraint. By the clustering property and Lemma 1.9(i) above, any other w ∈ C with these properties is contained within the Cδ-neighborhood of the interior of w_{1}. By Lemma 1.9(ii), b cannot be tangent to more than a bounded number of incomparable (δ, t)-rectangles which intersect the latter set. A symmetric argument for the circles with larger radius than b completes the proof. □

\underline{Lemma 1.15} (Clustering Lemma) Given a value of μ_{0} we can subdivide the white circles as

W = W_{g} ∪ W_{b}

where

1. W_{g} and B have no (δ, t)-rectangles of type (≥ μ_{0}, ≥ 1), and

2. W_{b} is the union of at most \frac{|W|}{μ_{0}}(log m)(log n) clusters.

\underline{Proof} We fix a 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 pairwise incomparable rectangles of type ≥ μ_{0}, ≥ 1 for W^{i} and B. Note first of all that κ(W) ≲ \frac{mn}{μ_{0}} by Lemma 1.6.

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 there is a set R_{i} of at most A\frac{m}{μ_{0}} log m rectangles such that each circle in W_{b}^{i} is tangent to a rectangle which is comparable to a rectangle in R_{i}.

Namely, let R be a set of pairwise incomparable rectangles of type (≥ μ_{0}, ≥ 1) for W_{i} and B with maximum possible cardinality k. There are two cases.

(i) If k < A\frac{m}{μ_{0}} log m then we let W_{b}^{i} be all circles in W^{i} which are tangent to a rectangle comparable to a rectangle in R_{i} and W^{i+1} = W^{i}\W_{b}^{i}. Evidently κ(W^{i+1}) = 0.

(ii) If k < A\frac{m}{μ_{0}} log m then we choose A\frac{m}{μ_{0}} log m rectangles from R at random. We let W_{b}^{i} be the circles tangent to a rectangle which is comparable to a rectangle in the random sample, and W^{i+1} = W^{i}\W_{b}^{i}. We will show that with high probability κ(W^{i+1}) ≤ \frac{k}{2}.

For this define, for each c ∈ W^{i},

P (w) = k^{−1}|{R ∈ R_{i} : w is tangent to a rectangle comparable to R}|

Then w is in W^{i+1} with probability at most (1 − P (w))^{A \frac{m}{μ_{0}} log m}. If P (w) ≥ \frac{1}{2} \frac{μ_{0}}{m} then this is at most m^{−CA}. Accordingly with high probability no circles with P (w) ≥ \frac{1}{2} \frac{μ_{0}}{m} belong to W^{i+1}.

Now let R_{i+1} be a maximal set of pairwise incomparable rectangles of type (≥ μ_{0}, ≥ 1) for W^{i+1} and B. Every rectangle in R_{i+1} is comparable to a rectangle in R_{i}, since otherwise we could have enlarged R_{i} by adjoining an incomparable member of R_{i+1} to it. Consider now the quantity

∑_{w∈W^{i+1}} |{R ∈ R_{i} : R is comparable to a rectangle tangent to w}|  (11)

This is equal to ∑_{w∈W^{i+1}} kP (w), so with high probability it is less than \frac{k}{2} \frac{μ_{0}}{m}. On the other hand, the quantity (11) can also be described as

∑_{R∈R_{i}} |{w ∈ W^{i+1} : w is tangent to a rectangle comparable to R}|

and therefore dominates μ_{0}|R_{i+1}|. We conclude 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 initally ≤ \frac{mn}{μ_{0}} and decreases each time by a factor of 2, we then have T ≲ log n. 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 O(log n) sets W_{b}^{i}, for each of which there is a collection of A\frac{m}{μ_{0}} log m rectangles such that each circle in W_{b}^{i} is tangent to a rectangle comparable to one of these. The lemma follows. □

We prove Lemma 1.4 in two steps: first we consider the case μ = ν = 1 and then the general case.

\underline{Lemma 1.16} For any ϵ > 0 the cardinality of a set of pairwise incomparable rectangles of type (≥ 1, ≥ 1) is

≤ Cϵ (((mn)^{\frac{3}{4} +ϵ} + m log n + n log m)  (12)

for a suitable constant Cϵ.

\underline{Proof} We fix ϵ and will argue by induction on mn. We note first of all that the case of small mn is a tautology, and may therefore assume that (12) has been proved for families of circles W′, B′ with |W′||B′| ≤ \frac{mn}{2}. If m ≤ n^{\frac{1}{3} +ϵ} (or vice versa) then the bound (12) follows directly from Lemma 1.10, since in this case mn^{\frac{2}{3}} ≤ (mn)^{\frac{3}{4} +ϵ}. We may therefore assume that

m^{\frac{1}{3} +ϵ} ≤ n ≤ m  (13)

Let R be a maximal set of pairwise incomparable (δ, t) rectangles of type (≥ 1, ≥ 1). Let μ_{0} = ν_{0} = (mn)^{\frac{1}{4}} , replace δ with C_{2}δ, and apply Lemma 1.15 twice with these

parameters, once as stated and once with the roles of W and B reversed. This gives decompositions W = W_{g} ∪ W_{b}, B = B_{g} ∪ B_{b}.

Any R ∈ R is a rectangle of type (≥ 1, ≥ 1) for one of the three pairs (W_{b}, B), (W, B_{b}) or (W_{g}, B_{g}). We show first that there are ≲ (log m)(log n)(mn)^{\frac{3}{4}} members of R which are rectangles of type (≥ 1, ≥ 1) for (W_{b}, B). Let R be such a rectangle. Then there is C, one of the clusters in 2. of Lemma 1.15, such that R is a ≥ 1, ≥ 1-rectangle for C and B. For given b ∈ B, by Lemma 1.14, there are at most a bounded number of incomparable (C_{2}δ, t)-rectangles which are C_{2}δ-tangent to some circle in C and to b. It follows using Lemma 1.2 that there are at most a bounded number of R ∈ R which are δ-tangent to some circle in C and to b. Hence the number of (δ, t) rectangles R ∈ R of type (≥ 1, ≥ 1) for C and B is ≲ n. Summing over the clusters, there are ≲ \frac{mn}{μ_{0}}(log m)(log n) rectangles of type (≥ 1, ≥ 1) for W_{b} and B, and this is ≲ (log m)(log n)(mn)^{\frac{3}{4}} by choice of μ_{0}.

Likewise there are ≲ (log m)(log n)(mn)^{\frac{3}{4}} members of R which are rectangles of type (≥ 1, ≥ 1) for (W, B_{b}). These contributions are small compared with Cϵ(mn)^{\frac{3}{4} +ϵ}, so it remains to consider (W_{g}, B_{g}).

Fix an appropriate r, which should be large compared with (log n)\frac{1}{ϵ}, but small compared with both \frac{n^{\frac{3}{4}}}{m^{\frac{1}{4}} log(mn)} and \frac{m^{\frac{1}{4}}}{n^{\frac{1}{12}} log(mn)}; this is possible by the assumption (13). Now

apply Lemma 1.12 to W_{g} and B_{g} with this r. We first show that |W_{g}^{∗}| < \frac{m}{100}.

We know by Lemma 1.13 that |W_{g}^{∗}| ≲ \frac{r|I^{}˜ (W_{g},B_{g})|}{n}. Since W_{g} and B_{g} have no (C_{2}δ, t)- rectangles of type (≥ μ_{0}, ≥ 1) nor of type (≥ 1, ≥ ν_{0}), we may apply Lemma 1.11 replacing δ with C_{2}δ. We conclude that

|Ĩ(W_{g}, B_{g})| ≲ mν_{0} log μ_{0} + nm^{\frac{2}{3}} μ0^{\frac{1}{3}} log ν_{0}
≤ m^{\frac{5}{4}} n^{\frac{1}{4}} log m + m^{\frac{3}{4}} n^{\frac{13}{12}} log n

and therefore

|W_{g}^{∗}| ≲ r(m^{\frac{5}{4}} n^{− \frac{3}{4}} log m + m^{\frac{3}{4}} n^{\frac{1}{12}} log n)  (14)

The right side of (14) is small compared with m by choice of r, so |W_{g}^{∗}| < \frac{m}{100} as claimed. We may therefore apply the inductive hypothesis obtaining that W_{g}^{∗} and B_{g} have fewer than Cϵ ((\frac{mn}{1000})^{\frac{3}{4} +ϵ} + m log n + n log m) rectangles of type (≥ 1, ≥ 1). Using (13) it

follows that this is ≤ \frac{1}{10}Cϵ(mn)^{\frac{3}{4} +ϵ}.

Now consider the individual cells W_{g}^{i}. For each of them, we let B_{g}^{i} be the circles b ∈ B_{g} for which there exists a rectangle R ∈ R tangent to b and to some w ∈ W_{g}^{i}. By Lemma 1.5 and part (ii) of Lemma 1.12 we have |B_{g}^{i}| ≲ \frac{n log n}{r}. Since r is large compared with log n, we conclude by the inductive hypothesis that W_{g}^{i} and B_{g}^{i} have at most

Cϵ ((C \frac{n log n}{r}|W_{g}^{i}|)^{\frac{3}{4} +ϵ} + |W_{g}^{i}| log(C \frac{n log n}{r}) + C \frac{n log n}{r} log(|W_{g}^{i}))

rectangles of type (≥ 1, ≥ 1). Summing up, we bound the number of rectangles of type (≥ 1, ≥ 1) for ∪_{i}W_{g}^{i} and B_{g} by

Cϵ ((C \frac{n log n}{r})^{\frac{3}{4} +ϵ} ∑_{i} |W_{g}^{i}|^{\frac{3}{4} +ϵ} + log(C \frac{n log n}{r}) ∑_{i} |W_{g}^{i}| + C \frac{n log n}{r} ∑_{i} log |W_{g}^{i}|) (15)

We use (i) of Lemma 1.12 and Holder’s inequality to bound the first term, and we also use (i) of Lemma 1.12 to bound the third term. For the middle term, we use that r is large compared with log n. This leads to

(15) ≤ Cϵ ((C \frac{n log n}{r})^{\frac{3}{4} +ϵ}m^{\frac{3}{4} +ϵ}(r^{3}a(r))^{\frac{1}{4} −ϵ} + m log n + C \frac{n log n}{r}r^{3}a(r) log m)

≤ Cϵ (\frac{(C log n)^{\frac{3}{4} +ϵ}a(r)^{\frac{1}{4} −ϵ}}{r^{4ϵ}} (mn)^{\frac{3}{4} +ϵ} + m log n + Cr^{2}a(r)n log n log m)

The last term is small compared with (mn)^{\frac{3}{4} +ϵ} since r < \frac{m^{\frac{1}{4}}}{n^{\frac{1}{12}}} and n ≤ m. The first

term is small compared with (mn)^{\frac{3}{4} +ϵ} since r is large compared with (log n)\frac{1}{ϵ}, so that the

coefficient \frac{(log n)^{\frac{3}{4} +ϵ}a(r)^{\frac{1}{4} −ϵ}}{r^{4ϵ}} is small. The middle term is also small compared with (mn)^{\frac{3}{4} +ϵ} since m ≤ n^{3}. We conclude that (15) is small compared with (mn)^{\frac{3}{4} +ϵ}, which concludes the proof. □

\underline{Proof of Lemma 1.4} This is now a simple random sampling argument. Let A be a large constant. Randomly choose A\frac{n}{ν} black and A\frac{m}{μ} white circles, with the convention that if ν ≤ 2A we choose all the black circles, and if μ ≤ 2A we choose all the white ones. Each fixed rectangle of type (≥ μ, ≥ ν) for W and B is then a (≥ 1, ≥ 1) rectangle for the samples with probability bounded below. Hence, if R is a set of pairwise incomparable (δ, t)-rectangles of type (≥ μ, ≥ ν) for W and B then with high probability R has a subset with proportional cardinality consisting of rectangles which are of type (≥ 1, ≥ 1) for the samples. Hence |R| ≤ Cϵ ((\frac{mn}{μν})^{\frac{3}{4} +ϵ} + \frac{m}{μ} log \frac{n}{ν} + \frac{n}{ν} log \frac{m}{μ}) by Lemma 1.16. □

We will apply Lemma 1.3 by means of a certain slightly weaker statement, Lemma 1.17 below, which is formulated in terms of Euclidean rectangles instead of the curvilinear “rectangles” which were convenient in the preceding proof; we remark that no further use of the curvilinear rectangles will be made in this paper. For a suitable constant C, a (Euclidean) rectangle R with dimensions δ × δ^{\frac{1}{2}} is \underline{δ-tangent} to a circle c if it is contained in the Cδ-neighborhood of c. If we are given collections W and B of white and black circles then R is of \underline{type (≥ μ, ≥ ν)} if it is tangent to ≥ μ white circles and to ≥ ν black ones. Two rectangles are incomparable if neither is contained in a suitable fixed multiple of the other.

\underline{Lemma 1.17} Let W and B satisfy (5), (6), and (7). The cardinality of a set R of pairwise incomparable (Euclidean) δ × δ^{\frac{1}{2}} -rectangles of type (≥ μ, ≥ ν) obeys the bound

|R| ≤ Cϵt^{− \frac{1}{2}} · δ^{−ϵ} ((\frac{mn}{μν})^{\frac{3}{4}} + \frac{m}{μ} + \frac{n}{ν})  (16)

\underline{Proof} Assume at first that W and B also satisfy (8). It is not difficult to see that if C_{1} was chosen large enough then for any R ∈ R there must be a (Cδ, t)-rectangle in the sense of Lemma 1.4 which contains R and which is tangent in the sense of Lemma 1.4 to any white or black circle to which R is tangent. Each such (Cδ, t)-rectangle contains ≲ t^{− \frac{1}{2}} incomparable R’s, and the bound (16) therefore follows from Lemma 1.4. To eliminate the assumption d(w, b) ≤ 100t, note that by (6) there must be a number τ ≥ t such that τ ≤ d(w, b) ≤ 100τ for all w and b, and apply the above reasoning replacing t by τ. □

\underline{Remark} We take t = 1 and m = n, μ = ν in this remark to avoid certain technical issues. The question arises as to whether or not the exponent \frac{3}{2} in the bound

∀ϵ ∃Cϵ : |R| ≤ Cϵδ^{−ϵ}(\frac{m}{μ})^{\frac{3}{2}}  (17)

of Lemma 1.17 is sharp or not. This appears to be a difficult question. One can show that \frac{3}{2} cannot be replaced by a number less than \frac{4}{3}, and it is also fairly clear that \frac{3}{2} is the best exponent that can be obtained by an argument based on the techniques of [3]. See [19], p. 143.

2. A dual formulation A \underline{δ-plate} is a δ × δ^{\frac{1}{2}} × 1-rectangle whose longest axis is a light ray and whose second

longest axis is tangent to the corresponding light cone, and a \underline{δ-tube} is a δ×δ×1-rectangle whose longest axis is a light ray. The \underline{direction} of a tube or plate is the direction of its longest axis. This direction will always be of the form (e, 1) with e ∈ S^{1}. We remark that it is sometimes necessary to replace the length parameter 1 by a fixed constant (e.g. this is the case in Lemma 2.2 below) and that we will often ignore technicalities of this kind. Two δ-plates or δ-tubes are \underline{comparable} if one is contained in the dilate of the other by a fixed constant C, and they are \underline{parallel} if their axis directions fail to be Cδ-separated for a suitable C. A family of δ-plates or δ-tubes is \underline{separated} if no more than C are comparable to any given one. We also fix a small number ϵ_{0} > 0 and then a much smaller number ϵ.

\underline{Lemma 2.1} Let W ⊂ ℝ^{3} and B ⊂ ℝ^{3} be δ-separated sets of points; assume that d(w, b) ≥ t and that d(w_{1}, w_{2}) ≤ t, d(b_{1}, b_{2}) ≤ t. Let R be a separated set of δ-plates, each containing at least μ points of W and ν points of B. Then for any ϵ > 0

|R| ≲ t^{− \frac{1}{2}} δ^{−ϵ} ((\frac{mn}{μν})^{\frac{3}{4}} + \frac{m}{μ} + \frac{n}{ν})

In particular, this is ≲ δ^{−ϵ}t^{− \frac{1}{2}} (\frac{max(m,n)}{min(μ,ν)})^{\frac{3}{2}} .

\underline{Proof} Since the plates have bounded diameter we can assume that all the plates and points are contained in the region 1 ≤ x_{3} ≤ C for suitable C. To any plate π we can associate a rectangle R_{π} in ℝ^{2} as follows: extend the plate to an infinite δ×δ^{\frac{1}{2}} -rectangular cylinder with the same axes, and then intersect this with the plane x_{3} = 0. It is not difficult to see that if a point (\overline{x}, x_{3}) ∈ W ∪ B belongs to π, then the circle with center \overline{x} and radius x_{3} must be δ-tangent to this δ × δ^{\frac{1}{2}} -rectangle in the manner discussed at the end of section 1. The result therefore follows from Lemma 1.17. □

\underline{Lemma 2.2} (i) Let P be a family of δ-plates. Then there is a separated family of (Cδ)^{\frac{1}{2}} -tubes T so that

• Each π ∈ P is contained in some τ(π) ∈ T , and for fixed τ the plates with τ(π) = τ are all parallel.

(ii) Let T be a family of δ^{\frac{1}{2}} -tubes. Then there is a separated family of (Cδ)^{\frac{1}{2}} -plates A so that

• Each τ ∈ T is contained in some Π(τ) ∈ A. If Π ∈ A then the directions of the tubes with Π(τ) = Π will all be of the form (e, 1) where e belongs to an arc of S^{1} of length Cδ^{\frac{1}{4}} .

\underline{Proof} This is almost a tautology. For (i), we choose for each π a roughly δ^{\frac{1}{2}} -tube τ containing π, and observe that if τ is replaced by any comparable tube, then after dilation by by a fixed constant the comparable tube will still contain π. Now pass to a maximal separated subset and let T be the dilations of the tubes in the subset. The fact that plates π with a given τ(π) are all “parallel” in our sense is a simple property of light rays. Part (ii) can be seen in a similar way. □

In, say, part (i) of the above lemma it is of course possible for π to be contained in more than one tube τ ∈ T ; if so we fix a definite choice of τ(π). In the sequel, given a family of δ-plates P, we will always fix a family of tubes T as in (i), and will denote T by T (P). For each τ ∈ T (Π) we define

XP(τ) = {π ∈ P : τ(π) = τ}

Thus P is the disjoint union of the XP(τ)’s where τ runs over T (P). Similarly, given a family of δ^{\frac{1}{2}} -tubes, we will fix a family of (Cδ)^{\frac{1}{2}} -plates A as in (ii), and will denote A by A(T ), and for Π ∈ A we will denote

YT (Π) = {τ ∈ T : Π(τ) = Π}

Let W be a set of points (of ℝ^{3}), P a set either of δ-plates or of δ-tubes. We let

I(W, P) = {(w, π) ∈ W × P : w ∈ π}

If t is fixed and if x is a point of ℝ^{3} then we let Q(x) be the t-cube containing x. In the sequel we will be working (analogously to [20]) with a relation ∼ between t-cubes for suitable t and elements of P.

If ∼ is such a relation, and if w ∈ W, π ∈ P, then we use the notation w ∼ π to mean that Q ∼ π, where Q is the t-cube containing w, and in addition w ∈ π.

A \underline{good incidence} is a pair (w, π), w ∼ π and a \underline{bad incidence} is a pair (w, π) ∈ I(W, P), w ≁ π. We use I_{b} for the set of bad incidences; sometimes we also use I_{b}(∼) if we want to emphasize the dependence on the relation ∼. In what follows, we always assume that on a logarithmic scale t is large compared with δ^{\frac{1}{2}} ; in practice t will be set equal to δ^{ϵ^{0}}.

The following property will be crucial. We note here that the value of C_{0} will be adjusted numerous times during the course of the proof.

\underline{Property (R)}: For each π ∈ P there are at most (log \frac{1}{δ})^{C_{0}} t-cubes Q’s such that Q ∼ Π.

The next few lemmas construct relations with this property. The main results which we need in subsequent sections are Lemmas 2.7 and 2.8 below.

\underline{Lemma 2.3} (i) If P is a δ-separated family of δ-plates and W is a δ-separated set of points then there is a relation satisfying (R) and with |I_{b}| ≲ δ^{−Cϵ}t^{−6}|W||P|^{\frac{1}{3}} .

(ii) If T is a δ-separated family of δ-tubes and W is a δ-separated set of points then there is a relation satisfying (R) and with |I_{b}| ≲ δ^{−Cϵ}t^{−5}|W||P|^{\frac{1}{2}} .

\underline{Remark} The exponents 6 and 5 in (i) and (ii) respectively can easily be improved substantially, but this would not lead to any improvement in the results of the paper. On the other hand, in part (a) it would be interesting to decide whether the exponent \frac{1}{3} in (i) can be improved to \frac{1}{p} for arbitrary p < 4 or even to \frac{1}{p} for some fixed p > 3; this is the same issue we have discussed in the remark at the end of section 1.

\underline{Proofs} From the form of the statements we can assume that all the plates and points are contained in a disc of fixed size.

(i) Define a relation ∼ as follows: for each π let Q(π) be the t-cube Q for which |W ∩ π ∩ Q| is maximum (if there are several such, then pick one arbitrarily), and declare π to be related to Q(π) and to its neighbors. Property (R) is clear, so it remains to estimate |I_{b}(∼)|.

By pigeonholing, we can find P′ ⊂ P and a positive integer ν so that the following hold:

1. |P′|ν ≳ (log \frac{1}{δ})^{−1}|I_{b}|.

2. If π ∈ P′ then |{w ∈ W : (w, π) ∈ I_{b}(∼)}| ∈ [ν, 2ν].

If π ∈ P′, then π intersects ≲ t^{−1} t-cubes. Hence there is a t-cube Q′(π) with Q′(π) ≁ π such that |W ∩ Q′(π) ∩ π| ≳ tν. Q(π) is at distance at least t from Q′(π); and by the maximality property in its definition, Q(π) must also satisfy |W ∩ Q(π) ∩ π| ≳ tν.

In view of the assumption that everything is contained in a fixed disc there are ≲ t^{−6} possible pairs of t-cubes (Q, Q′), so some pair must be (Q(π), Q′(π)) for ≳ t^{6}|P′| choices of π. Applying Lemma 2.1 to W ∩ Q(π) and W ∩ Q′(π), we obtain

t^{6}|P′| ≲ δ^{−ϵ}t^{− \frac{1}{2}} (\frac{|W|}{tν})^{\frac{3}{2}}
≲ δ^{−ϵ}t^{− \frac{1}{2}} (\frac{|W|}{t|I_{b}|/|P′|log\frac{1}{δ}})^{\frac{3}{2}}

so that

|I_{b}| ≲ log\frac{1}{δ}δ^{− \frac{2}{3}ϵ}t^{− \frac{16}{3}} |W||P′|^{\frac{1}{3}}
≲ δ^{−ϵ}t^{−6}|W||P|^{\frac{1}{3}}

as claimed.

(ii) This is done the same way, but we need a substitute for Lemma 2.1. For this we use the following easy bound: suppose we have two δ-separated sets of points W and B with d(w, b) ≥ t for each w ∈ W, b ∈ B, and a set of tubes T such that each τ ∈ T is incident to μ points or W and to ν points of B. Then

|T | ≲ t^{−1} \frac{|W||B|}{μν}  (18)

To prove (18) we argue as follows (see [19], p. 137-8). Consider all triples (τ, w, b) where τ ∈ T , w ∈ W, b ∈ B and τ is incident to both w and b. There are at least |T |μν such triples but at most t^{−1}|W||B|, since two t-separated points can have at most t^{−1} tubes^{1} in common. This proves (18).

The proof of (ii) is now entirely analogous to the proof of (i). We define ∼ via: for each τ ∈ T let Q(τ) be the t-cube Q for which |W ∩ Q ∩ τ| is maximum. Declare τ to be related to Q(τ) and to its neighbors. Let I_{b} be the set of all bad incidences. We can find T′ ⊂ T and ν so |T′|ν ≳ (log \frac{1}{δ})^{−1}|I_{b}| and each plate τ ∈ T′ has about ν bad incidences. Hence there is a cube Q′(π) containing ≳ tν points w ∈ W such that (w, π) is a bad incidence. Some pair of t-cubes must be (Q(π), Q′(π)) for at least t^{6}|T′| choices of τ. Applying (18), we obtain

1The bound is t^{−1} instead of t^{−2} because of the light ray structure.

t^{6}|T′| ≲ t^{−1} (\frac{|W|}{tν})^{2}
≲ t^{−1} (\frac{|W|}{t|I_{b}|/|T′|log\frac{1}{δ}} )^{2}

so that

|I_{b}| ≲ log\frac{1}{δ}t^{− \frac{9}{2}} |W||T′|^{\frac{1}{2}}
≲ δ^{−ϵ}t^{−5}|W||T |^{\frac{1}{2}}
□

Lemma 2.3 leads immediately to a certain refinement of itself. If ∼ is a relation as above, and h is a number then we define W^{h} = \{w ∈ W : w belongs to between \frac{h}{2} and h plates from P}, and I_{b}^{h} = {(w, π) ∈ I_{b} : w ∈ W^{h}}. We also make the analogous definitions for tubes.

\underline{Lemma 2.4} (i) If P is a δ-separated family of δ-plates and W is a δ-separated set of points then there is a relation satisfying (R) and with |I_{b}^{h}| ≲ δ^{−Cϵ}t^{−6}|W^{h}||P|^{\frac{1}{3}} for all h.

(ii) If T is a δ-separated family of δ-tubes and W is a δ-separated set of points then there is a relation satisfying (R) and with |I_{b}| ≲ δ^{−Cϵ}t^{−5}|W^{h}||T |^{\frac{1}{2}} for all h.

\underline{Proof} We do (i); (ii) is exactly the same. We may clearly assume h is a power of 2, and also that h ≤ Cδ^{− \frac{1}{2}} since otherwise W^{h} = ∅. For each dyadic integer h = 2^{j} ≤ Cδ^{− \frac{1}{2}} we apply Lemma 2.5 to P and W^{h}, obtaining a sequence of relations ∼_{j}. We then define Q ∼ π if q ∼_{j} π for some j. Property (R) will still hold, since there are ≲ log \frac{1}{δ} values of h, and the rest follows from the definitions and the bound in Lemma 2.3. □

We now prove a result for tubes which is stronger than the one in Lemmas 2.3(ii) and 2.4(ii); this will follow roughly by combining Lemma 2.4(i) for the family of plates A(T ) and Lemma 2.4(ii) for the families of tubes YT (Π).

\underline{Lemma 2.5} Let W ⊂ ℝ^{3} be a δ^{\frac{1}{2}} -separated set of points; T a family of δ^{\frac{1}{2}} -tubes with cardinality k; assume that for each plate Π ∈ A(T ) we have |YT (Π)| ≤ m. Then there is a relation ∼ satisfying (R) and with

|I_{b}^{μ}| ≲ δ^{−Cϵ}t^{−6}m^{\frac{1}{12}} k^{\frac{1}{6}} μ^{\frac{1}{2}} |W^{μ}|  (19)

for each μ.

\underline{Proof} We assume at first that

|YT (Π)| ∈ [\frac{m}{2}, m]

for each Π ∈ A(T ). Notice that this implies

|A(T )| ≲ \frac{k}{m}  (20)

We also first \underline{fix} μ and construct a relation ∼ satisfying (19) for the given value of μ. This relation ∼ is defined as follows.

1. Apply Lemma 2.4(i) to the family A(T ) obtaining a relation ∼_{1} between plates from A(Π) and t-cubes.

2. Apply Lemma 2.4(ii) to the family YT (Π) for each Π ∈ A(T) obtaining, for each Π, a relation ∼_{Π} between tubes from YT (Π) and t-cubes.

3. Define Q ∼ τ if either Q ∼_{1} Π(τ) or Q ∼_{Π(τ)} τ.

Property (R) is clear from the definition 3. and the corresponding property for ∼_{1} and ∼_{Π}. We now prove (19).

Fix a positive integer a and define

W^{μ,a,Π} = {w ∈ W^{μ} : ∑_{τ∈YT (Π)} χ_{τ}(w) ≥ a}

I_{b}(a, Π) = {(w, τ) : τ ∈ YT (Π), w ∈ W^{μ,a,Π}, w ∈ τ and w ≁_{Π} τ}

By Lemma 2.4(ii) we have

|I_{b}(a, Π)| ≲ δ^{−Cϵ}t^{−5}m^{\frac{1}{2}} |W^{μ,a,Π}|

Hence also

∑_{Π∈A(T )} |I_{b}(a, Π)| ≲ δ^{−Cϵ}t^{−5}m^{\frac{1}{2}} ∑_{Π} |W^{μ,a,Π}|

≲ δ^{−Cϵ}t^{−5}m^{\frac{1}{2}} \frac{μ}{a}|W^{μ}|  (21)

The last line followed since a point of W^{μ} is in W^{μ,a,Π} for ≲ \frac{μ}{a} choices of Π.

Now fix a and b and define W^{μ}(a, b) to be the set of points w ∈ W^{μ} such that, for at least b plates Π ∈ A(T ), we have both

(i). (w, Π) ∈ I_{b}(∼_{1}) and

(ii). There are at least a tubes τ ∈ YT (Π) such that (w, τ) is a ∼_{Π}-bad incidence.

Notice that if w and Π are as in (i), (ii), then w ∈ W^{μ,a,Π}, and furthermore if τ is as in (ii), then (w, τ) ∈ I_{b}(a, Π). We may therefore estimate W^{μ}(a, b) using (21). We obtain

|W^{μ}(a, b)| ≤ (ab)^{−1} ∑_{Π} |I_{b}(a, Π)|
≲ δ^{−Cϵ}t^{−5}m^{\frac{1}{2}} (a^{2}b)^{−1}μ|W^{μ}|  (22)

On the other hand, using property (i), and then Lemma 2.4(i) and estimate (20) we obtain

|W^{μ}(a, b)| ≤ b^{−1}|I_{b}(∼_{1})|
≲ δ^{−Cϵ}t^{−6}b^{−1}(\frac{k}{m})^{\frac{1}{3}} |W(μ)|  (23)

Now define

W^{μ}(ν) = {w ∈ W^{μ} : |{τ : (w, τ) ∈ I_{b}(∼)}| ≥ ν}

If w ∈ W^{μ}(ν) then by pigeonholing there are dyadic values of a and b with ab ≈ (log \frac{1}{δ})^{−1}ν such that, for at least b plates Π ∈ A(T ), there are at least a tubes τ ∈ YT (Π) such that (w, τ) ∈ I_{b}(∼). Thus w belongs to W^{μ}(a, b) with these values of a and b. Using (23),

(22) and summing over dyadic a we get that

ν|W^{μ}(ν)| ≲ ∑_{a} min (δ^{−Cϵ}t^{−6}a(\frac{k}{m})^{\frac{1}{3}} |W(μ)|, δ^{−Cϵ}t^{−5}m^{\frac{1}{2}} a^{−1}μ|W(μ)|)

≲ δ^{−Cϵ}t^{−6} (a_{0}(\frac{k}{m})^{\frac{1}{3}} + \frac{m^{\frac{1}{2}} μ}{a_{0}}) |W(μ)|

for any given a_{0}. Optimizing in regard to a_{0} we get

ν|W^{μ}(ν)| ≲ δ^{−Cϵ}t^{−6}m^{\frac{1}{12}} k^{\frac{1}{6}} μ^{\frac{1}{2}} |W^{μ}|

Summing over dyadic ν we obtain

|I_{b}^{μ}| ≲ ∑_{ν} ν|W^{μ}(ν)|
≲ δ^{−Cϵ}t^{−6}m^{\frac{1}{12}} k^{\frac{1}{6}} μ^{\frac{1}{2}} |W^{μ}|

which is (19).

We can clearly now construct a relation ∼ satisfying (19) for \underline{all} μ by the argument in the proof of Lemma 2.4, so it remains only to remove the assumption that |YT (Π)| ≥ \frac{m}{2}. For this, we partition T as ∪_{j:2^{j}≤m}T_{j} where

T_{j} = ∪_{|YT (Π)|∈(2^{j−1},2^{j}]^{Π∈A(T )}}YT (Π)

Then we apply what we have already done to construct a relation ∼_{j} between tubes from T_{j} and t-cubes satisfying (R) and satisfying (19) with m replaced by 2^{j} on the right hand side. Then we can define a relation ∼ between T and t-cubes by taking the union of the relations ∼_{j}. It clearly satisfies (R), and it satisfies (19) in view of the favorable dependence on m there. □

Notice that the hypothesis about the cardinality of X(Π) is always satisfied with m ≈ δ^{− \frac{1}{2}} , since a Cδ^{\frac{1}{2}} -plate contains O(δ^{− \frac{1}{2}} ) separated δ^{\frac{1}{2}} -tubes. Thus the following statement is immediate from (19):

\underline{Lemma 2.6} Let W be a \sqrt{δ}-separated set of points; T a family of δ^{\frac{1}{2}} -tubes with cardinality k. Then there is a relation satisfying (R) and with

|I_{b}^{μ}| ≲ δ^{−Cϵ}t^{−6}δ^{− \frac{1}{24}} k^{\frac{1}{6}} μ^{\frac{1}{2}} |W^{μ}|  (24)

for each μ.

What we actually need in the sequel is versions of the above lemmas incorporating “Schwartz tails”, which can now be obtained in a routine way.

We make some further definitions. If P is a family of tubes or plates we define

ΦP = ∑_{π∈P} ϕ_{π}  (25)

where ϕ_{π} was defined after the introduction. Given a relation ∼ between t-cubes and elements of P, we let

ΦP^{b}(x) = ∑_{π≁Q(x)} ϕ_{π}(x)  (26)

be the corresponding sum involving only tubes unrelated to the t-cube Q(x) containing x. We note that if P is a set of δ-plates then for any δ-cube Δ we have

max_{Δ} ΦP ≤ C min_{Δ} ϕP  (27)

where C depends only on the choice of M. There is a corresponding fact for families of δ^{\frac{1}{2}} -tubes, where Δ may then be taken to be a δ^{\frac{1}{2}} -cube.

If W is a set of points, T a set of tubes or plates, ∼ a relation as above, μ > 0 then we define

J_{b} = ∑_{w∈W} ΦT^{b} (w)

\underline{Lemma 2.7} Let W be a \sqrt{δ}-separated set, T a set of C\sqrt{δ}-tubes with cardinality k, μ a number, assume that ΦT ≤ μ on W. Then there is a relation between t-cubes and elements of T satisfying (R) and with

|J _{b}^{μ}| ≤ δ^{−Cϵ}t^{−6}δ^{− \frac{1}{24}} k^{\frac{1}{6}} μ^{\frac{1}{2}} |W| + δ^{100}|W|  (28)

\underline{Proof} If T ≥ 1 is dyadic then subdivide in T \sqrt{δ} cubes and define W^{T} to be a subset of W containing (exactly) one point in each such cube which contains a point of W; thus W^{T} is essentially a T \sqrt{δ}-separated set. If w ∈ W then we use below the notation w_{T} for the point of W^{T} which belongs to the same T \sqrt{δ} cube as w.

If τ ∈ T then let τ^{T} be the dilation of τ by T and T^{T} the family {τ^{T} }. T^{T} is evidently not separated, so we let T _{0}^{T} be a maximal separated subset. It is then clear that, for any τ ∈ T , τ^{T} will be contained in a suitable fixed dilate τ_{∗}^{AT}, where τ_{∗}^{T} ∈ T _{0}^{T}, and that there are only boundedly many possible choices for τ^{∗}.

We now apply a rescaled version of Lemma 2.6 to T _{0}^{T} and W^{T} for each dyadic T with T \sqrt{δ} ≤ δ^{Cϵ_{0}} for a suitable constant C, obtaining a family of relations which we call ≃_{T}; thus ≃_{T} is a relation between tubes from T _{0}^{T} and t-cubes. We then define a corresponding relation ∼_{T} between tubes from T and t-cubes via τ ∼ Q if τ_{∗}^{T} ≃_{T} Q for some β^{T} ∈ T _{0}^{T} with τ^{T} ⊂ τ_{∗}^{AT}, and we define τ ∼ Q if τ ∼_{T} Q for some T. Since there are only logarithmically many dyadic scales we see that the relation ∼ has property (R). It remains to check (28).

Suppose that w ∈ W and that τ ≁ Q(w). We express

ϕ_{τ}(w) ≤ ∑_{T:T \sqrt{δ}≤δ^{Cϵ_{0}}} T^{−M}χ_{τ_{T}} (w)

plus a small error. The error is bounded by a high negative power of the quantity δ^{−1} + dist(w, τ), and is therefore negligible for present purposes since the right side of (28) is ≥ δ^{100}|W|; we will in fact neglect this error term in the calculations below.

For each given τ and T choose τ_{∗}^{T} as above. We know that τ_{∗}^{T} is not ≃_{T} -related to the t-cube containing w. Accordingly, we can bound

ΦT^{b} (w) ≤ ∑_{T} T^{−M} ∑_{τ} |{τ_{∗}^{T} : (w_{T} , τ_{∗}^{T}) is a ≃_{T} -bad incidence}|

Any given point can arise as w_{T} for at most O(T^{3}) choices of w, and any given tube in T _{0}^{T} can arise as τ_{T}^{∗} for at most O(T^{3}) choices of τ. Accordingly,

J_{b} ≤ ∑_{T} T^{6−M}|{(x, σ) ∈ W^{T} × T _{0}^{T} : (x, σ) is a ≃_{T} -bad incidence}|  (29)

Further, if w ∈ W then w is at most a CμT^{M}-fold point for T^{T}, since ΦT (x) ≤ μ. A fortiori x is at most a CμT^{M}-fold point for T _{0}^{T}. We conclude using Lemma 2.6 for the relation ≃_{T} (and (29)) that

J_{b} ≤ ∑_{T} T^{6−M}δ^{−Cϵ}t^{−6}δ^{− \frac{1}{24}} k^{\frac{1}{6}} (T^{M}μ)^{\frac{1}{2}} |W|  (30)

We have overestimated in several of the factors here; for example, k bounds the cardinality of T _{0}^{T}.

For large M we have a favorable dependence on T here, and may sum up to obtain the lemma. □

Lemma 2.7 was of course based on Lemma 2.6. We can do the same in the context of Lemma 2.3(i), obtaining the following lemma whose detailed proof we omit.

\underline{Lemma 2.8} Let P be a set of δ-plates; W a δ-separated set in ℝ^{3}. Then there is a relation satisfying (R) and with

J_{b} ≤ δ^{−Cϵ}t^{−6}|W| |P|^{\frac{1}{3}}
□

In the sequel, we will apply Lemma 2.7 to T (P) where P is a set of δ-plates. We will be using the following terminology. If P is a set of δ-plates then we will say that P is \underline{type r} if for each τ ∈ T (P) the cardinality of XP(τ) is between r and 2r. For a general P, the \underline{type r component} of P is the subset P_{r} ⊂ P defined by

P_{r} = ∪(XP(τ) : r ≤ |XP(τ)| ≤ 2r)

Evidently P_{r} is of type r.

We insert the following fact here for future reference.

\underline{Lemma 2.9} Assume that P is type r. Then for any δ^{\frac{1}{2}} -cube Q we have

∫ ΦPϕ_{Q} ≲ δ^{\frac{1}{2}} r ∫ Φ_{T (P)}ϕ_{Q}

\underline{Proof} First consider the version of this where we ignore the tail of ϕ_{Q}, i.e. the estimate

∫_{Q} ΦP ≲ δ^{\frac{1}{2}} r ∫_{Q} Φ_{T (P)}  (31)

If a tube τ contains r parallel plates π_{j}, then we have ∑_{j=1}^{r} ϕ_{π_{j}}(x) ≲ rδ^{\frac{1}{2}} ϕ_{τ}(x) if x is not in the double of the infinite tube coaxial with τ. This gives (31) if Q doesn’t intersect the double of the infinite tube coaxial with τ. On the other hand, if Q is a δ^{\frac{1}{2}} - cube intersecting τ then ∫_{Q} ϕ_{τ} ≳ |Q| and ∫_{Q} ϕ_{π_{j}} ≲ δ^{\frac{1}{2}} |Q| for each j, and a corresponding statement is valid if Q intersects the infinite tube coaxial with τ since ϕ_{τ} and ϕ_{π_{j}} die at the same rate as one moves away from τ in its axis direction. This gives (31) in the remaining cases. The lemma follows since ϕ_{Q} is essentially a sum of indicator functions of δ^{\frac{1}{2}} -cubes. □.

In addition, we will need to deal with certain technical issues involving the distinction between measure and entropy; the following definition and lemma give an easy way of doing this.

\underline{Definition} A set E is δ-evenly covered if there is a constant a such that |E ∩ Δ| ∈ [aδ^{3}, 2aδ^{3}] for each δ-cube Δ which intersects E.

Notice that if E is δ-evenly covered and if F ⊂ E then we have

|F | ≲ aδ^{3}Eδ(F)  (32)

Conversely, if F is the intersection of E with the union of some collection of δ-cubes then we have

|F | ≳ aδ^{3}Eδ(F)  (33)

\underline{Lemma 2.10} Let E be a set with the property that for each 1-cube Q, E ∩ Q is either empty or of measure ≥ δ^{100000}. Then E has a δ-evenly covered subset E′ with

|E′| ≳ (log \frac{1}{δ})^{−1}|E|.

\underline{Proof} This is just a pigeonhole argument. Let S_{n} be the union of the δ-cubes Δ with the property that |Δ∩E| ∈ [2^{−(n+1)}δ^{3}, 2^{−n}δ^{3}]. Then S_{n} = ∅ if n < 0. Let N = C(log \frac{1}{δ})^{−1} for a suitable C. Then |(∪_{n≥N}S_{n}) ∩ E| ≤ 2^{−N} ≤ \frac{1}{2}|E| by the hypothesis concerning the measure of the intersection of E with 1-cubes. So there must be a choice of n ≤ N such

that |S_{n} ∩ E| ≳ (log \frac{1}{δ})^{−1}|E|. We now set E′ = S_{n} ∩ E. □

3. Localization property In this section and subsequently, we make the convention that the parameters N and

δ always satisfy N = δ^{−1}. In this section we always let t = δ^{ϵ^{0}} where ϵ_{0} was defined after the introduction.

If π is a 1×δ^{\frac{1}{2}} ×δ-plate, with respective axes e_{1}, e_{2}, e_{3}, then we let π_{∗} be a rectangle and centered at the point Ne_{3} with axes e_{1}, e_{2}, e_{3} and respective axis lengths C_{1}, C_{1}N^{\frac{1}{2}} , N, where C_{1} is a large constant. Thus π^{∗} is approximately dual to π and is contained in Γ_{N}(C) for suitable C.

An \underline{N-function} is a function f which has a decomposition

f = ∑_{π∈P} f_{π}

where P = P(f) is a separated family of δ-plates and 1. |f_{π}| ≤ ϕ_{π}.

2. f̂_{π} is supported on π^{∗}.

Such a decomposition is of course not unique; however, given an N-function we will always fix a family of plates for f, and the associated functions f_{π}. The properties of plate families derived in section 2 should evidently translate into properties of N-functions and we now carry this out. We note that we will use cancellation here, via the fact that functions with disjoint Fourier supports are orthogonal.

Let f be an N-function with family of plates P, and let P̃ be a subset of P. The \underline{subfunction} of f corresponding to P̃ is fP^{}˜=^{def} ∑_{π∈P^{}}˜ f_{π}.

Let Wλ = \{x : |f(x)| ≥ λ\}. We say that \underline{(f, Wλ) localizes} if there are a collection of functions f_{Q} (Q runs over t-cubes) such that

1. Each f_{Q} is the restriction to Q of the subfunction of f corresponding to a certain collection of plates P_{Q}, and

∑_{Q} |P_{Q}| ≤ (log \frac{1}{δ})^{C}|P|  (34)

2. The (Lebesgue) measure of the set ∪_{Q}\{x ∈ Q : |f_{Q}(x)| ≥ (log \frac{1}{δ})^{−C}λ\} is at least

(log \frac{1}{δ})^{−C}|Wλ|.

In the subsequent arguments we will use the terminology “A (log \frac{1}{δ})^{−C}-fraction of W”. This means a subset of W with measure ≥ (log \frac{1}{δ})^{−C}|W|.

The following lemmas are what we have been leading up to.

\underline{Lemma 3.1} Let f be an N-function with plate family P and assume that

|P| ≤ t^{300}λ^{3}  (35)

Then (f, Wλ) localizes.

Before giving the proof we record a simple uncertainty principle estimate which will be used several times below. Namely, suppose that suppf̂ is contained in |ξ| ≤ N and that |f(a)| ≥ λ. Then, for a suitable fixed constant C

If |x − a| ≤ (CN∥f∥_{∞})^{−1}λ then |f(x)| ≥ \frac{λ}{2}  (36)

by Bernstein’s inequality and the mean value theorem.

\underline{Proof of Lemma 3.1} Let k = |P(f)|, and notice that since k is an integer we necessarily have λ > 1. Since |f_{π}| ≤ ϕ_{π} and P is separated we also have λ ≤ ΦP(x) ≲ δ^{− \frac{1}{2}} if x ∈ Wλ. We now make a “pigeonhole” type reduction.

\underline{Claim} There are a value of μ ∈ [\frac{λ}{2}, Cδ^{− \frac{1}{2}} ] and a set W ⊂ W \frac{λ}{2} so that the following

hold:

1. |W| ≥ (log \frac{1}{δ})^{−C}|Wλ|.

2. If x ∈ W then ΦP(x) ∈ [μ, 2μ].

3. W is δ-evenly covered.

For this, we show first that there is a set W_{0} ⊂ W \frac{λ}{2} such that |W_{0}| ≥ (log \frac{1}{δ})^{−C}|Wλ|

and

(∗) W_{0} intersects each 1-cube either in measure ≥ δ^{5} or not at all.

Namely, we have ∥f∥_{∞} ≲ δ^{− \frac{1}{2}} , so, by (36), if Wλ intersects a 1-cube Q then |W \frac{λ}{2} ∩

Q| ≳ δ^{\frac{9}{2}} . We may therefore take W_{0} to be the set ∪(Q ∩ W \frac{λ}{2} : Q ∩ Wλ ≠ ∅).

Next we choose a value of μ ≥ λ so that W_{1}, a ≳ (log \frac{1}{δ})^{−1} proportion of W_{0} consists of points with ΦP(x) ∈ [μ, 2μ]. We discard from W_{1} the set ∪(Q∩W_{1} : |Q∩W_{1}| ≤ δ^{100}}, where Q runs over 1-cubes, and denote the remaining set by W_{2}. It is easily seen using (∗) that W_{2} still has measure ≳ (log \frac{1}{δ})^{−1}|W_{0}|. We then apply Lemma 2.10 to obtain a δ-evenly covered subset and let W be this subset. That proves the claim.

We apply Lemma 2.8 to the plate family P and a maximal δ-separated subset of W, obtaining a relation ∼ which satisfies ∫_{X} ΦP^{b} ≤ δ^{−Cϵ}t^{−6}k^{\frac{1}{3}} |X|. Here X is the union of the δ-cubes containing points of W, and we have used (27). By the even covering property we can replace X by W here; see (32), (33). Thus

∫W ΦP^{b} ≤ δ^{−Cϵ}t^{−6}k^{\frac{1}{3}} |W|

It follows by the assumption (35) that

∫W ΦP^{b} ≤ t^{90}λ|W|

Accordingly there is a subset W^{∗} ⊂ W with proportional measure and so that at each point x ∈ W^{∗} we have, say,

ΦP^{b}(x) ≤ tλ  (37)

Now we define two functions f^{∗} and h as follows: On each given t-cube Q

f^{∗} = ∑_{π∼Q} f_{π}

h = ∑_{π≁Q} f_{π}

Notice that even though f^{∗} is not a subfunction of f, nevertheless its restriction to any t-cube Q is the restriction to Q of a subfunction of f. Evidently f^{∗} + h = f, and |h| is ≲ ΦP^{b} and therefore much less than λ on W^{∗} so |f^{∗}| must be ≥ \frac{λ}{2} on W^{∗}. Let f_{Q} be f^{∗} restricted to Q, and P_{Q} the plates related to Q. The bound (34) follows from the property (R), so we are done. □

Lemma 3.1 does not suffice for what we want to do, because the assumption (35) is too strong^{2}. Accordingly we now give another lemma of the same type; notice that the hypothesis (38) below is considerably weaker than (35) when λ is close to its maximum possible value δ^{− \frac{1}{2}} .

\underline{Lemma 3.2} Let f be an N-function with plate family P and assume that

|P| ≤ t^{10000}δ^{\frac{11}{4}} λ^{9}  (38)

Then either (f, Wλ) localizes, or else there are a subfunction f^{∗} of f, and a subset W ⊂ W \frac{λ}{2} with |W| ≥ (log \frac{1}{δ})^{−C}|Wλ|, so that |f^{∗}| ≥ (log \frac{1}{δ})^{−C}λ on W, and so that

∥ψ_{Δ}f^{∗}∥_{2}^{2} ≤ t^{1000}δ^{\frac{5}{2}} λ^{3}  (39)

2We note though that any improvement on the exponent \frac{1}{3} in Lemma 2.3(i) would lead to a version of Lemma 3.1 where the exponent 3 is replaced by some q > 3, and that this would allow us to prove Theorem 1 for some p without using Lemma 3.2.

for each δ^{\frac{1}{2}} -cube Δ.

\underline{Proof} This is related to the proof of Lemma 3.1, but we use Lemma 2.7 instead of Lemma 2.8. We note that necessarily λ > 1, and as with Lemma 3.1 we begin the proof with some pigeonhole reductions. Let k = |P|. Let P_{r} be the type r component of P. For some r we must have |\{x ∈ Wλ : |f_{P_{r}}(x)| ≥ (log \frac{1}{δ})^{−1}λ\}| ≳ (log \frac{1}{δ})^{−1}|Wλ|. With this value of r, let T = T (P_{r}) (= {τ ∈ T (P) : r ≤ |XP(τ)| ≤ 2r}). We clearly have

|T | ≤ \frac{k}{r}  (40)

A minor variant on the proof of the claim in the proof of Lemma 3.1 now shows the following:

There are a value of r, a value of μ ≲ δ^{− \frac{1}{2}} , and a subset W ⊂ W \frac{λ}{2} ∩ \{x : |f_{P_{r}}(x)| ≥

\frac{1}{2}(log \frac{1}{δ})^{−1}λ\} so that the following hold:

1. |W| ≥ (log \frac{1}{δ})^{−C}|Wλ|.

2. If x ∈ W then ΦT (x) ∈ [μ, 2μ].

3. W is \sqrt{δ}-evenly covered.

Now we consider cases.

Case 1: λ ≥ t^{−1500}δ^{− \frac{1}{24}} (\frac{k}{r})^{\frac{1}{6}} μ^{\frac{1}{2}} .

Case 2: λ ≤ t^{−1500}δ^{− \frac{1}{24}} (\frac{k}{r})^{\frac{1}{6}} μ^{\frac{1}{2}} .

In case 1., let S be the set of all \sqrt{δ}-cubes which intersect W, and let X = ∪_{Q∈S}Q. Then ΦT (x) is approximately equal to μ for any x ∈ X by (27) and the subsequent discussion. It follows using Lemma 2.8 (and (40)) that we have a relation between tubes from T and t-cubes satisfying (R) and

∫_{X} ΦT^{b} ≤ δ^{−Cϵ}t^{−6}δ^{− \frac{1}{24}} (\frac{k}{r})^{\frac{1}{6}} μ^{\frac{1}{2}} |X| + δ^{100}|X| ≲ t^{1000}λ|X|

The second inequality here follows from the hypothesis of case 1 (and the fact that λ ≥ 1). We conclude that there is a subset S_{1} of S with proportional cardinality so that ΦT^{b} ≤ tλ on the union of the S_{1}-cubes. Let W_{1} = W ∩ ∪_{Δ∈S_{1}}Δ. Then |W_{1}| is comparable to |W| in view of the even covering property (see (32), (33)). On the other hand, we can define a relation between plates π ∈ P_{r} and t-cubes via π ∼ Q if τ(π) ∼ Q. Now we proceed as in the proof of Lemma 3.1: define two functions f^{∗} and h as follows. On each given t-cube Q, define f^{∗} and h via

f^{∗} = ∑_{π∼Q^{π∈P r}} f_{π}

h = ∑_{π≁Q^{π∈P r}} f_{π}

Then f^{∗} +h = f_{P_{r}}, and |h| is ≲ ΦT^{b} and is therefore much less than (log \frac{1}{δ})^{−1}λ on W_{1}. So |f^{∗}| must be ≳ (log \frac{1}{δ})^{−1}λ on W_{1}. The bound (34) follows from property (R), so (f, Wλ) localizes.

In case 2. we will show that f_{P_{r}} satisfies (39). The functions ψ_{Δ}f_{π} (for fixed Δ and variable π ∈ P_{r}) are essentially orthogonal^{3}. Accordingly

∥ψ_{Δ}f_{P_{r}}∥_{2}^{2} = ∥ ∑_{π∈P_{r}} ψ_{Δ}f_{π}∥_{2}^{2}
≲ ∑_{π∈P_{r}} ∥ψ_{Δ}f_{π}∥_{2}^{2}  (41)

≲ ∫ ∑_{π∈P_{r}} |f_{π}|^{2}ϕ_{Δ}

≲ ∫ Φ_{P_{r}}ϕ_{Δ}  (42)

On the other hand the hypothesis of case 2 says that

r ≤ t^{−9000}kδ^{− \frac{1}{4}} μ^{3}λ^{−6}
≲ t^{−9000}kδ^{− \frac{1}{4}} δ^{− \frac{3}{2}} λ^{−6}  (43)

For the four inequalities below, we use (42) and Lemma 2.9, then a simple calculation based on the fact that ΦT is pointwise ≲ δ^{− \frac{1}{2}} , then (43) and finally the hypothesis (38):

∥ψ_{Δ}f_{P_{r}}∥_{2}^{2} ≲ δ^{\frac{1}{2}} r ∫ ΦT ϕ_{Δ}

3By this we mean that (41) is valid, which follows in a standard way: roughly, for any pair of incomparable π’s, either the Fourier supports are disjoint or their x-supports are disjoint up to tails, since the support of ψ̂_{Δ} has diameter ≤ N^{\frac{1}{2}} . See also section 4 below.

≲ rδ^{\frac{3}{2}}
≲ t^{−9000}δ^{− \frac{1}{4}} kλ^{−6}
≲ t^{1000}δ^{\frac{5}{2}} λ^{3}

The lemma is proved. □

4. Properties of ∥ ∥_{p,mic} The purpose of this section is to derive certain properties of the norms ∥ ∥_{p,mic} which

we need below. The arguments here are quite standard. What we do is (1) to track down the behavior of ∥ ∥_{p,mic} under various kinds of rescaling and (2) to relate ∥f∥_{p,mic} to N-functions. In this section t is arbitrary, i.e. may not be equal to δ^{ϵ^{0}}.

The following “interpolation inequality” is essentially obvious:

∥f∥_{p,mic} ≲ ∥f∥2^{\frac{2}{p}} ∥f∥_{∞,mic}^{1− \frac{2}{p}}, p ≥ 2  (44)

Next we estimate ∥f∥_{p,mic} for N-functions; we will prove a converse to Lemma 4.1 in Lemma 4.4 below.

\underline{Lemma 4.1} If f is an N-function with plate family P then (for p ≥ 2)

∥f∥_{p,mic} ≲ (N^{− \frac{3}{2}} |P|)^{\frac{1}{p}}  (45)

\underline{Proof} By (44) it suffices to do the cases p = 2 and p = ∞. The case p = 2 follows from the essential orthogonality of the functions f_{π} and the case p = ∞ follows by writing down the equation

Ξ_{Θ} ∗ f = ∑_{π} Ξ_{Θ} ∗ f_{π}

and observing that for fixed Θ the π’s with Ξ_{Θ} ∗ f_{π} ≠ 0 must all be roughly parallel, and therefore have bounded overlap. □

In Lemma 4.4 below we will prove a type of converse to Lemma 4.1.

If f̂ is supported in Γ_{N}(C) then (for cubes Q of side t ≤ 1) we let f_{Q} = (ψ_{Q}f) ∘ a_{Q}.

\underline{Lemma 4.2} f̂_{Q} is supported in Γ_{tN}(C′) where C′ depends on C, and

∥f_{Q}∥_{p,mic} ≲ t^{−( \frac{1}{2} + \frac{1}{p})}∥f∥2^{\frac{2}{p}} ∥f∥_{∞,mic}^{1− \frac{2}{p}}  (46)

provided p ≥ 2. In addition, if t ≥ N^{− \frac{1}{2}} then

∥f_{Q}∥_{2} ≲ N^{\frac{1}{4}} ∥f∥_{∞,mic}  (47)

\underline{Proof} The Fourier support statement is clear using the compact Fourier support of ψ. To prove (46), first consider the case p = 2, where (46) is basically the uncertainty principle. Namely, f_{Q} = ψ · (f ∘ a_{Q}), with { \atop f ∘ a_{Q}} supported in the Ct-neighborhood of Γ_{tN}. A standard argument with Schur’s test shows that ∥f_{Q}∥_{2} ≲ t^{\frac{1}{2}} ∥f ∘ a_{Q}∥_{2} = t^{−1}∥f∥_{2}.

We also have ∥f_{Q}∥_{∞,mic} ≲ t^{− \frac{1}{2}} ∥f∥_{∞,mic} since each sector of angular length (tN)^{− \frac{1}{2}} intersects O(t^{− \frac{1}{2}} ) sectors of angular length N^{− \frac{1}{2}} . Using (44) we get

∥f_{Q}∥_{p,mic} ≲ ∥f_{Q}∥2^{\frac{2}{p}} ∥f_{Q}∥_{∞,mic}^{1− \frac{2}{p}}
≲ t^{−( \frac{1}{2} + \frac{1}{p})}∥f∥2^{\frac{2}{p}} ∥f∥_{∞,mic}^{1− \frac{2}{p}}

as claimed.

To prove (47) we use that the Fourier supports of the functions ψ_{Q} · (Ξ_{Θ} ∗ f) for fixed Q and varying Θ (Θ is an arc of angular length N^{− \frac{1}{2}} ) have bounded overlap in view of the assumption t ≥ N^{− \frac{1}{2}} . Thus

∥f_{Q}∥_{2}^{2} ≲ ∑_{Θ} ∥ψ · (Ξ_{Θ} ∗ f) ∘ a_{Q})∥_{2}^{2}
≲ ∑_{Θ} ∥f∥_{∞,mic}^{2} · ∥ψ∥_{2}^{2}
≲ N^{\frac{1}{2}} ∥f∥_{∞,mic}^{2}
□

We will also need to use a rescaling argument with suitable Lorentz transformations which we record next. Suppose that Θ is an arc of length ρ > N^{− \frac{1}{2}} centered at a point e ∈ S^{1}. Let ω be a vector orthogonal to (e, 1) and (e, −1) and let T_{Θ} be the transformation (a scalar multiple of a Lorentz transformation) mapping (e, 1) to ρ^{−2}(e, 1), (e, −1) to (e, −1), and ω to ρ^{−1}ω. Let T_{Θ}^{−t} be the inverse (transpose).

\underline{Lemma 4.3} Let Θ be an arc of S^{1} with angular length ρ, Ξ_{Θ} an angular frequency cutoff^{4} corresponding to Θ. Suppose that û is supported on the set {ξ ∈ Γ_{N}(1) : Ξ̂_{Θ}(ξ) ≠ 0}. Let v = u ∘ T _{Θ}^{−t}. Then v̂ is supported on Γ_{ρ^{2}N}(C) for suitable C and

∥v∥_{∞,mic} ≲ ∥u∥_{∞,mic}

\underline{Proof} This follows from the fact that sectors of Γ_{N} of angular length N^{− \frac{1}{2}} contained in Θ correspond to sectors of Γ_{ρ^{2}N} of angular width (ρ^{2}N)^{\frac{1}{2}} under T_{Θ}; see for example [17] or [20] for further details regarding this sort of calculation. □

\underline{Lemma 4.4} Suppose that f̂ is supported on Γ_{N}(A). Then f is the sum of at most C = C_{A} functions f̃ each of which has a decomposition f̃ = ∑λ λfλ, where the sum is over dyadic λ satisfying

λ ≲ ∥f∥_{∞,mic}  (48)

Each fλ is an Ñ-function with Ñ ∈ [C^{−1}N, CN], and

∑ ; λ λ^{p}δ^{\frac{3}{2}} |P(fλ)| ≲ ∥f∥_{p,mic}^{p}  (49)

for each fixed p ∈ [2, ∞).

\underline{Remark} We have to allow several f̃’s, due to the fact that we have defined N-function using a fixed choice of the constant C_{1}. However, it is clear that this does not cause any difficulties and we will ignore this technicality when Lemma 4.4 is applied, namely in the proof of Lemma 4.6 and in the final step of the proof of Theorem 1.

\underline{Proof} By the form of the statement we can assume that Ξ_{Θ} ∗ f is nonzero for only one choice of Θ (thus ∥f∥_{p,mic} = ∥f∥_{p}), and since we are allowing several f̃’s it is also easy to see that we can assume the support of f̂ is contained in (say) N ≤ |ξ| ≤ \frac{11}{10}N. We fix a plate π so that the middle half of π^{∗} contains the intersection of the region N ≤ |ξ| ≤ \frac{11}{10}N with the Fourier support of Ξ_{Θ}. This is possible if the constant in the definition of π^{∗} was chosen large enough. Taking ψ_{j} = ψ_{π_{j}} we obtain a family of functions {ψ_{j}} with the following properties:

4Of the usual type as discussed in the introduction. Our notation here is a little inconsistent since Θ does not have length N^{− \frac{1}{2}} .

1. For each j ψ_{j} = η_{j}^{2} where η̂_{j} is supported in the dilation by a small factor α << 1 of the rectangle dual to π and centered at the origin.

2. The functions {ψ_{j}} form a partition of unity.

3. |η_{j}| ≤ Cϕ_{π_{j}}; here {π_{j}} runs through a tiling of ℝ^{3} by translates of π.

We decompose f = ∑_{j} ψ_{j}f. Then we define

fλ = λ^{−1} ∑_{j:∥η_{j}f∥_{∞}∈[λ,2λ]} ψ_{j}f  (50)

It is clear that fλ vanishes identically if λ is large compared with ∥f∥_{∞} = ∥f∥_{∞,mic}. Further, we evidently have |ψ_{j}f| ≲ λϕ_{π_{j}}, and the support of ψ̂_{j}f is contained in the dilation of π^{∗} by a factor \frac{1}{2} + Cα, so the definition (50) exibits fλ as an N-function. We now make the estimate (49). By Bernstein’s inequality

∥η_{j}f∥_{∞}^{p} ≲ |π^{∗}|∥η_{j}f∥_{p}^{p} = δ^{− \frac{3}{2}} ∥η_{j}f∥_{p}^{p}

Hence

∑ ; λ λ^{p}δ^{\frac{3}{2}} |Pλ| ≲ ∑_{j} ∥η_{j}f∥_{p}^{p} ≲ ∥f∥_{p}^{p}

as claimed. □

In practice we will need a “localized” version of Lemma 4.4. We first prove a sublemma. Fix t ∈ [CN^{−1}, 1]. Let {ψ_{Q}} be the partition of unity associated to the covering by t-cubes obtained from ψ by scaling as previously defined. In the next lemma we abuse notation a bit by letting Ξ_{Θ} be functions whose Fourier transforms agree on Γ_{N}(A) with an angular partition of unity subordinate to a covering J involving Γ_{N} by arcs of angular length (tN)^{− \frac{1}{2}} (rather than N^{− \frac{1}{2}} ).

\underline{Lemma 4.5} With this notation, if f̂ is supported in Γ_{N}(1) then the Fourier support of (ψ_{Q}f) ∘ a_{Q} is contained in the C-neighborhood of Γ_{tN} and

∑_{Q} ∥(ψ_{Q}f) ∘ a_{Q}∥_{p,mic}^{p} ≲ t^{−3} ∑_{Θ∈J} ∥Ξ_{Θ} ∗ f∥_{p}^{p}  (51)

\underline{Proof} The Fourier support of ψ_{Q} is contained in a disc of size t^{−1} ≤ (\frac{N}{t})^{\frac{1}{2}} . Accordingly, for each Θ ∈ J the cardinality of the set of J (Θ) = {Θ′ ∈ J : ∃Q : Ξ_{Θ}∗(ψ_{Q}·Ξ_{Θ′} ∗f) ≠ 0} is bounded and similarly |{Θ : Θ′ ∈ J (Θ)}| is bounded. We therefore have

∑_{Θ∈J} ∑_{Q} ∥Ξ_{Θ} ∗ (ψ_{Q}f)∥_{p}^{p} ≲ ∑_{Θ} ∑_{Q} ∑_{Θ′∈J (Θ)} ∥Ξ_{Θ} ∗ (ψ_{Q} · Ξ_{Θ′} ∗ f)∥_{p}^{p}

≲ ∑_{Q} ∑_{Θ′} ∥ψ_{Q}Ξ_{Θ′} ∗ f∥_{p}^{p}
≲ ∑_{Θ′} ∥Ξ_{Θ′} ∗ f∥_{p}^{p}

which gives (51) after rescaling. □

\underline{Lemma 4.6} Suppose that f̂ is supported in Γ_{N}(1) and ∥u∥_{∞,mic} = 1. Fix λ ≥ δ^{10}. Then for t ∈ [CN^{−1}, 1] we have the following^{5}. Consider a decomposition of ℝ^{3} in t-cubes Q as above. There is a value of λ_{∗} ∈ [C^{−1}δ^{2ϵ}λt^{\frac{1}{2}} , C(tN)^{\frac{1}{2}} ] and a family of tN-functions f_{Q} with plate families P_{Q}, so that

1) A (log \frac{1}{δ})^{−C} fraction of the set \{|f| ≥ λ\} is contained in ∪_{Q}a_{Q}^{−1}(\{|f_{Q}| ≥ λ^{∗}\}).

2) ∑_{Q} |P_{Q}| ≤ δ^{−Cϵ}(\frac{λ_{∗}}{λ})^{p}(\frac{N}{t})^{\frac{3}{2}} ∑_{Θ} ∥Ξ_{Θ} ∗ f∥_{p}^{p} for any given p ≥ 2; here Θ runs over a family of arcs of angular length (tN)^{− \frac{1}{2}} as in Lemma 4.5.

3) |P_{Q}| ≤ δ^{−Cϵ}(\frac{λ_{∗}}{λ})^{2}(\frac{N}{t})^{\frac{3}{2}} ∥ψ_{Q}f∥_{2}^{2} for each Q.

\underline{Proof} Decompose f = ∑_{Q} ψ_{Q}f where Q runs over t-cubes. We have a bound

|(ψ_{Q}f)(x)| ≲ δ^{− \frac{1}{2}} (1 + t^{−1}dist(x, Q))^{−M}  (52)

for each fixed M and all x, in view of the Schwartz decay of ψ and the fact that ∥f∥_{∞} ≲ δ^{− \frac{1}{2}} ∥f∥_{∞,mic}. It follows in a standard way that for any given ϵ

{|f| ≥ λ} ⊂ ∪_{Q}{|ψ_{Q}f| ≥ Cϵ^{−1}δϵλ}  (53)

For each Q apply Lemma 4.4 at scale tN to (ψ_{Q}f) ∘ a_{Q}, obtaining a decomposition^{6}

(ψ_{Q}f) ∘ a_{Q} = ∑_{h} hg_{h}^{Q}

5Lemma 4.6 will be applied with t = N^{− \frac{1}{2}} .

6See the remark after the statement of Lemma 4.4.

Here the g_{h}^{Q} are tN-functions and

h ≲ ∥(ψ_{Q}f) ∘ a_{Q}∥_{∞,mic}
≲ t^{− \frac{1}{2}} ∥f∥_{∞,mic}
= t^{− \frac{1}{2}}  (54)

using (48) and then (46). For each Q, since λ ≥ δ^{10} there are ≲ log \frac{1}{δ} values of h such that |g_{h}^{Q}(x)| ≥ δ^{100}λ for some x. It follows that there is a value of h = h(Q) so that a ≳ (log \frac{1}{δ})^{−C} proportion of the set where |(ψ_{Q}f) ∘ a_{Q}| ≥ δϵλ is contained in the set where |hg_{h}^{Q}| ≥ δ^{2ϵ}λ. We have ∥g_{h}^{Q}∥_{∞} ≲ (tN)^{\frac{1}{2}} ∥g_{h}^{Q}∥_{∞,mic} ≲ (tN)^{\frac{1}{2}} and therefore

h(tN)^{\frac{1}{2}} ≳ δ^{2ϵ}λ  (55)

We can pigeonhole to get a fixed value of h so that part 1) of the lemma holds with f_{Q} = g_{h}^{Q} and λ_{∗} = δ^{2ϵ} \frac{λ}{h}. Inequalities (54) and (55) imply that δ^{2ϵ}t^{\frac{1}{2}} λ ≲ λ^{∗} ≲ (tN)^{\frac{1}{2}} . It remains to estimate the cardinality of P_{Q}, the plate family for g_{h}^{Q}.

Applying (49) on scale tN and using that h = δ^{2ϵ} \frac{λ}{λ_{∗}} we have

|P_{Q}| ≲ (tN)^{\frac{3}{2}} (δ^{2ϵ} \frac{λ}{λ_{∗}})^{−p}∥(ψ_{Q}f) ∘ a_{Q}∥_{p,mic}^{p}  (56)

Summing over Q and using Lemma 4.5 we get

∑_{Q} |P_{Q}| ≲ t^{−3}(tN)^{\frac{3}{2}} (δ^{2ϵ} \frac{λ}{λ_{∗}})^{−p} ∑_{Θ} ∥Ξ_{Θ} ∗ f∥_{p}^{p}

which is part 2. of the lemma. For 3., we just apply (56) with p = 2 and then use that ∥(ψ_{Q}f) ∘ a_{Q}∥_{2}^{2} = t^{−3}∥ψ_{Q}f∥_{2}^{2}. □

5. Proof of Theorem 1 In this section we always let t = δ^{ϵ^{0}}. We will carry out an induction argument of the following type. Fix p > 74 and suppose

that for functions with Fourier support in Γ_{N}(1) and with ∥f∥_{∞,mic} ≤ 1 we have

|\{|f| > λ\}| ≤ δ^{−α} \frac{∥f∥_{2}^{2}}{λ^{4}(λ\sqrt{δ})^{p−4}}  (57)

provided δ is small enough. Then, provided δ is small enough, for functions with Fourier support in Γ_{N}(1) and with ∥f∥_{∞,mic} ≤ 1 we also have

|\{|f| > λ\}| ≤ δ^{−β} \frac{∥f∥_{2}^{2}}{λ^{4}(λ\sqrt{δ})^{p−4}}  (58)

for any fixed β > (1 − \frac{ϵ_{0}}{4} )α.

We will see at the end of the section that once we establish (57)⇒(58) it is easy to obtain Theorem 1.

As a preliminary to the proof of (57)⇒(58) we make the following remarks.

1. Suppose that (57) holds for functions with Fourier support in Γ_{N}(1) and with ∥f∥_{∞,mic} ≤ 1. Then, for any fixed C (57) holds also for functions with Fourier support in Γ_{N}(C) and with ∥f∥_{∞,mic} ≤ 1, provided we include a constant factor A(C) on the right hand side. This is because the dilation of Γ_{N}(C) by a small fixed constant factor will be covered by a bounded number of sets of the form ΓÑ(1). This remark allows us to ignore the constant factors arising in some of the previous lemmas, e.g. Lemmas 4.3 and 4.4.

2. If (57) holds, then (for the same class of f) the corresponding strong type estimates

∥f∥_{p}^{p} ≲ log \frac{1}{δ}δ^{−α}δ^{2− \frac{p}{2}} ∥f∥_{2}^{2}  (59)

also hold. This follows since there are only logarithmically many relevant dyadic values for λ; |f(x)| is necessarily less than δ^{− \frac{1}{2}} , and on the other hand (59) with f replaced by (say) min(|f|, δ^{− \frac{1}{4}} ) is trivial.

We now prove a lemma.

\underline{Lemma 5.1} Fix p and α and assume we know (57). Let f be Fourier-supported in Γ_{N}(1) and such that ∥f∥_{∞,mic} = 1. Then for any λ ≥ δ^{10} there is a value λ_{∗} ∈ (λδ^{\frac{1}{4} +ϵ}, δ^{− \frac{1}{4}} ), and a collection of N^{\frac{1}{2}} -functions \{f_{Δ}\} so that

1. A (log \frac{1}{δ})^{−C} fraction of \{|f| ≥ λ\} is contained in ∪_{Δ}a_{Δ}^{−1}(\{|f_{Δ}| ≥ λ^{∗}\}).

2. The following estimates hold for any given ϵ > 0:

|P(f_{Δ})| ≤ δ^{−Cϵ}(\frac{λ_{∗}}{λ})^{2}δ^{− \frac{9}{4}} ∥ψ_{Δ}f∥_{2}^{2}  (60)

∑_{Δ} |P(f_{Δ})| ≤ δ^{−Cϵ}δ^{− \frac{9}{4}} · δ^{− \frac{α}{2}} \frac{∥f∥_{2}^{2}}{(\frac{λ}{λ_{∗}})^{4}(\frac{λ}{λ_{∗}} δ^{\frac{1}{4}} )^{p−4}}  (61)

\underline{Proof} We apply Lemma 4.6 with t = N^{− \frac{1}{2}} . The decomposition there satisfies 1., and it satisfies the estimate (60) (for this, apply 3. of Lemma 4.6 with t = N^{− \frac{1}{2}} ) and also, by 2. of Lemma 4.5, the estimate

∑_{Δ} |P(f_{Δ})| ≤ δ^{−Cϵ}δ^{− \frac{9}{4}} (\frac{λ_{∗}}{λ})^{p} ∑_{Θ} ∥Ξ_{Θ} ∗ f∥_{p}^{p}  (62)

where Θ runs over a family of N^{− \frac{1}{4}} -arcs. We now show that (62) and the inductive hypothesis (57) imply (61).

Namely, for each Θ a suitable Lorentz rescaling of Ξ_{Θ}∗f has Fourier support in Γ_{N^{\frac{1}{2}}} (C) and behaves as described by Lemma 4.3. So we can apply the inductive hypothesis (57) on scale N^{\frac{1}{2}} . We conclude by (59) that

∥Ξ_{Θ} ∗ f∥_{p}^{p} ≤ δ^{−Cϵ}δ^{− \frac{α}{2}} δ^{1− \frac{p}{4}} ∥Ξ_{Θ} ∗ f∥_{2}^{2}

Inserting this bound into (62) we obtain

∑_{Δ} |P(f_{Δ})| ≤ δ^{−Cϵ}δ^{− \frac{α}{2}} δ^{− \frac{9}{4}} \frac{∑_{Θ} ∥Ξ_{Θ} ∗ f∥_{2}^{2}}{(\frac{λ}{λ_{∗}})^{4}(\frac{λ}{λ_{∗}} δ^{\frac{1}{4}} )^{p−4}}

≈ δ^{−Cϵ}δ^{− \frac{α}{2}} δ^{− \frac{9}{4}} \frac{∥f∥_{2}^{2}}{(\frac{λ}{λ_{∗}})^{4}(\frac{λ}{λ_{∗}} δ^{\frac{1}{4}} )^{p−4}}

as claimed. □

\underline{Lemma 5.2} Assume (57) for δ small enough and that f is an N-function with associated family of plates P satisfying (35). Then (58) holds (for small δ) if β > (1 − ϵ_{0})α, provided we replace ∥f∥_{2}^{2} on the right hand side with δ^{\frac{3}{2}} |P|.

\underline{Proof} Let W be the set where |f| ≥ λ. Apply Lemma 3.1 with t = δ^{ϵ^{0}}. Thus

|W| ≤ (log \frac{1}{δ})^{C}| ∪_{Q} W_{Q}|, with W_{Q} = \{x ∈ Q : |f_{Q}(x)| ≥ (log \frac{1}{δ})^{−C}λ\}, where f_{Q} are

subfunctions of f and (34) holds.

Now, for each Q, apply the inductive hypothesis (57) to g_{Q} =^{def} (ψ_{Q}f_{Q}) ∘ a_{Q} with N replaced by tN, and λ replaced by (log \frac{1}{δ})^{−C}λ. We have ∥g_{Q}∥_{∞,mic} ≲ t^{− \frac{1}{2}} by Lemma 4.2 so we obtain

t^{−3}|W_{Q}| ≤ |\{|g_{Q}| > (log \frac{1}{δ})^{−C}λ\}|
≤ (\frac{t}{δ} )^{\frac{p}{2} −2} (\frac{t}{δ} )^{α} ∥g_{Q}∥_{∞,mic}^{p−2}∥g_{Q}∥_{2}^{2} (λ(log \frac{1}{δ})^{−C})^{−p}
≤ (\frac{t}{δ})^{\frac{p}{2} −2}(\frac{t}{δ})^{α}t^{−( \frac{p}{2} −1)}∥g_{Q}∥_{2}^{2}(λ(log \frac{1}{δ})^{−C})^{−p}
= (log \frac{1}{δ})^{Cp}δ^{−α}δ^{−( \frac{p}{2} −2)}t^{α−1}λ^{−p}∥g_{Q}∥_{2}^{2}

On the other hand, we have ∥g_{Q}∥_{2}^{2} ≲ t^{−2}∥f_{Q}∥_{2}^{2} by (46) with p = 2, and ∥f_{Q}∥_{2}^{2} ≲ δ^{\frac{3}{2}} |P_{Q}| by (45). Accordingly

∑_{Q} ∥g_{Q}∥_{2}^{2} ≤ t^{−2}δ^{\frac{3}{2}} (log \frac{1}{δ})^{C}|P|

by (34). It follows that

|W| ≤ (log \frac{1}{δ})^{C} ∑_{Q} |W_{Q}|
≤ (log \frac{1}{δ})^{C}δ^{−α}δ^{−( \frac{p}{2} −2)}t^{α+2}λ^{−p} ∑_{Q} ∥g_{Q}∥_{2}^{2}
≤ (log \frac{1}{δ})^{C}δ^{−α}δ^{−( \frac{p}{2} −2)}t^{α}λ^{−p} · δ^{\frac{3}{2}} |P|

The lemma follows since t = δ^{ϵ^{0}}. □

The next step is

\underline{Lemma 5.3} Assume (57) for δ small enough and that f is an N-function satisfying (38). Then (58) holds (for small δ) if β > (1 − \frac{ϵ_{0}}{2} )α, provided we replace ∥f∥_{2}^{2} on the right hand side with δ^{\frac{3}{2}} |P(f)|.

The proof of this is similar to the proof of Lemma 5.2, except we use Lemma 3.2 instead of Lemma 3.1, and obtain the localization effect on one of two scales, corresponding to the two possibilities in Lemma 3.2.

\underline{Proof of Lemma 5.3} By assumption Lemma 3.2 is applicable to the set W = \{|f| ≥ λ\}. If (f, W) localizes, then we obtain (58) for any β > (1 − ϵ_{0})α by the proof of Lemma 5.2, since the hypothesis (35) there was used only to guarantee the localization property.

On the other hand suppose we are in the second case of Lemma 3.2. Thus we have a subfunction f^{∗} of f and a subset W^{∗} ⊂ W with |W^{∗}| ≥ (log \frac{1}{δ})^{−C}|W| so that |f^{∗}| ≥ (log \frac{1}{δ})^{−C}λ on W^{∗}, and such that

∥ψ_{Δ}f^{∗}∥_{2}^{2} ≤ t^{1000}δ^{\frac{5}{2}} λ^{3}  (63)

for each δ^{\frac{1}{2}} -cube Δ.

Now apply Lemma 5.1 to f_{∗}. The resulting functions f_{Δ} and parameter λ_{∗} satisfy

1. \{|f| ≥ λ\}| ≲ (log \frac{1}{δ})^{C}| ∪_{Δ} a_{Δ}^{−1}(\{|f_{Δ}| ≥ λ^{∗}\})|.

2. |P(f_{Δ})| ≤ δ^{−Cϵ}t^{1000}λ_{∗}^{3}

3. ∑_{Δ} |P(f_{Δ})| ≤ δ^{−Cϵ}δ^{− \frac{9}{4}} · δ^{− \frac{α}{2}} _{\frac{∥f∥_{2}^{2}}{( \frac{λ}{λ_{∗}})^{4}( \frac{λ}{λ_{∗}} δ^{\frac{1}{4}} )^{p−4}}} .

Namely, 1. and 3. are immediate from 1. of lemma 5.1 and from (61). To obtain 2., we start from (60) and then substitute in (63); thus,

|P(f_{Δ})| ≤ δ^{−Cϵ}(\frac{λ_{∗}}{λ})^{2}δ^{− \frac{9}{4}} ∥ψ_{Δ}f^{∗}∥_{2}^{2}
≤ δ^{−Cϵ}(\frac{λ_{∗}}{λ})^{2}δ^{− \frac{9}{4}} · t^{1000}δ^{\frac{5}{2}} λ^{3}
≤ δ^{−Cϵ}t^{1000}λ_{∗}^{3}

since λ ≤ δ^{− \frac{1}{4} −ϵ}λ_{∗}.

The f_{Δ}’s are \sqrt{N}-functions, and property 2. implies that they satisfy (35) with λ_{∗} and \sqrt{N} replacing λ and N. Accordingly by Lemma 5.2

|\{|f_{Δ}| ≥ λ_{∗}\}| ≤ δ^{−Cϵ}δ^{− \frac{1}{2} (1−ϵ_{0})α} \frac{δ^{\frac{3}{4}} |P(f_{Δ})|}{λ_{∗}^{4}(λ_{∗}δ^{\frac{1}{4}} )^{p−4}}  (64)

Scale back down and sum over Δ. This gives

|\{|f| ≥ λ\}| ≤ (log \frac{1}{δ})^{C} ∑_{Δ} δ^{\frac{3}{2}} |\{|f_{Δ}| ≥ λ_{∗}\}|

≤ δ^{−Cϵ}δ^{\frac{3}{2}} ∑_{Δ} δ^{− \frac{1}{2} (1−ϵ_{0})α} \frac{δ^{\frac{3}{4}} |P(f_{Δ})|}{λ_{∗}^{4}(λ_{∗}δ^{\frac{1}{4}} )^{p−4}}

≤ δ^{−Cϵ}δ^{\frac{3}{2}} δ^{− \frac{1}{2} (1−ϵ_{0})α} \frac{δ^{\frac{3}{4}} δ^{−Cϵ}δ^{− \frac{9}{4}} · δ^{− \frac{α}{2}} ∥f∥_{2}^{2}}{λ_{∗}^{4}(λ_{∗}δ^{\frac{1}{4}} )^{p−4}(\frac{λ}{λ_{∗}})^{4}(\frac{λ}{λ_{∗}} δ^{\frac{1}{4}} )^{p−4}}

≤ δ^{−Cϵ}δ^{−( \frac{1}{2} (1−ϵ_{0})α+ \frac{α}{2} )} · \frac{δ^{\frac{3}{2}} |P|}{λ^{4}(λ\sqrt{δ})^{p−4}}

The four inequalities followed respectively from property 1., from (64), from property 3., and from Lemma 4.1 with p = 2. The lemma is proved. □

\underline{Proof of Theorem 1} As discussed at the beginning of the section the main step is to show that if (57) is true for functions Fourier supported in some^{7} Γ_{N}(1) with ∥f∥_{∞,mic} ≤ 1 then so is (58). Fix then a function f Fourier supported in some Γ_{N}(1) and with ∥f∥_{∞,mic} ≤ 1.

We observe to begin with that (58) follows from Tchebyshev’s inequality if

λ^{2}(λ\sqrt{δ})^{p−4} < 1

or equivalently if we do \underline{not} have

λ ≥ δ^{− \frac{1}{2} + \frac{1}{p−2}}  (65)

We will therefore assume that (65) holds.

Apply Lemma 5.1 to f. The resulting functions f_{Δ} satisfy

1. |\{|f| ≥ λ\}| ≲ (log \frac{1}{δ})^{C}| ∪_{Δ} a_{Δ}^{−1}(\{|f_{Δ}| ≥ λ^{∗}\})|.

2. |P(f_{Δ})| ≤ δ^{−Cϵ}t^{10001}δ^{\frac{11}{8}} λ_{∗}^{9}.

3. ∑_{Δ} |P(f_{Δ})| ≤ δ^{−Cϵ}δ^{− \frac{9}{4}} · δ^{− \frac{α}{2}} _{\frac{∥f∥_{2}^{2}}{( \frac{λ}{λ_{∗}})^{4}( \frac{λ}{λ_{∗}} δ^{\frac{1}{4}} )^{p−4}}} .

Namely, 1. and 3. follow immediately from 1. in Lemma 5.1 and from (61). We now verify 2. Namely, if Δ is a \sqrt{δ}-disc then by (47) and rescaling we have

∥ψ_{Δ}f∥_{2}^{2} ≲ δ

Hence by (60) and then (65) we have

|P(f_{Δ})| ≤ δ^{−Cϵ}(\frac{λ_{∗}}{λ})^{2}δ^{− \frac{5}{4}}
≤ δ^{−Cϵ}λ_{∗}^{2}δ^{− \frac{1}{4} − \frac{2}{p−2}}  (66)

7N is always taken sufficiently large.

We also have

λ_{∗} ≥ δ^{Cϵ}δ^{− \frac{1}{4} + \frac{1}{p−2}}  (67)

because λ_{∗} ≥ δ^{Cϵ}δ^{\frac{1}{4}} λ and λ satisfies (65). Since p > 74, (67) implies that

δ^{− \frac{1}{4} − \frac{2}{p−2}} λ_{∗}^{2} ≤ δ^{κ} · δ^{\frac{11}{8}} λ_{∗}^{9}

for a fixed κ = κ_{p} > 0. Combining this inequality with (66) proves 2. provided ϵ_{0} (and of course ϵ) have been chosen small enough.

We may therefore apply Lemma 5.3 to the f_{Δ}’s replacing δ with \sqrt{δ} and λ with λ_{∗}, obtaining that if γ > (1 − \frac{ϵ_{0}}{2} )α then

|\{|f_{Δ}| ≥ λ^{∗}\}| ≤ δ^{− \frac{γ}{2}} \frac{δ^{\frac{3}{4}} |P(f_{Δ})|}{λ_{∗}^{4}(λ_{∗}δ^{\frac{1}{4}} )^{p−4}}

We now repeat the last part of the proof of Lemma 5.3. Scaling back down and summing over Δ gives

|\{|f| ≥ λ\}| ≤ (log \frac{1}{δ})^{C} ∑_{Δ} δ^{\frac{3}{2}} |\{|f_{Δ}| ≥ λ_{∗}\}|

≤ δ^{−Cϵ}δ^{\frac{3}{2}} ∑_{Δ} δ^{− \frac{γ}{2}} \frac{δ^{\frac{3}{4}} |P(f_{Δ})|}{λ_{∗}^{4}(λ_{∗}δ^{\frac{1}{4}} )^{p−4}}

≤ δ^{−Cϵ}δ^{\frac{3}{2}} δ^{− \frac{γ}{2}} \frac{δ^{\frac{3}{4}} δ^{−Cϵ}δ^{− \frac{9}{4}} · δ^{− \frac{α}{2}} ∥f∥_{2}^{2}}{λ_{∗}^{4}(λ_{∗}δ^{\frac{1}{4}} )^{p−4}(\frac{λ}{λ_{∗}})^{4}(\frac{λ}{λ_{∗}} δ^{\frac{1}{4}} )^{p−4}}

≤ δ^{−Cϵ}δ^{− \frac{1}{2} (γ+α)} · \frac{∥f∥_{2}^{2}}{λ^{4}(λ\sqrt{δ})^{p−4}}

Thus (58) holds for any β > (1 − \frac{ϵ_{0}}{4} )α, as was to be proved.

We conclude by iterating (57)⇒(58) that (57) holds for arbitrary α > 0 provided ∥f∥_{∞,mic} ≤ 1. It remains to pass from this estimate to Theorem 1. It is not hard to see using the partition of unity {ψ(x − j)}_{j∈ℤ^{3}} and estimates like (52) that it suffices to prove Theorem 1 locally; we omit the details of this reduction. Assume then that f is Fourier-supported in Γ_{N}(1) and that ∥f∥_{p,mic} ≤ 1, and let Q_{0} be the unit cube; we need to show that ∥f∥_{L^{p}(Q_{0})} ≲ δ^{−( \frac{1}{2} − \frac{2}{p} +ϵ)}.

Using Lemma 4.4 we decompose f = ∑λ λfλ where ∥fλ∥_{∞,mic} ≲ 1 and (by Lemma 4.1) ∥fλ∥_{2}^{2} ≲ λ^{−p}; the sum is over dyadic values of λ bounded above by a negative power of δ. By (57), for any fixed ϵ > 0 we have

∥fλ∥_{p} ≲ δ^{−ϵ}δ^{−( \frac{1}{2} − \frac{2}{p})}λ^{−1}

for each λ. In view of the bound on λ, we can sum over (dyadic) λ ≥ δ^{100}, obtaining

∥ ∑_{λ>δ^{100}} λfλ∥_{p} ≲ δ^{−ϵ} log(\frac{1}{δ})δ^{−( \frac{1}{2} − \frac{2}{p})}

On the other hand, the function

∑_{λ≤δ^{100}} λfλ

is clearly bounded pointwise by δ^{50}, and therefore has a similar bound in L^{p} norm on the unit cube. The proof is complete. □

\underline{Remark} In the preceding argument we have changed scales from δ to \sqrt{δ} twice; once in the final argument, and once in the proof of Lemma 5.3. This was necessary in order to obtain Theorem 1 for some p. The reader will be able to see this by going through the logic setting λ = δ^{− \frac{1}{2}} , since in this case it is only the condition (38) that can be obtained by a single rescaling, and not the condition (35).

6. Proofs of the corollaries In this section Γ is redefined to be the full (rather than forward) light cone and

similarly with Γ_{N} etc. It is easy to see that Theorem 1 remains true, since the statement is invariant under complex conjugation which interchanges the forward and backward cones on the Fourier side. Let χ : ℝ → ℝ be a fixed Schwartz function which is nonzero on the interval [1, 2] and whose Fourier transform has compact support. Let η be a radial Schwartz function in ℝ^{2} whose Fourier transform is supported in \frac{1}{2} ≤ |ξ| ≤ 2 and such that

∑_{n≥0} η̂(\frac{ξ}{2^{n}})

is equal to 1 outside a compact set. Define η_{n}(x) = 2^{2n}η(2^{n}x). We start with the following observation.

\underline{Lemma 6.1} Assume that u is a solution of the wave equation,

□u = 0, u(·, 0) = f, \frac{∂u}{∂t}(·, 0) = g  (68)

Assume that the supports of f̂ and ĝ are contained in the annulus N ≤ |ξ| ≤ 2N. Let V (x, t) = η_{n} ∗ (χ(t)u(x, t)), where the convolution is in the space variables. Then the support of Ĝ is contained in Γ_{N}(C), N = 2^{n}, and (for p ≥ 2)

∥V ∥_{p,mic} ≲ ∥f∥_{p} + N^{−1}∥g∥_{p}  (69)

\underline{Proof} The Fourier support statement follows since the Fourier support of u is contained in Γ_{N} and χ̂ has compact support. The bound (69) also differs only notationally from known bounds. We give the argument assuming g = 0. It suffices to prove (69) when p = ∞, since it is obvious when p = 2. Let A be a smooth cutoff to a sector of angular width N^{− \frac{1}{2}} and Ξ the inverse Fourier transform of A. Then

Ξ ∗ V (x, t) = ∫ χ(t)A(ξ)η̂(\frac{ξ}{2^{n}}) ∫ e^{2πi(x−y)·ξ} cos(2πt|ξ|)f(y)dydξ

so it suffices to show that for fixed t and y the L^{1}(dx) norm of the function

B(x) = ∫ A(ξ)η̂(\frac{ξ}{2^{n}})e^{2πi((x−y)·ξ+t|ξ|)}dξ

is bounded by a constant. However, the latter statement is well known (cf. [11]) and follows from stationary phase. □

\underline{Proof of Corollary 2} For the local smoothing statement, we apply Theorem 1 to the function V in Lemma 6.1 obtaining

∥η_{n} ∗ (χ(t)u(x, t))∥_{p} ≤ Cϵ2^{n( \frac{1}{2} − \frac{2}{p} +ϵ)}(∥f∥_{p} + 2^{−n}∥g∥_{p})

hence by the Fourier support of V, if α > \frac{1}{2} − \frac{2}{p} then for some ϵ we also have

∥η_{n} ∗ (χ(t)u(x, t))∥_{p} ≲ 2^{−nϵ}(∥f∥_{p,α} + ∥g∥_{p,α−1})

We then sum over n to get the result.

For the cone multiplier statement, it suffices to show that if ρ̂δ is a smooth cutoff (with the natural bounds) to the δ-neighborhood of Γ_{1} then ∥ρδ ∗ f∥_{p} ≲ δ^{−α}∥f∥_{p} assuming the indicated conditions on p and α. Rescaling, it suffices to show that if R̂_{N} is a smooth cutoff to the 1-neighborhood of Γ_{N} then

∥R_{N} ∗ f∥_{p} ≲ N^{α}∥f∥_{p}  (70)

However, we have ∥R_{N} ∗ f∥_{p,mic} ≲ ∥f∥_{p}, since Ξ_{Θ} ∗ f for fixed Θ is obtained from f by convolution with a function with bounded L^{1} norm, and one can interpolate with L^{2}. So (70) follows from Theorem 1. □

In principle Corollary 3 follows from Theorem 1 using the relation between circular means and cone Fourier transforms or solutions of the wave equation; see [11] and [13]. However, there is no reference for the exact statement that we need so we give the argument. Let σ_{t} be normalized length measure on the circle of radius t. We will use the asymptotics

σ̂_{t}(ξ) = 2\sqrt{2π}(\frac{t}{|ξ|})^{\frac{1}{2}} cos(2πt|ξ| − \frac{π}{4} ) + O(|ξ|^{− \frac{3}{2}} )  (71)

We always assume 1 ≤ t ≤ 2; thus this asymptotics is uniform in t as |ξ| → ∞. We define σ_{t}^{n} : ℝ^{2} → ℝ by

σ_{t}^{n} = (2\sqrt{2π})^{−1}t^{− \frac{1}{2}} η_{n} ∗ σ_{t}

\underline{Lemma 6.2} Let n ∈ ℤ^{+} and let N = 2^{n}. Let f be a function in ℝ^{2} with ∥f∥_{∞} = 1, and define u : ℝ^{3} → ℝ by

u(x, t) = χ(t)σ_{t}^{n} ∗ f(x)

Then, when \frac{1}{2} ≤ t ≤ 2, we have u = v + w where

1. v̂ is supported in Γ_{N}(C) and v satisfies the estimates

∥v∥_{∞,mic} ≲ N^{− \frac{1}{2}}  (72)
∥v∥_{2}^{2} ≲ N^{−1}∥f∥_{2}^{2}  (73)

2. ∥w∥_{∞} ≲ N^{− \frac{1}{2}} ∥f∥_{2} \underline{Proof} Using the asymptotics (71) we express

σ̂_{t}^{n}(ξ) = η̂(\frac{ξ}{N})|ξ|^{− \frac{1}{2}} cos(2πt|ξ| − \frac{π}{4} ) + g_{t}(ξ)

where

∥ĝ_{t}∥_{L^{2}(dξ)} ≲ N^{− \frac{1}{2}}  (74)

for each fixed t. Define w(x, t) = χ(t)g_{t} ∗ f(x) and v = u − w. Part 2. of the lemma follows from (74) by Plancherel and Cauchy-Schwartz.

To prove 1. we express v as the real part of the function

V (x, t) = e^{−i \frac{π}{4}} χ(t) ∫ η̂(\frac{ξ}{N})|ξ|^{− \frac{1}{2}} e^{2πi(x·ξ+t|ξ|)}f̂(ξ)dξ

therefore

V̂ (ξ, τ) = e^{−i \frac{π}{4}} χ̂(τ − |ξ|)η̂(\frac{ξ}{N})|ξ|^{− \frac{1}{2}} f̂(ξ)

which is clearly supported in Γ_{N}(C), and is also clearly bounded by N^{− \frac{1}{2}} |f̂(ξ)|, which proves the estimate (73). It remains to prove (72). Let A be a smooth cutoff to a sector of angular width N^{− \frac{1}{2}} and Ξ the inverse Fourier transform of A. Then

Ξ ∗ V (x, t) = ∫ e^{−i \frac{π}{4}} χ(t)A(ξ) ∫ η̂(\frac{ξ}{N})|ξ|^{− \frac{1}{2}} e^{2πi((x−y)·ξ+t|ξ|)}f(y)dydξ

so it suffices to show that for fixed t and y the L^{1}(dx) norm of the function

B(x) = ∫ A(ξ)η̂(\frac{ξ}{N})|ξ|^{− \frac{1}{2}} e^{2πi((x−y)·ξ+t|ξ|)}dξ

is ≲ N^{− \frac{1}{2}} . However, the latter estimate is the same estimate from [11] that was used in the proof of Lemma 6.1. The lemma follows. □

\underline{Proof of Corollary 3} We can assume that the set F of Corollary 2 is contained in the region 1 ≤ t ≤ 2 and then also that the set E is contained in a large fixed disc.

We will use the notation in Lemma 6.1. Let f be a (measurable) function in ℝ^{2} with ∥f∥_{∞} ≤ 1. Assume {x : f(x) ≠ 0} ⊂ E. We will show first that the three dimensional Lebesgue measure of

Yλ =^{def} {(x, t) ∈ ℝ^{2} × [1, 2] : |σ_{t}^{n} ∗ f(x)| ≥ λ}

is ≤ Cϵλ^{−75}N^{−2+ϵ}|E| provided that λ is large compared with N^{− \frac{1}{2}} .

Namely, there is a function F : ℝ^{3} → ℝ with the following properties:

1. suppF̂ ⊂ Γ_{N}(C)

2. ∥F ∥_{∞,mic} ≲ N^{− \frac{1}{2}} and ∥F ∥_{2} ≲ N^{− \frac{1}{2}} |E|^{\frac{1}{2}} .

3. If (x, t) ∈ Yλ then |F (x, t)| ≥ \frac{λ}{2}.

This is clear by setting F equal to the function v in Lemma 6.2; we just note that 3. follows from the estimate in 2. of the lemma since λ >> N^{− \frac{1}{2}} and E is contained in a large fixed disc. It follows by (44) that

∥F ∥_{p,mic} ≲ N^{− \frac{1}{2}} |E|^{\frac{1}{p}}

hence by Theorem 1, if p = 75 and ϵ > 0 then

∥F ∥_{p} ≲ N^{− \frac{2}{p} +ϵ}|E|^{\frac{1}{p}}

Using Tchebyshev’s inequality and 3. we get

|Yλ| ≲ λ^{−p}N^{−2+ϵ}|E|

as claimed. Taking now f = χ_{E} we obtain the following:

For any ϵ > 0 there is η > 0 such that

|\{(x, t) ∈ ℝ^{2} × [\frac{1}{2}, 2] : |η_{n} ∗ σ_{t} ∗ χ_{E}(x)| ≥ \frac{1}{2}N^{−η}\}| ≤ N^{−2+ϵ}|E|  (75)

We can convert this to an entropy estimate on scale δ = N^{−1}:

For any ϵ > 0 there is η > 0 such that

Eδ (\{(x, t) ∈ ℝ^{2} × [\frac{1}{2}, 2] : |η_{n} ∗ σ_{t} ∗ χ_{E}(x)| ≥ N^{−η}\}) ≤ N^{1+ϵ}|E|  (76)

Namely, since ∥η_{n} ∗ σ_{t} ∗ χ_{E}∥_{∞} ≲ 1, (36) shows that if |η_{n} ∗ σ_{t} ∗ χ_{E}(x)| ≥ N^{−η} then |η_{n} ∗ σ_{t} ∗ χ_{E}(y)| ≥ \frac{1}{2}N^{−η} for all y belonging to the \frac{1}{2}N^{−(1+η)}-disc centered at x. This shows that (75) implies (76).

Now let α be a small positive number, let U be an open set with measure < α, K ⊂ ℝ^{2} × [1, 2] a set such that each circle C(x, t), (x, t) ∈ K, intersects U in measure ≥ C > 0. If (x, t) ∈ K then ∑_{n} |η_{n} ∗ σ_{t} ∗ χ_{U}(x)| ≳ 1, so for some n we have

|η_{n} ∗ σ_{t} ∗ χ_{U}(x)| ≳ n^{−2}  (77)

Notice that (77) cannot hold for any x if n is too small, i.e. if n^{−2}2^{−n} >> α. For the remaining values of n, we have shown above that the set where (77) holds has 2^{−n}-entropy ≲ 2^{(1+ϵ)n}|E| < 2^{(1+ϵ)n}. We now have a covering of K by a family of discs of dyadic radius 2^{−n} satisfying 2^{−n}n^{−2} ≲ α, and involving less than 2^{(1+ϵ)n} discs of any given radius 2^{−n}.

If E has Lebesgue measure zero then we can apply the preceding to an arbitrarily small open neighborhood U of E. We conclude that any set in ℝ^{2} ×ℝ consisting of circles which intersect E in outer measure ≥ C must have Hausdorff 1 + ϵ-dimensional measure zero. Letting C → 0 we obtain the result. □

Theorem 1 and its proof also imply various “L^{p} → L^{q} inequalities for the wave equation relative to fractal measures” which generalize the Strichartz inequality as well as the L^{3} estimate proved in [18]. To motivate this recall for example the well-known Strichartz inequality into L_{t}^{2}(L_{x}^{∞}): for solutions of (68), locally in time we have

∥u∥_{L_{t}^{q}(L_{x}^{∞})} ≲ ∥f∥_{W^{2,s}} + ∥g∥_{W^{2,s−1}}  (78)

provided \frac{3}{4} < s ≤ 1, q < \frac{1}{1−s}. This can be considered as an L^{q} estimate

∥u∥_{L^{q}(dμ)} ≲ ∥f∥_{W^{2,s}} + ∥g∥_{W^{2,s−1}}  (79)

for u with respect to a measure μ whose t-projections are Lebesgue and whose marginals are Dirac measures, which is a one dimensional measure. A similar interpretation is valid for the result of [18]. One can ask for analogues of all these inequalities in general dimension α, i.e. estimates from L^{p} Sobolev spaces to L^{q}(μ) for the solution of □ with the right dependence on the dimensionality of μ. For general α and p this appears hard; when α = 3 and p ≠ 2 it is essentially the local smoothing conjecture. The following result is satisfactory only when α ≤ 1; when α is large (i.e. in case (i)) it is a minor variant on Theorem 1 and the value of p is far from sharp. We refer also to [9] for a discussion concerning estimates of this type.

\underline{Proposition 6.1} Suppose that μ be a measure in ℝ^{3} supported in the unit disc and μ(D(x, r)) ≤ r^{α} for all discs D(x, r) of radius r ≤ 1. Then for functions with Fourier support in Γ_{N}(1)

∥u^{∗}∥_{L^{p}(μ)} ≲ N^{\frac{1}{2} +σ}∥u∥_{p,mic}  (80)

provided that σ > \frac{1−α}{p} and one of the following is true: (i) p > 77 − α or (ii) p > 3 and α < 1 + ϵ_{p,σ}, where ϵ_{p,σ} > 0 is a small (computable) constant.

We will use induction on N to show that for some fixed positive κ

μ(\{x : u(x) ≥ λ\} ≤ δ^{κ} \frac{δ^{−pσ}∥u∥_{p,mic}^{p}}{(λ\sqrt{δ})^{p}}  (81)

for functions with Fourier support in Γ_{N}(1). Since there are only logarithmically many relevant dyadic values for λ this weak type estimate implies (80).

Assume that (81) has been proved on scales much less than N and fix u. Suppose at first that it u an N-function with plate family P. In this case, we will show that

μ(\{x : |u(x)| ≥ λ\} ≤ δ^{ζ} \frac{δ^{−pσ}|P|δ^{\frac{3}{2}}}{(λ\sqrt{δ})^{p}}  (82)

Here ζ is a fixed positive number depending on α and p.

We first look at what follows formally from Theorem 1. Namely, if q > 74 then Theorem 1 and Lemma 4.1 imply for any given τ > 0 that

|\{x : |u(x)| ≥ λ\}| ≲ δ^{−τ}δ^{2}(λ\sqrt{δ})^{−q}|P|δ^{\frac{3}{2}}

The bound (36) implies that \{|u| ≥ λ\} behaves essentially like a collection of δ^{\frac{3}{2}} λ-discs. Using the dimension assumption on μ it therefore follows that

μ(\{x : u(x) ≥ λ\}) ≲ δ^{τ}(δ^{\frac{3}{2}} λ)^{α−3}δ^{2}(λ\sqrt{δ})^{−q}|P|δ^{\frac{3}{2}}
= δ^{τ}δ^{α−1}(λ\sqrt{δ})^{α−q−3}|P|δ^{\frac{3}{2}}  (83)

In case (i), this is clearly smaller that the right hand side of (82), so (82) is proved (without an induction argument) in that case.

In case (ii), we need to combine (83) with an induction argument based on the localization property in Lemma 3.1. There is a technical issue that arises, namely that we must redefine “localization” so that the measure in 2. of the definition is μ rather than Lebesgue measure. However, it is easy to see that Lemma 3.1 remains valid with this change. We now distinguish two cases.

1. (u, W) doesn’t localize (to scale t = δ^{ϵ^{0}}).

2. (u, W) localizes.

In case 1., by Lemma 3.1 we have |P| ≥ t^{C}λ^{3}. Fix a sufficiently small number η > 0. If λ < δ^{η− \frac{1}{2}} then provided the numbers ϵ_{p,σ} and ζ have been chosen small enough the estimate (82) follows since the right hand side of (82) is greater than 1. On the other hand, if λ > δ^{η− \frac{1}{2}} then the bound (83) implies (82).

It remains to consider case 2. In this case we let u_{Q} and P_{Q} be as in the definition^{8} of localization. We consider the rescalings

(ψ_{Q}u_{Q}) ∘ a_{Q}

Fix Q. Using Lemma 4.2 and then Lemma 4.1, we see that (ψ_{Q}u_{Q}) ∘ a_{Q} is Fourier supported in Γ_{tN}(C) and

∥(ψ_{Q}u_{Q}) ∘ a_{Q}∥_{p,mic}^{p} ≲ t^{−( \frac{p}{2} +1)}∥u_{Q}∥_{2}^{2}∥u_{Q}∥_{∞,mic}^{p−2}
≲ t^{−( \frac{p}{2} +1)}δ^{\frac{3}{2}} |P_{Q}|  (84)

Define a new measure ν by rescaling the restriction of μ to Q and then multiplying by t^{−α}, i.e.

dν(x) = t^{−α}χ_{0}(x)dμ(a_{Q}x)

where χ_{0} is the indicator function of the unit cube. Then ν is supported in the unit cube and satisfies ν(D(x, r)) ≲ r^{α}. We apply the inductive hypothesis on scale tN, and then substitute in the bound (84). This gives

ν(\{x : |(ψ_{Q}u_{Q}) ∘ a_{Q}(x)| ≥ λ\}) ≲ \frac{(\frac{δ}{t})^{κ−pσ}∥(ψ_{Q}u_{Q}) ∘ a_{Q}(x)∥_{p,mic}^{p}}{(λ\sqrt{\frac{δ}{t}})^{p}}

≲ \frac{(\frac{δ}{t})^{κ−pσ}t^{−( \frac{p}{2} +1)}δ^{\frac{3}{2}} |P_{Q}|}{(λ\sqrt{\frac{δ}{t}})^{p}}

Using that ψ_{Q} is bounded away from zero on Q and undoing the rescaling we arrive at

μ(\{x ∈ Q : |u_{Q}(x)| ≥ λ\}) ≲ t^{α} \frac{(\frac{δ}{t})^{κ−pσ}t^{−( \frac{p}{2} +1)}δ^{\frac{3}{2}} |P_{Q}|}{(λ\sqrt{\frac{δ}{t}})^{p}}

8To avoid any possible confusion we note that u here plays the role of f in that definition.

Summing over Q gives

∑_{Q} μ(\{x ∈ Q : |u_{Q}(x)| ≥ λ\}) ≤ (log \frac{1}{δ})^{C}t^{α} \frac{(\frac{δ}{t})^{κ−pσ}t^{−( \frac{p}{2} +1)}δ^{\frac{3}{2}} |P_{Q}|}{(λ\sqrt{\frac{δ}{t}})^{p}}

By the (current) definition of the localization property we then have

μ(\{x : u(x) ≥ λ\}) ≲ (log \frac{1}{δ})^{C} ∑_{Q} μ(\{x ∈ Q : |u_{Q}(x)| ≥ λ\})

≤ (log \frac{1}{δ})^{C}t^{α} \frac{(\frac{δ}{t})^{κ−pσ}t^{−( \frac{p}{2} +1)}δ^{\frac{3}{2}} |P_{Q}|}{(λ\sqrt{\frac{δ}{t}})^{p}}

The exponent of t here is α + pσ − 1, which is positive. Since t = δ^{ϵ^{0}}, it follows that (82) holds provided ζ has been chosen less than (α + pσ − 1)ϵ_{0}.

Now we can simply repeat the last steps in the proof of Theorem 1, decomposing a general function as a sum of N-functions and applying (82) to each. The conclusion (81) follows provided κ has been chosen less than ζ. □

Proposition 6.1 can be used to obtain various more explicit estimates. Let u be a solution of the wave equation (68). Using Lemma 6.1 and Proposition 6.1 one easily gets

∥u∥_{L^{p}(μ)} ≲ ∥f∥_{p, \frac{1}{2} +σ} + ∥g∥_{p,− \frac{1}{2} +σ}  (85)

with the same conditions on σ, p and μ. In particular this gives

\underline{Corollary 6.1} Assume p > 3. Consider a solution of (68). Then for measures with compact support and satisfying μ(D(x, r)) ≤ r^{α} we have

∥u∥_{L^{p}(μ)} ≲ ∥f∥_{p, \frac{1}{2}} + ∥g∥_{p,− \frac{1}{2}}  (86)

provided α > 1, and

∥u∥_{L^{p}(μ)} ≲ ∥f∥_{p, \frac{1}{2} +σ} + ∥g∥_{p,− \frac{1}{2} +σ}

provided α ≤ 1 and σ > \frac{1−α}{p}.

The proof is as indicated above. In particular, we have

∥u∥_{L_{t}^{p}(L_{x}^{∞})} ≲ ∥f∥_{p, \frac{1}{2} +ϵ} + ∥g∥_{p,− \frac{1}{2} +ϵ}

for any ϵ > 0 by the case α = 1 and finite propagation speed, so Corollary 6.1 includes the result of [18]. Note also that (86) gives an L^{p} version of Corollary 3.

\underline{Remark} It is also possible to derive Strichartz type estimates from Proposition 6.1, e.g. the following statement: suppose s ∈ (\frac{3}{4}, 1] and α > 4(1 − s). Then, for any q < \frac{α}{1−s},

∥u∥_{L^{q}(μ)} ≲ ∥f∥_{W^{2,s}} + ∥g∥_{W^{2,s−1}}  (87)

provided μ is a measure with fixed compact support and with μ(D(x, r)) ≤ r^{α}. This follows by decomposing u in N-functions as in Lemma 4.4 and applying Proposition 6.1 in an appropriate way. We omit the argument since it appears to us that (87) would also follow from (78) using a suitable version of the Marstrand projection theorem. However, it is possible that issues of this type could be of some interest when s < \frac{3}{4}.

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