AN INFORMAL PROOF OF WOLFF’S LOCAL SMOOTHING ESTIMATE

TERENCE TAO

Abstract. We give an informal proof of Wolff’s recent local smoothing estimate [2].

1. Introduction

Let N ≫ 1 be a large number. Let ϕ be a solution to the free wave equation in R^{2+1} with frequency N, and localized to spatial scale 1. By the uncertainty principle, the spatial scales thus range from the finest scale 1/N to the coarsest scale 1, while the frequency scales range from the finest scale 1 to the coarsest scale N.

We can write ϕ as a linear combination ∑_{P} c_{P} ϕ_{P} of Knapp examples, where the plates P are 1 × N^{−1/2} × N^{−1} boxes arranged along light rays, and ϕ_{P} is an L^{∞}- normalized wave supported near P with Fourier support in the dual 1 × N^{1/2} × N region. (We shall also refer to ϕ_{P} as a plate). This can be achieved by first decomposing ϕ in Fourier space along sectors, and then decomposing in physical space to the maximum amount allowed by the Uncertainty principle.

The local smoothing conjecture asserts that^{1}

∥ϕ∥_{L_{x,t}^{4}} ≲ ∥(∑_{P} |c_{P} ϕ_{P} |^{2})^{1/2}∥_{L_{x,t}^{4}}.

On the other hand, from the trivial observation that there can be at most N^{1/2} plates P intersecting at a point, we have

∥ϕ∥_{L_{x,t}^{∞}} ≲ N^{1/2}∥ sup_{P} |c_{P} ϕ_{P} |∥_{L_{x,t}^{∞}}.

Thus local smoothing implies

∥ϕ∥_{L_{x,t}^{p}} ≲ N^{1/2−2/p}∥(∑_{P} |c_{P} ϕ_{P} |^{p/2})^{2/p}∥_{L_{x,t}^{p}}

for 4 ≤ p ≤ ∞. We may try to strengthen this to

∥ϕ∥_{L_{x,t}^{p}} ≲ N^{1/2−2/p}∥(∑_{P} |c_{P} ϕ_{P} |^{p})^{1/p}∥_{L_{x,t}^{p}}  (1)

though one can show this strengthening is only possible when p ≥ 6.

In [2], Wolff showed

Theorem 1.1. [2] (1) holds for p ≥ 74.

1991 Mathematics Subject Classification. 42B15, 35L05.

1All epsilon powers of N shall be completely ignored, and spatial norms are always assumed to be localized to scale 1. Oh yes, and we’re ignoring Schwartz tails too.

The purpose of this note is to give an informal sketch of the argument. We shall gloss over many technical details, such as the mechanics of pigeonholing and of the induction-on-scales argument. We shall also be rather careless in our use of the uncertainty principle, orthogonality, etc.

Since ∥ϕ_{P} ∥_{L_{x,t}^{p}} ∼ N^{−3/2p}, we can rewrite (1) as

∥ϕ∥_{L_{x,t}^{p}} ≲ N^{1/2−2/p}N^{−3/2p}(∑_{P} |c_{P} |^{p})^{1/p}.

We pass to a restricted weak-type version of this estimate. Suppose that P ranges over a collection ¶ of M plates, and that c_{P} = 1 for these plates. The above estimate is then equivalent (modulo epsilons) to

|\{|ϕ| ≥ λN^{1/2}\}| ≲ λ^{p}N^{−3}(M/\sqrt{N}).

The bound for λ ≫ 1 is trivial from the L^{∞} theory. Since p is so large, this bound is really only interesting when λ ∼ 1, thus it becomes

|\{|ϕ| ∼ N^{1/2}\}| ≲ N^{−3}(M/\sqrt{N}).

Roughly speaking, the point is as follows. If one takes \sqrt{N} plates, then one can make them focus on a N^{−1}-cube, which has volume N^{−3}. More generally, if one has M plates for some M ≳ \sqrt{N}, then one can make them focus (with the maximal multiplicity of N^{1/2} on M/\sqrt{N} separate N^{−1}-cubes. But one cannot do any better than this (informally, one cannot use a plate to achieve maximal multiplicity in more than one location, on the average at least).

One may think of the ϕ_{P} as being like characteristic functions χ_{P}, although this is not completely accurate as they also retain some oscillation (For instance, the ϕ_{P} are all orthogonal). However, we won’t use this oscillation very much (in particular we don’t use any L^{4} theory).

Define a focussing cube to be a N^{−1} cube where |ϕ| ∼ N^{1/2}. In other words, a focussing cube is a place where N^{1/2} of the plates ϕ_{P} come together without any cancellation. Our task is thus to show that every plate can only contain at most one focussing cube, on the average.

In the above example, M plates have been grouped into M/\sqrt{N} bunches of \sqrt{N} plates, each of which only interact at a very small scale (a cube of sidelength 1/N, to be exact). This behaviour turns out to be absolutely typical: the only way that every plate can contain a focussing cube is if the plates organize themselves into groups which interact at a lower scale; by induction we can go all the way down to the smallest scale 1/N with this argument. Another way of saying this is that if we exclude smaller-scale interactions, then not every plate can contain a focussing cube. If we can prove this statement, then there is a reasonably standard induction-on-scales argument which can give back the whole thing.

Okay, so now suppose for contradiction that we have M plates, each of which contains a focussing cube, and such that a significant fraction of all interactions are at the largest scale (in that we can’t decompose these plates into bunches, each of which only interact in a cube of length ≪ 1).

Now we need to introduce some intermediate scales to analyze the superposition of these M plates. Currently these plates point in one of a N^{−1/2}-separated set of directions. Let’s introduce a sparser N^{−1/4}-separated set of directions {ω}. For each such direction ω, let ¶_{ω} denote the collection of all plates P which point in a direction within N^{−1/4} of ω; thus the ¶_{ω} form a partition of ¶.

The entire sum ∑_{P ∈¶} ϕ_{P} is essentially constant on 1/N-cubes. Now fix ω, and consider the ω component ∑_{P ∈¶_{ω}} ϕ_{P} of the previous sum. This sum is not only constant on 1/N-cubes, it is in fact constant on N^{−1/2}×N^{−3/4}×N^{−1} boxes oriented in the direction ω. This can be seen in a number of ways, such as using a Lorentz transform to thicken the angular aperture from N^{−1/4} back up to ∼ 1.

In light of this, we shall refer to these N^{−1/2}×N^{−3/4}×N^{−1} boxes as microplates. The sum ∑_{P ∈¶_{ω}} ϕ_{P} cannot exceed N^{1/4} on any one of these microplates. Call a microplate with direction ω focussing if ∑_{P ∈¶_{ω}} ϕ_{P} ∼ N^{1/4}.

From Theorem 1.1 applied at the smaller scale of \sqrt{N}, and then Lorentz transformed, we see that every plate contains at most one focussing microplate on the average.

Now consider a focussing cube. This cube has to be incident with N^{1/2} plates with essentially no cancellation. A little thought then reveals that these plates must be organized into N^{1/4} groups, each of which lies within N^{−1/4} of a direction ω. Because of the lack of cancellation, we thus see that this focussing cube is the superposition of N^{1/4} focussing microplates with no cancellation.

From Theorem 1.1 applied at the smaller scale of \sqrt{N} (and rescaled) we see that every focussing microplate contains at most one focussing cube on the average.

Now one could try to combine these two observations together and conclude the desired conclusion that every plate contains at most one focussing cube. Unfortunately we cannot quite close the induction like this because we lose a little control on the constants. But what this analysis does indicate is that everything must be absolutely sharp in order to have a counterexample, thus every plate should contain exactly one focussing microplate, which in turn should contain exactly one focussing cube. Also, if one considers the focussing microplates within a single N^{−1/2}-cube, and rescales this back to the unit cube, this should now be a critical example for the scale N^{1/2} problem.

Also, if one can somehow show that the interactions of the focussing microplates inside a single N^{−1/2} cube are actually happening at some strictly smaller scale, then one can indeed close the induction properly. So we can assume that a significant fraction of the microplate interactions in a N^{−1/2}-cube occur at the coarsest scale N^{−1/2}.

Also, from the above analysis we see that there are exactly M/\sqrt{N} focussing cubes.

Now let’s analyze what happens on a typical N^{−1/2} cube Q which contains at least one focussing cube. First of all, let’s try to control the number of focussing microplates here. The total number of plates that could possibly intersect Q is at most N (N^{1/2} possible directions, and for each direction there are N^{1/2} possible locations). Each plate has an energy (i.e. spatial L^{2} norm squared) of N^{−3/2}, so the total energy here is at most N^{−1/2} by orthogonality. On the other hand, each focussing microplate eats up N^{1/2}N^{−7/4} units of energy. Thus the total number of focussing microplates in Q cannot exceed N^{3/4}. (This is the ONLY place where we use any oscillation in the plates!)

On the other hand, the total number of focussing microplates in Q cannot be significantly smaller than N^{3/4}. This is because of the following combinatorial lemma (which is the key geometric ingredient in Wolff’s paper), which we discuss in the next section:

Lemma 1.2. Suppose there are M distinct plates which contain M/\sqrt{N} focussing cubes at the coarsest scale. Then M ≳ N^{3/2}.

Of course, one could argue that one only needs N^{1/2} plates focussing at a point to create a focussing cube, but this interaction is at a very small scale, and doesn’t count. A slightly more precise way to formulate “at the coarsest scale” is to allow one to remove from each plate an exceptional set of diameter ≪ 1, and only to consider the multiplicity away from this exceptional set.

Since the focussing microplates need a multiplicity of N^{1/4} in order to create focussing cubes, the claim follows by applying Lemma 1.2 at the scale of the microplates. Thus we must have exactly N^{3/4} microplates in each N^{−1/2}-cube of interest, which when one chases back the previous argument shows that there must be exactly N plates in each such cube. In particular, the cubes are absolutely filled with plates! We will come back to this point shortly.

Note that any improvement of N^{3/2} to N^{3/2+δ} in the above Lemma would allow us to declare victory at this point. Unfortunately, it is not known if one can actually obtain this improvement, so we have to do substantially more work.

Now recall that the situation inside each N^{−1/2}-cube needs to be a critical example to Theorem 1.1. Thus we may as well rescale this cube to the unit scale and work entirely in this cube. After replacing \sqrt{N} with N, we thus see that there must be exactly N^{3/2} plates in this unit cube (because there were N^{3/4} microplates in the old N^{−1/2} cube).

Having done this, we can run all the above arguments again, and conclude once again that every N^{−1/2}-cube which contains at least one focussing cube must have N^{3/4} microplates and N plates.

Now it’s convenient to switch from plates to tubes. Define a tube to be a N^{−1/2}× N^{−1/2}×1 box oriented in a null ray. Every tube could potentially carry N^{1/2} plates. Now remember how every N^{−1/2}-cube was filled with N plates? This forces the N plates in question to be organized into N^{1/2} tubes, each of which is absolutely filled with N^{1/2} plates. Since there are N^{3/2} plates in total, there are N tubes in total, and every N^{−1/2}-cube of interest is incident with \sqrt{N} tubes.

This is consistent with the easy

Lemma 1.3. Given any M distinct tubes (not necessary in null directions), the multiplicity at the coarsest scale cannot exceed M^{1/2}.

Proof This is the “bush” argument. Suppose there are Q N^{−1/2}-cubes where the multiplicity is μ. Then a little combinatorics shows that each tube should contain about Qμ/M cubes on the average. Now form a bush of tubes around a single μ-multiplicity cube. If one only considers interactions away from the center of the bush, one thus obtains Qμ^{2}/M distinct cubes. Since there are only Q cubes, this can only happen if μ ≲ M^{1/2}.

However one can use the fact that the tubes point in null directions to squeeze a little more out of this. Basically the point is that Lemma 1.3 can only be sharp if all the Q cubes are contained in a bush, which is then contained in a light cone. But this argument can be applied starting from any μ-multiplicity tube, which means that all the Q cubes are actually contained inside several light cones simultaneously, and this quickly leads to a degenerate or impossible case. (In [2] a more precise bound is given by combining Lemma 1.3 at one scale with a generalization of Lemma 1.2 at the complementary scale, but we will not discuss that here).

It only remains to show Lemma 1.2.

2. Proof of Lemma 1.2

For each focussing cube, we associate an annulus of thickness N^{−1} and radius ∼ 1, simply by dropping the backward light cone from this cone to the time slice t = −1. Similarly we associate each plate with an N^{−1} × N^{−1/2} rectangle by extending it back to the same time slice. The situation is now as follows. We have M/\sqrt{N} distinct annuli and M distinct rectangles; each annuli contains N^{1/2} rectangles, i.e. each annulus is absolutely filled with rectangles. From elementary combinatorics we thus see that each rectangle is tangent to ∼ 1 annuli, on the average at least.

We shall actually prove

Lemma 2.1. Suppose we have two large collections W, B of annuli which are well separated in either radii or centers. Then there are at most (|W ||B|)^{3/4} rectangles which are tangent to both an annulus in W and an annulus in B.

If one applies this lemma, one sees that

M ≲ (M/\sqrt{N} M/\sqrt{N})^{3/4}

which leads to the desired bound M ≳ N^{3/2}. Proof This is achieved by induction. The claim is clear when W, B are small.

Pick a reasonably small number r (something like r ∼ (|W ||B|)^{1/100}). There is a tool known as the cell decomposition which allows one to subdivide the collection W into about r^{3} sub-collections W_{i}, each of which is only tangent to at most |B|/r annuli from B. (Actually, there are some error terms that become a nuisance if too many annuli are simultaneously tangent, but this is the gist of it. If one has many annuli tangent at a single rectangle, then one can win by sorting the annuli into “clusters” and using rather crude estimates). Assuming this cell decomposition, the lemma then follows for the following reason. The worst case is when the W_{i} are spread out evenly, so that |W_{i}| ∼ |W |/r^{3}. Applying the inductive hypothesis, the number of rectangles tangent to an annulus in W_{i} and an annulus in B is (|W |/r^{3}|B|/r)^{3/4} = (|W ||B|)^{3/4}/r^{3}, and the induction follows by summing up over i. (The induction can close because we only used the hypothesis once at a smaller scale. The reason why we ran into trouble before is that we tried to obtain the N-scale step from two applications of the \sqrt{N}-scale step, which is not allowed).

[Actually, to be totally fair, this induction argument only works when W and B are both large. There are some important cases in which the cell decomposition is counter-productive, and one has to use some other facts, such as a quantitative version of Appolonius’s theorem (which asserts that given three circles in general position, there are at most O(1) circles which are tangent to all three).]

So it remains to prove the cell decomposition. The rigorous form of this is quite tricky, but one can describe the decomposition informally without too much difficulty. Every annulus can be associated with a double light cone whose intersection with the t = −1 plane is the annulus. We thus have “white” cones coming from W and “black” cones coming from B. We shall be somewhat careless about identifying annuli and cones.

Pick r black cones at random. This carves up R^{2+1} into about r^{3} regions Ω_{i}. (To see this, add in the cones one at a time. It’s enough to show that each new cone cuts through about r^{2} existing regions in R^{2+1}. A little thought shows that this is the same as showing that r conic sections in the plane determine about r^{2} regions in R^{2}. But if one adds the conic sections one at a time one sees that each conic section cuts through r existing regions (because any two conic sections intersect in O(1) points at most), and we are done).

For each region Ω_{i}, let W_{i} be all the white annuli whose vertex lies inside Ω_{i}. These W_{i} essentially form a partition of W (there are some cases to consider when the vertex lies on the boundary between two Ω_{i}, but we shall ignore these cases, which one can do if r is sufficiently small). We only have to explain why the annuli coming from W_{i} are only tangent to about |B|/r annuli from B.

Let B_{i} be all the annuli which are tangent to something from W_{i}. Suppose for contradiction that B_{i} has far more than |B|/r elements. Then the law of large numbers says that with extremely large probability, B_{i} must contain one of the randomly selected r black cones. But such a cone cannot cut into Ω_{i}, since Ω_{i} would then divide into smaller regions. Because of this, this cone cannot be tangent to any cone in W_{i}, a contradiction.

References

[1] T. Wolff, A sharp bilinear restriction estimate, to appear, IMRN.

[2] T. Wolff, Local smoothing estimates type estimates on L^{p} for large p, to appear.

Department of Mathematics, UCLA, Los Angeles, CA 90024

   E-mail address: tao@math.ucla.edu