LECTURE NOTES 7 FOR 254B

TERENCE TAO

1. Known progress on the Kakeya conjectures

In the last set of notes we studied the following partial Kakeya conjectures.

For any p ≥ n/(n − 1), let K(p) denote the statement that

∥ ∑_{ω∈Ω} χ_{T_{ω}}∥_{p} ≲ δ^{\frac{n}{p} −(n−1)+ε}  (1)

for all ε > 0, 0 < δ ≪ 1 and any collection T_{ω} of 1 × δ tubes which point in δ-separated directions. It is conjectured that K(p) holds for all p ≥ n/(n−1); since K(∞) is trivial, lower vlaues of p clearly imply higher values of p. Here is what is known:

• Córdoba’s argument (1977) gives K(2) for all n.

• Drury’s argument (1983) gives K((n + 1)/(n − 1)) for all n. A sharper “xray” version of this was given by Christ, Rubio de Francia, and Duandikoetxea (1985).

• Wolff’s argument (1995) gives K((n + 2)/n).

• Bourgain (1998) has shown that there exists a c > 1/2 such that K(p) holds for all p′ ≤ cn, if n is sufficiently large.

Let d(n) and d_{M}(n) denote the least Hausdorff and Minkowski dimension of Besicovitch sets in R^{n}, thus d_{M}(n) ≥ d(n). It is conjectured that both quantities are equal to n. If K(p) holds, then d_{M}(n), d(n) ≥ p′. Here is what is known on these problems:

• We trivially have d(1) = 1, d(n + 1) ≥ d(n), and d(n + m) ≤ d(n) + d(m) for both types of dimension. In particular, the conjecture is proven if we can show that d(n)/n → 1 as n → ∞.

• Córdoba’s argument (1977) gives d(2) = 2 (though I believe this observation was first made by Davies (1971)).

• Bourgain’s “bush” argument (1991) gives d(n) ≥ (n + 1)/2; in the same paper he shows that in fact d(n) ≥ (n + 1)/2 + ε_{n} for some ε_{n} > 0. (An alternate proof of this is later given by Schlag (1998)).

• Wolff’s argument (1995) gives d(n) ≥ (n + 2)/2.

• Bourgain (1998) showed that d(n) ≥ 13n/25 + 12/25.

• Katz, Laba, and myself (1999) have shown that d_{M}(3) > 5/2+10^{−10}, which is slightly better than Wolff’s bound in three dimensions.

As one can see, there is still much progress to be made on these conjectures.

2. Localized restriction theorems

In previous sections, we saw how the restriction conjecture implied the Kakeya conjecture. It is not known if one can completely reverse this implication, and say that the Kakeya conjecture implies the restriction conjecture. (Of course, we believe both conjectures to be true, so the two would then be equivalent in a strictly logical sense!).

However, some partial progress has been achieved. The benchmark Tomas-Stein restriction theorem has been improved in a number of ways, and many of these new estimate rely crucially on the modern Kakeya estimates such as Wolff’s estimate.

However, to leap from Kakeya estimates to restriction estimates one must first introduce a notion of a local restriction estimate as a kind of stepping stone between the two. Recall that R(p → q) denotes the estimate

∥f̂∥_{L^{p}(S^{n−1})} ≲ ∥f∥_{q}

for all functions f on R^{n}. In adjoint form, we write this as

∥ĝdσ∥_{L^{q′} (R^{n})} ≲ ∥g∥_{L^{p′} (S^{n−1})}

and write this as R^{∗}(q′ → p′); this is of course logically equivalent to R(p → q).

The estimate R^{∗}(q′ → p′) is estimating the quantity ĝdσ on the entirety of R^{n}. The local estimates shall only estimate this quantity on a ball. Namely, for any α ≥ 0, we let R^{∗}(q′ → p′, α) denote the estimate

∥ĝdσ∥_{L^{q′} (B(0,R))} ≲ R^{α}∥g∥_{L^{p′} (S^{n−1})}

for all g on S^{n−1} and all R ≫ 1, where B(0, R) is the ball of radius R around the origin. Clearly low α is better than high α, and when α = 0 we recover the global restriction estimate R^{∗}(q′ → p′). Note that the center of the ball B(0, R) is not important, since we can freely modulate g by a phase to translate ĝdσ wherever we want. So once we prove this estimate for origin-centered balls we get it for all balls.

Hence it is easier to prove local restriction estimates than global ones. Also, there are some exponents which have good local restriction estimates that one cannot see on the global level. Remember that one could not get a restriction estimate in L^{2}(R^{n}) because of Plancherel’s theorem (ĝdσ is never in L^{2})? Well, this difficulty goes away if we localize on a ball (although we must of course lose some power of R).

Lemma 2.1. R^{∗}(2 → 2, 1/2) holds.

Proof We want to show that

∥ĝdσ∥_{L^{2}(B(0,R))} ≲ R^{1/2}∥g∥_{L^{2}(S^{n−1})}.

We can write the left-hand side as

∥ĝdσχ_{B(0,R)}∥_{2},

but this cutoff χ_{B(0,R)} is too rough. Let’s use a smoother cutoff

∥ĝdσψ(x/R)∥_{2},

where ψ is a Schwarz function with compactly supported Fourier transform which is bounded away from zero on the unit ball. (Exercise: show such a function exists). Clearly this expression bounds the previous one. By Plancherel’s theorem, we thus have to show

∥(gdσ) ∗ ψ̂_{R}∥_{2} ≲ R^{1/2}∥g∥_{L^{2}(S^{n−1})},

where ψ̂_{R}(ξ) = R^{n}ψ̂(Rξ). Once again, we prove L^{2} estimates by interpolating between L^{1} and L^{∞} estimates. (This trick, combined with a Plancherel-type identity, is often a very good way to get L^{2} estimates). The function ψ̂_{R} is an approximation to the identity, and has L^{1} norm O(1). Thus from Young’s inequality or Fubini’s theorem we have

∥(gdσ) ∗ ψ̂_{R}∥_{1} ≲ ∥g∥_{L^{1}(S^{n−1})},

So by interpolation, we just need to show

∥(gdσ) ∗ ψ̂_{R}∥_{∞} ≲ R∥g∥_{L^{∞}(S^{n−1})}.  (2)

In other words, we want to show

| ∫_{S^{n−1}} g(ω)R^{n}ψ̂(R(ξ − ω)) dω| ≲ R∥g∥_{∞}

for all ξ.

Fix ξ. Replacing everything by absolute values and estimating g by its sup norm, it suffices to show

∫_{S^{n−1}} |ψ̂(R(ξ − ω))| dω ≲ R^{1−n}.

However, since ψ̂ is rapidly decreasing, we can estimate the left-hand side by

∫_{S^{n−1}} \frac{1}{(1 + R|ξ − ω|)^{N}} dω ≲ R^{1−n}.

I’ll leave it as an exercise to verify this inequality. (Hint: first do the case when |ξ − ω| ≤ 1/R, and then go out in dyadic shells 2^{k−1}/R ≤ |ξ − ω| ≤ 2^{k}/R.)

Note that this result holds under any type of hypersurface S, even a hyperplane. In fact S could have codimension k, as long as we have α = k/2 instead of α = 1/2. It is variously known as the Sobolev trace lemma (which in its more traditional form, states that an H^{1/2+ε} function is L^{2} when restricted to any surface), the energy estimate, or even “the easy L^{2} estimate”.

Now local restriction estimates are nice, but global ones are the true goal. (This is particularly important in PDE: showing, say, existence globally in time is far sexier than mere local-time existence). So the question is, can we convert local restriction estimates, such as the one proved above, to obtain good global restriction estimates?

As a benchmark, we recall the Tomas-Stein inequality, which becomes in our new notation

R^{∗}(2 → \frac{2(n + 1)}{n − 1}).

The important exponent here is the second one, 2(n + 1)/(n − 1). We would like to lower this if at all possible. Of course, to do so we will need more than just L^{2} control on the function g (we know Tomas-Stein is sharp if we keep g in L^{2}), so we will have to raise the 2 exponent a little.

By modifying the Tomas-Stein argument, we can achieve precisely this, if we are given a sufficiently good local restriction estimate. However, the argument is a bit clunky and artificial, and the exponents obtained are horribly ugly. However, we don’t know of a better one (although we can improve the ∞ exponents somewhat). If you can find one, then you can snatch the world record for restriction theorems of the sphere (as of May 1999, at least :-).

Proposition 2.2 (Bourgain, 1991). Suppose that we have the local restriction estimate R^{∗}(∞ → p, α) for some α > 0 and αp < (n+1)/2. Then we have R^{∗}(∞ → q) whenever

q > 2 + \frac{p}{\frac{n+1}{2} − αp}.

Proof Fix p, α, q. We have to show that

∥ĝdσ∥_{q} ≲ ∥g∥_{∞}

for all g. Since every L^{∞} function is an average of characteristic functions, it suffices to show that

∥{ \atop χ_{Ω}dσ}∥_{q} ≲ 1

for all Ω ⊂ S^{n−1}. Clearly we can restrict ourselves to the case when |Ω| ∼ 1, since for small Ω we can write the characteristic function as a difference of two sets.

From the identity

∥{ \atop χ_{Ω}dσ}∥_{q}^{q} = ∫_{0}^{∞} qλ^{q−1}|\{x : |{ \atop χ_{Ω}dσ}(x)| > λ\}| dλ

(Exercise!), and the fact that { \atop χ_{Ω}dσ} is clearly bounded from the definition of Fourier transform, we see that it suffices to show the weak-type estimate

|\{x : |{ \atop χ_{Ω}dσ}(x)| > λ\}| ≲ λ^{−q+ε}

for all 0 < λ ≤ 1.

Actually, we’ll just prove the slight variant

|E| ≲ λ^{−q+ε}  (3)

where E is the set

E = \{x : Re({ \atop χ_{Ω}dσ})(x) > λ\},

and leave the full proof as an exercise.

The next step is duality. From the construction of E, we have

Re(∫_{E} { \atop χ_{Ω}dσ}) > λ|E|.

From Plancherel this becomes (ignoring the distinction between Fourier transform and inverse Fourier transform)

Re(∫_{Ω} χ̂_{E}(ω) dω) > λ|E|.

From Hölder’s inequality (this is quite inefficient, but the Tomas-Stein argument badly needs an L^{2} norm somewhere!), the left-hand side is bounded in magnitude by

Re(∫_{Ω} χ̂_{E}(ω) dω) ≤ ∥χ̂_{E}∥_{L^{2}(S^{n−1})}.

Thus we have an L^{2} lower bound, which we square as

λ^{2}|E|^{2} ≤ ∥χ̂_{E}(ω)∥_{L^{2}(S^{n−1})}^{2}.

The right-hand side can be rewritten as

∫ χ̂_{E}\overline{χ̂_{E}}dω.

Now we do a Tomas-Stein style trick, and use Plancherel again, to get to

λ^{2}|E|^{2} ≤ ⟨χ_{E}, χ_{E} ∗ d̂ω⟩.

Now we estimate the right-hand side. Well, we’ve done this once already, in Lecture Notes 2, and so we’ll just borrow the estimates we had there. As in those notes, we break up

d̂ω = ϕd̂ω + ∑_{k>0} ψ_{k}d̂ω,

and estimate the right-hand side of the above by the triangle inequality as

|⟨χ_{E}, χ_{E} ∗ ϕd̂ω⟩| + ∑_{k>0} |⟨χ_{E}, χ_{E} ∗ ψ_{k}d̂ω⟩|.  (4)

For the first term, we simply estimate this by the L^{1} norm of χ_{E} ∗ ϕd̂ω, which is O(|E|) by Young’s inequality. For the latter term

|⟨χ_{E}, χ_{E} ∗ ψ_{k}d̂ω⟩|  (5)

there are two estimates, just as in Tomas’s argument.

The first estimate is an L^{2} estimate. By Cauchy-Schwarz, (5) is bounded by

|E|^{1/2}∥χ_{E} ∗ ψ_{k}d̂ω∥_{2}.

But by estimate (12) in Lecture notes 2, this is bounded by

|E|2^{k/2}.

(One can also get this from R^{∗}(2 → 2, 1/2) and Plancherel - exercise!)

As in Tomas’s argument, the second estimate is an L^{∞} based estimate using the decay of the Fourier transform d̂ω. But there is a twist (otherwise we’d be repeating Tomas’s argument essentially verbatim!). We can estimate (5) by

|E|∥χ_{E} ∗ ψ_{k}d̂ω∥_{∞}.

Now let’s estimate the normed expression. We know that ψ_{k}d̂ω is O(2^{−(n−1)k/2}) from the decay of the Fourier transform. But we also know that this function is supported on the ball B(0, C2^{k}). Thus we can estimate the above by

|E|2^{−(n−1)k/2}∥χ_{E} ∗ χ_{B(0,C2^{k})}∥_{∞} = 2^{−(n−1)k/2}|E| sup_{x∈R^{n}} |E ∩ B(x, C2^{k})|.

Observe that if we naively estimated this supremum by |E|, we would be doing almost exactly what Tomas did. So we do something different instead here, and use the local restriction hypothesis.

Fix x ∈ R^{n}. From our hypothesis R^{∗}(∞ → p, α) we have

∥{ \atop χ_{Ω}dσ}∥_{L^{p}(B(x,C2^{k})} ≲ 2^{αk}.

Since the function { \atop χ_{Ω}dσ} has real part > λ on E, we thus have

λ|E ∩ B(x, C2^{k})|^{1/p} ≲ 2^{αk}.

Inserting this back into the above, we obtain the icky

(5) ≲ 2^{−( \frac{n−1}{2} −αp)k}λ^{−p}|E|.

Inserting all these estimates back into (4) we obtain

λ^{2}|E|^{2} ≲ |E| + ∑_{k>0} min(2^{k}|E|, 2^{−( \frac{n−1}{2} −αp)k}λ^{−p}|E|).

The expressions 2^{k} and 2^{−( \frac{n−1}{2} −αp)k}λ^{−p} meet when

2^{k} = λ^{− _{\frac{p}{\frac{n+1}{2} −αp}}} = λ^{2−q+ε}

by the hypothesis on q. So we get

λ^{2}|E|^{2} ≲ |E| + λ^{2−q+ε}|E|.

Since 2 − q + ε < 0, we thus obtain the desired result (3).

Now let’s test this numerology. Let’s feed in the local restriction estimate R^{∗}(2 → 2, 1/2), which of course implies R^{∗}(∞ → 2, 1/2). Computing the numerology, we obtain R^{∗}(∞ → 2(n + 1)/(n − 1) − ε) - in other words, we just get a wimpy version of Tomas-Stein.

Now suppose we tried to feed in the best restriction estimate we know of so far, which is Tomas-Stein R^{∗}(2 → 2(n + 1)/(n − 1)). Of course this implies R^{∗}(∞ → 2(n + 1)/(n − 1), 0). Turning the crank, we see once again that we just obtain R^{∗}(∞ → 2(n + 1)/(n − 1) − ε) again.

What about interpolation? If we interpolate between the trace lemma and Tomas-Stein we get

R^{∗}(2 → p, \frac{1}{2} + \frac{n + 1}{2}(\frac{1}{p} − \frac{1}{2})),

which of course implies

R^{∗}(∞ → p, \frac{1}{2} + \frac{n + 1}{2}(\frac{1}{p} − \frac{1}{2})).  (6)

Turning the crank one last time, we see - once again - that we only get the weak Tomas-Stein estimate R^{∗}(∞ → 2(n + 1)/(n − 1) − ε).

So we can’t pull ourselves off of Tomas-Stein purely by bootstrapping. But rather than be discouraged, we have seen a very important moral. If we want to get a better global restriction theorem than Tomas-Stein, all we need to do is decrease the α exponent on (6). In other words, we just need to beat some interpolant between Tomas-Stein and the trace lemma. This is much easier to do (from a logical perspective) than trying to beat Tomas-Stein directly, and this represents progress.

On the flip side, it will turn out that because this machine is so clunky, even quite dramatic improvements to (6) will only end up with very incremental improvements to Tomas-Stein. The machine is very lossy.

3. Local restriction estimates from Kakeya estimates

In the previous section we showed how local restriction estimates can be converted into global ones. Of course, to use this one must first figure out how to prove local restriction estimates. At present, we basically have only one way to push restriction estimates beyond the Tomas-Stein exponent, which is based on the Kakeya estimates and the decomposition of an R-ball into \sqrt{R}-balls.

To be precise, let f be an L^{∞} function on the sphere S^{n−1}, normalized so that ∥f∥_{∞} = 1, and let R ≫ 1 be a large number. We are asking for bounds of the form

∥f̂dσ∥_{L^{p}(B(0,R))} ≲ R^{α}

where 2 ≤ p ≤ 2(n + 1)/(n − 1), and α is small.

The first thing to do is to exploit the uncertainty principle. Since we are localizing space to scale R, we expect to have a frequency uncertainty of 1/R. To capture this, we’re going to blur out f, which is currently supported on a sphere, to an annulus of thickness 1/R.

More precisely, we estimate

∥f̂dσ∥_{L^{p}(B(0,R))} ≲ ∥f̂dσψ_{R}∥_{p} = ∥{ \atop fdσ ∗ ψ̂_{R}}∥_{p}

as in the previous section. If we write F = R^{−1}fdσ∗ψ̂_{R}, then by (2) F is a bounded function on the 1/R-annulus of the unit sphere. Our problem now becomes:

Problem 3.1. Let F be a function on the 1/R-neighbourhood of the unit sphere such that ∥F ∥_{∞} ≲ 1. Can we show that

∥F̂∥_{p} ≲ R^{−1+α}?  (7)

Now we do something akin to what we did when doing the converse implication, deriving Kakeya from restriction. Namely, we break up the aforementioned annulus into little disks. A little investigation using Pythagoras’s theorem (or more precisely, that the sphere touches its tangent planes to second order everywhere), shows that the largest disks one can efficiently cover the 1/R-annulus by are 1/R × 1/\sqrt{R} × … × 1/\sqrt{R} disks. So let’s partition the annulus into about R^{(n−1)/2} such disks D_{α}, and let F = ∑_{α} F_{α} be a decomposition of F into these disks. (Since we only have L^{∞} control on F anyway, there is no distinction between a smooth partition and a rough one really). Let ω_{α} ∈ S^{n−1} denote the center of each disk.

We’re trying to show

∥ ∑_{α} F̂_{α}∥_{p} ≲ R^{−1+α}.  (8)

The trouble is that the sum on the left is the sum of many terms, each of which oscillate, and so we expect a lot of cancellation which we have to control. It’s not obvious how to deal with this; however, after a lot of work people eventually realized that the way to deal with this expression is to interpose a square function, namely

∥(∑_{α} |F̂_{α}|^{2})^{1/2}∥_{p}.  (9)

This quantity is much easier to estimate accurately than (8), because the F̂_{α} terms don’t cancel each other.

If we had randomized signs in front of F̂_{α}, then the two quantities above are related by Khinchin’s inequality. Unfortunately, we don’t have that, but we still hope to get a bound of the form

∥ ∑_{α} F̂_{α}∥_{p} ≲ R^{β}∥(∑_{α} |F̂_{α}|^{2})^{1/2}∥_{p}  (10)

for some (hopefully) small value of β. If we had (10), then the only thing we’d have left to do is estimate (9).

Actually, it’s conjectured that β can be arbitrarily small if 2 ≥ p ≥ 2n/(n − 1). We don’t have much evidence for this claim (which would be a rather amazing thing if true, and would imply just about all the other extant conjectures in the field), although it’s easy to prove when p = 2 from orthogonality. As further evidence, we have

Proposition 3.2 (Fefferman,Cordoba). (10) is true when n = 2, p = 4, and β = 0.

Proof We have to show that

∥ ∑_{α} F̂_{α}∥_{4} ≲ ∥(∑_{α} |F̂_{α}|^{2})^{1/2}∥_{4}.  (11)

Of course, as with all things done in two dimensions, this is a consequence of the special properties of the exponent 2n/(n − 1) = 4. By the triangle inequality, we may restrict our attention to those disks D_{α} - which are now rectangles - which are within 1/10 of the vertical.

Raising both sides of (11) to the fourth power, the left hand side can be written as

∥ ∑_{α} ∑_{β} F̂_{α}F̂_{β}∥_{2}^{2}

while the right-hand side can be written as

∑_{α} ∑_{β} ∥F̂_{α}F̂_{β}∥_{2}^{2}.

So basically we just want to show that the functions F̂_{α}F̂_{β} are more or less orthogonal. But this follows from an observation of Fefferman, that the Fourier supports of these functions (i.e. the supports of F_{α} ∗ F_{β}) are essentially disjoint as α, β varies. (This has to do with the fact that any two circles intersect in at most two points).

In higher dimensions we have so far been unable to replicate this trick. Instead, we have a partial version in which we have a non-zero loss β:

Proposition 3.3 (Bourgain, 1991). We have (10) for all 2(n + 1)/(n − 1) ≥ p ≥ 2 and β = \frac{n−1}{4}(\frac{1}{2} − \frac{1}{p}).

Proof Let’s first try to estimate (10) on a fairly small ball, or more specifically a \sqrt{R} ball:

∥ ∑_{α} F̂_{α}∥_{L^{p}(B(0,\sqrt{R}))}.

As usual, we estimate this by the smoother

∥ ∑_{α} F̂_{α}ψ\sqrt{R}∥_{L^{p}(B(0,\sqrt{R}))} = ∥{ \atop ∑_{α} F_{α} ∗ ψ̂\sqrt{R}}∥_{L^{p}(B(0,\sqrt{R}))}.

The expression inside the Fourier transform is supported on an annular neighbourhood of the unit sphere of thickness O(R^{−1/2}). We now apply

Lemma 3.4. For all r ≫ 1 and all functions F supported on an annular neighbourhood of thickness O(r^{−1}) around the unit sphere, we have

∥F̂∥_{L^{p}(B(0,r))} ≲ r^{\frac{n+1}{2} ( \frac{1}{p} − \frac{1}{2} )}∥F ∥_{2}.

Proof From polar co-ordinates we can write F as an integral of spherical measures:

F = ∫_{1−C/r}^{1+C/r}(F dσ_{ρ})dρ

where dσ_{ρ} is induced Lebesgue measure on the sphere S_{ρ}. From Minkowski we thus have

∥F̂∥_{L^{p}(B(0,r))} ≲ ∫_{1−C/r}^{1+C/r} ∥{ \atop F dσ_{ρ}}∥_{L^{p}(B(0,r))}dρ.

Applying (6), slightly rescaled by ρ ∼ 1, we estimate this by

r^{\frac{1}{2} + \frac{n+1}{2} ( \frac{1}{p} − \frac{1}{2} )} ∫_{1−C/r}^{1+C/r} ∥F ∥_{L^{2}(S_{ρ})} dρ.

By Cauchy-Schwarz this is estimated by

r^{\frac{n+1}{2} ( \frac{1}{p} − \frac{1}{2} )}(∫_{1−C/r}^{1+C/r} ∥F ∥_{L^{2}(S_{ρ})}^{2} dρ)^{1/2},

and the claim follows from another appeal to polar co-ordinates.

From this lemma, we have

∥ ∑_{α} F̂_{α}∥_{L^{p}(B(0,\sqrt{R}))} ≲ R^{\frac{n+1}{4} ( \frac{1}{p} − \frac{1}{2} )}∥ ∑_{α} F_{α} ∗ ψ̂\sqrt{R}∥_{2}.

The functions F_{α} ∗ ψ̂\sqrt{R} are supported on essentially disjoint \sqrt{R} balls, so we have

∥ ∑_{α} F_{α} ∗ ψ̂\sqrt{R}∥_{2} ≲ (∑_{α} ∥F_{α} ∗ ψ\sqrt{R}∥_{2}^{2})^{1/2}.

By Plancherel, this is

(∑_{α} ∥F̂_{α}ψ\sqrt{R}∥_{2}^{2})^{1/2}.

We interchange the l^{2} sum and the L^{2} norm:

∥(∑_{α} |F̂_{α}ψ\sqrt{R}|^{2})^{1/2}∥_{2}.

We split this as

∥(∑_{α} |F̂_{α}|^{2})^{1/2}|ψ\sqrt{R}|^{1/2}|ψ\sqrt{R}|^{1/2}∥_{2}.

The reason we’re splitting this ψ up is because we’re going to make it do double duty. Firstly, we need to replace the L^{2} norm by the higher L^{p} norm, and that’s achieved by Hölder:

∥(∑_{α} |F̂_{α}|^{2})^{1/2}|ψ\sqrt{R}|^{1/2}|ψ\sqrt{R}|^{1/2}∥_{2} ≤ ∥(∑_{α} |F̂_{α}|^{2})^{1/2}|ψ\sqrt{R}|^{1/2}∥_{p}∥|ψ\sqrt{R}|^{1/2}∥_{q}

where 1/q = 1/2 − 1/p. But ψ\sqrt{R} is about 1 on the ball B(0, \sqrt{R}) and is rapidly decreasing away from this ball, hence we have

∥|ψ\sqrt{R}|^{1/2}∥_{q} ≲ R^{n/2q} = R^{\frac{n}{2} ( \frac{1}{2} − \frac{1}{p})}.

Putting all this together we obtain

∥ ∑_{α} F̂_{α}∥_{L^{p}(B(0,\sqrt{R}))} ≲ R^{\frac{n−1}{4} ( \frac{1}{2} − \frac{1}{p})}∥(∑_{α} |F̂_{α}|^{2})^{1/2}|ψ\sqrt{R}|^{1/2}∥_{p}.

Replacing F̂_{α} by a translate of F̂_{α}, we thus see that

∥ ∑_{α} F̂_{α}∥_{L^{p}(B(x_{0},\sqrt{R}))} ≲ R^{\frac{n−1}{4} ( \frac{1}{2} − \frac{1}{p})}∥(∑_{α} |F̂_{α}|^{2})^{1/2}|ψ\sqrt{R}(· − x_{0})|^{1/2}∥_{p}

for all x_{0} ∈ R^{n}.

Now cover R^{n} by \sqrt{R}-balls B(x_{i}, \sqrt{R}). Applying this estimate to each such ball, raising to the p^{th} power, and summing, we obtain

∑_{i} ∥ ∑_{α} F̂_{α}∥_{L^{p}(B(x_{i},\sqrt{R}))}^{p} ≲ R^{\frac{n−1}{4} ( \frac{1}{2} − \frac{1}{p})p} ∑_{i} ∥(∑_{α} |F̂_{α}|^{2})^{1/2}|ψ\sqrt{R}(· − x_{i})|^{1/2}∥_{p}^{p}.

The left-hand side is essentially ∥ ∑_{α} F̂_{α}∥_{p}^{p}. The summation on the right-hand side can be rearranged as

∫ (∑_{α} |F̂_{α}|^{2})^{p/2} ∑_{i} |ψ\sqrt{R}(· − x_{i})|^{p/2}.

Since ψ\sqrt{R}(· − x_{i}) is rapidly decreasing away from B(x_{i}, \sqrt{R}), and the balls are essentially disjoint, we have

∑_{i} |ψ\sqrt{R}(· − x_{i})|^{p/2} ≲ 1.

Inserting this back into the previous estimate we obtain (10).

Incidentally, this bound is sharp at the endpoints p = 2 and p = 2(n + 1)/(n − 1), but probably is not sharp in between. (Annoyingly, though, we don’t know of a way to improve upon the β factor in the proposition, although this proposition has been improved in other directions (see e.g. [9], [10])).

So now we’ve replaced (8) with (9), albeit at the cost of some R^{β}. Now we need to estimate (9). The first thing to do is understand what |F̂_{α}| looks like. (Note that we no longer care about the phase of F̂_{α}, only the magnitude. This is the advantage of moving to the square function).

The most obvious bound on the magnitude of F̂_{α} comes from Plancherel’s theorem. Since F_{α} is bounded and is supported on a disk which has volume about R^{−(n+1)/2}, we have

∥F̂_{α}∥_{2} = ∥F_{α}∥_{2} ≲ R^{−(n+1)/4}.

However, we can say more than this by taking better advantage of the localization of F_{α}.

For simplicity let’s assume that the disk D_{α} is centered at ω_{α} = e_{n}. This means that the disk has thickness 1/R in the e_{n} direction and radius 1/\sqrt{R} in the other directions. We can therefore write

F_{α}(\underline{x}, x_{n}) = G_{α}(\sqrt{R}\underline{x}, R(x_{n} − 1)),

where G_{α} is a bounded function supported on a ball B(0, C). This implies that

F̂_{α}(\underline{ξ}, ξ_{n}) = R^{−(n+1)/2}e^{−2πiξ_{n}} Ĝ_{α}(R^{−1/2}\underline{ξ}, R^{−1}ξ_{n}).

In particular, we have

|F̂_{α}(\underline{ξ}, ξ_{n})| = R^{−(n+1)/2}|Ĝ_{α}(R^{−1/2}\underline{ξ}, R^{−1}ξ_{n})|.

So it suffices to understand |Ĝ_{α}|.

Since G_{α} is clearly in L^{2}, Ĝ_{α} is also in in L^{2}. However, we can say a lot more due to the compact support of G_{α}. Indeed, we see that ξ^{β}G_{α} is in L^{2} for any multi-index β, which implies that

∥∇^{β}Ĝ_{α}∥_{2} ≲ C_{β}

for all β. To exploit this, we use the Sobolev inequality:

∥f∥_{∞} ≲ ∑_{0≤|β|≤n} ∥∇^{β}f∥_{2}.

(The number of derivatives used here is not optimal, but that’s irrelevant for our purposes). Actually, we will use a localized form of this, which states that

∥f∥_{L^{∞}(Q)} ≲ ∑_{0≤|β|≤n} ∥∇^{β}f∥_{L^{2}(Q)}

on any unit cube. (This can be accomplished by writing f(x) for x ∈ Q as the average of f on Q, plus an integral of various derivatives of f, and estimate everything using Cauchy-Schwarz.

Squaring this estimate, and using the fact that there are only O(1) β’s here, we obtain

∥f∥_{L^{∞}(Q)}^{2} ≲ ∑_{0≤|β|≤n} ∥∇^{β}f∥_{L^{2}(Q)}^{2}.

Summing this as Q ranges over a cover of R^{n}, we obtain

∑_{Q} ∥f∥_{L^{∞}(Q)}^{2} ≲ ∑_{0≤|β|≤n} ∥∇^{β}f∥_{2}^{2}.

Applying this with f = Ĝ_{α}, we obtain

∑_{Q} ∥Ĝ_{α}∥_{L^{∞}(Q)}^{2} ≲ 1.

In other words, we have the pointwise estimate

|Ĝ_{α}| ≲ ∑_{Q} c_{Q}χ_{Q}

where c_{Q} are positive constants such that

∑_{Q} c_{Q}^{2} = 1.

Unwrapping this and seeing what it bodes for |F̂_{α}|, we obtain

|F̂_{α}| ≲ R^{−(n+1)/2} ∑_{T//ω_{α}} c_{T} χ_{T} ,

where T ranges over a finitely overlapping cover of R^{n} by \sqrt{R} × R tubes oriented in the ω_{α} = e_{1} direction, and

∑_{T//ω_{α}} c_{T}^{2} = 1.  (12)

Thus we may estimate (9) by

∥(∑_{α} |F̂_{α}|^{2})^{1/2}∥_{p} ≲ R^{−(n+1)/2}∥(∑_{α} ∑_{T//ω_{α}} c_{T}^{2} χ_{T})^{1/2}∥_{p}

which we can rewrite as

R^{−(n+1)/2}∥ ∑_{α} ∑_{T//ω_{α}} c_{T}^{2} χ_{T} ∥_{p/2}^{1/2}.

Now we do a probability theory trick. For each α, let T_{α} be a random tube in the direction ω_{α}, such that any given tube T has a probability c_{T}^{2} of being selected. This is legitimate by (12). It’s easy to see that

E(χ_{T_{α}}(x)) = ∑_{T//ω_{α}} c_{T}^{2} χ_{T} (x),

so

E(∑_{α} χ_{T_{α}}) = ∑_{α} ∑_{T//ω_{α}} c_{T}^{2} χ_{T} .

From Minkowski’s inequality, we thus have

∥ ∑_{α} ∑_{T//ω_{α}} c_{T}^{2} χ_{T} ∥_{p/2} ≤ E(∥ ∑_{α} χ_{T_{α}}∥_{p/2}).

(recall that p/2 ≥ 1).

To estimate this last quantity, assume that we have a Kakeya estimate K(p/2) at exponent p/2. This implies that

∥ ∑_{T} χ_{T} ∥_{p/2} ≲ δ^{\frac{2n}{p} −(n−1)−ε}

whenever T are a collection of 1×δ tubes pointing in different directions. We apply this with δ = 1/\sqrt{R} to obtain

∥ ∑_{T} χ_{T} ∥_{p/2} ≲ R^{− \frac{n}{p} + \frac{n−1}{2} +ε}.

However, our tubes T_{α} are not 1×1/\sqrt{R} tubes, but rather R×sqrtR tubes, although they do actually point in different directions. So we scale things up by a factor of R, to get

∥ ∑_{α} χ_{T_{α}}∥_{p/2} ≲ R^{− \frac{n}{p} + \frac{n−1}{2} +ε}R^{2n/p}.

This is uniform for all choices of T_{α}, so we may take expectations and obtain

∥ ∑_{α} ∑_{T//ω_{α}} c_{T}^{2} χ_{T} ∥_{p/2} ≲ R^{− \frac{n}{p} + \frac{n−1}{2} +ε}R^{2n/p}.

Putting this into the previous equation and simplifying, we obtain

∥(∑_{α} |F̂_{α}|^{2})^{1/2}∥_{p} ≲ R^{−1}R^{\frac{n}{2p} − \frac{n−1}{4} +ε}.

Combining this with (10) with β = \frac{n−1}{4}(\frac{1}{2} − \frac{1}{p}), we get (7) for

α = \frac{n + 1}{4p} − \frac{n − 1}{8} + ε.  (13)

Recall for comparison that (6) only gave

α = \frac{n + 1}{2p} − \frac{n − 1}{4}.

In other words, we have managed to halve the value of α (assuming a Kakeya bound at p/2). Roughly speaking, this is because we are only applying the (monstrously inefficient) estimate (6) at the \sqrt{R}-scale rather than at the R-scale.

Now we can put all these results together and get an improvement to Tomas-Stein. We illustrate the case n = 3, in which case Tomas-Stein gives

R^{∗}(2 → 4).

We start with the Wolff estimate, which is

K(5/3).

To apply this, we we pick p = 10/3. By the above discussion and (13), we thus get (7) for α = 1/20 + ε, i.e. we obtain

R^{∗}(∞ → 10/3, \frac{1}{20} + ε).

Applying Lemma 2.2, we thus get

R^{∗}(∞ → 42/11 + ε)

which has a slightly better second exponent than Tomas-Stein.

As you can see there is a lot of heavy-duty arithmetic involved here, and the final number obtained, i.e. 42/11 = 4 − 2/11, is somewhat disappointing - remember that we eventually want to get all the way to R^{∗}(∞ → 3 + ε)! (Note: one can modify the rotation theory argument in the previous set of notes, and show that R^{∗}(∞ → 2n/(n−1)+ε) implies the full restriction conjecture). Given that the one really non-trivial input we used - namely, Wolff’s Kakeya result d(3) ≥ 5/2 - was already halfway between the trivial bound d(3) ≥ 2 and the conjectured d(3) = 3, we seem to have lost quite a bit here. Recent improvements [9], [10] have tightened things up to (but not including) 26/7 = 4−2/7, mainly by using bilinear techniques (as in the proof of the two-dimensional restriction theorems in Lecture Notes 5), a later improvement by Wolff of his own Kakeya estimate, and a bilinear variant of Tomas-Stein, both of which I’ll discuss in later notes.

References

[1] J. Bourgain, Besicovitch-type maximal operators and applications to Fourier analysis, Geom. and Funct. Anal. 22 (1991), 147–187.

[2] J. Bourgain, Estimates for cone multipliers, Operator Theory: Advances and Applications, 77 (1995), 41–60.

[3] J. Bourgain, Some new estimates on oscillatory integrals, Essays in Fourier Analysis in honor of E. M. Stein, Princeton University Press (1995), 83–112.

[4] J. Bourgain, On the dimension of Kakeya sets and related maximal inequalities, preprint.

[5] N. Katz, I. Laba, T. Tao, An improved bound for the Minkowski dimension of Besicovitch sets in R^{3}, submitted to Annals of Math.

[6] A. Moyua, A. Vargas, L. Vega, Schrödinger Maximal Function and Restriction Properties of the Fourier transform, International Math. Research Notices 16 (1996).

[7] A. Moyua, A. Vargas, L. Vega, Restriction theorems and Maximal operators related to oscillatory integrals in R^{3}, to appear, Duke Math. J.

[8] T. Tao, The Bochner-Riesz conjecture implies the Restriction conjecture, to appear, Duke Math J.

[9] T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), pp. 967–1000.

[10] T. Tao, A. Vargas, A bilinear approach to cone multipliers I. Restriction Estimates, to appear.

[11] T. H. Wolff, An improved bound for Kakeya type maximal functions, Revista Mat. Iberoamericana. 11 (1995). 651–674.

[12] T. H. Wolff, A mixed norm estimate for the x-ray transform, Revista Mat. Iberoamericana. 14 (1998), 561-600.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555

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