LECTURE NOTES 5 FOR 254B

TERENCE TAO

1. Two dimensional restriction theorems

Recall that the restriction conjecture for the sphere states that one has the estimate

∥f̂∥_{L^{q}(S^{n−1})} ≲ ∥f∥_{p}

whenever p < 2n/(n + 1) and q ≤ (n − 1)p′/(n + 1). Let us write this in dual form as

Conjecture 1.1 (Restriction conjecture, adjoint form). One has the estimate

∥ĝdσ∥_{q} ≲ ∥g∥_{L^{p}(S^{n−1})} (1)

whenever q > 2n/(n − 1) and q ≥ (n + 1)p′/(n − 1).

The Tomas-Stein theorem gives some partial progress on this conjecture - it shows that the conjecture is true when p = 2. But this does not imply the whole conjecture in any dimension n ≥ 2.

In the two dimensional case, the two conditions on the exponents become q > 4 and q ≥ 3p′. The purpose of this section is to show that the full restriction conjecture is indeed true when n = 2. This is basically due to the special properties of the exponent 4; in higher dimensions one can try to repeat the same argument, but the results obtained do not improve upon Tomas-Stein, and can even be worse.

The proof of the two-dimensional restriction theorem is due to Fefferman and Córdoba, but I have modified the argument to emphasize a more “modern” outlook.

First of all, we make the minor change of replacing the circle S^{1} with the quartercircle, in order to avoid some nuisances involving antipodal points. Of course, the restriction theorem for the quarter-circle implies that for the full circle by the triangle inequality.

To prove (1), we first observe that if this estimate is true for some value of q, then it is automatically true for all higher values of q, by interpolation with the easy case q = ∞. Thus it suffices to check the case when q = 3p′. From the Marcinkiewicz interpolation theorem, it suffices to verify the restricted weak-type estimate

λ|\{|{ \atop χ_{Ω}dσ}| ≥ λ\}|^{1/q} ≲ |Ω|^{1/p},

where Ω is an arbitrary subset of the circle S^{1}. Actually, we don’t have to go quite this weak; I’ll actually prove the restricted estimate

∥{ \atop χ_{Ω}dσ}∥_{q} ≲ |Ω|^{1/p}.

Now for a simple, but powerful idea. Let’s square the estimate!

∥{ \atop χ_{Ω}dσ}{ \atop χ_{Ω}dσ}∥_{q/2} ≲ |Ω|^{2/p}. (2)

Since q > 4, q/2 is between two and infinity. So suddenly we are interested in finding estimates for bilinear quantities such as

∥f̂dσĝdσ∥_{2}

and

∥f̂dσĝdσ∥_{∞},

where f and g are some functions on the circle. This may appear to have made the problem more complex rather than less - since our expressions now depend on two functions rather than one - but there are many things you can do in this setting that seem to have no analogue in the linear setting.

The latter quantity is easy enough to estimate, thanks to the trivial estimate ∥f̂dσ∥_{∞} ≲ ∥f∥_{1}. So we have

∥f̂dσĝdσ∥_{∞} ≲ ∥f∥_{1}∥g∥_{1}. (3)

Now let’s look at the other end, namely the L^{2} estimates. Here we encounter an interesting phenomenon. It is difficult to obtain estimates for this quantity for general f and g. But if we know that f and g are separated in angle, then we can estimate this nicely. Namely:

Lemma 1.2. Suppose f and g are supported on distinct θ-arcs of the circle, whose separation is also comparable to θ. Then

∥f̂dσĝdσ∥_{2} ≲ θ^{−1/2}∥f∥_{2}∥g∥_{2}. (4)

Proof By Plancherel’s theorem, the estimate is equivalent to

∥(fdσ) ∗ (gdσ)∥_{2} ≲ θ^{−1/2}∥f∥_{2}∥g∥_{2}.

We will obtain this estimate by bilinear interpolation between

∥(fdσ) ∗ (gdσ)∥_{1} ≲ ∥f∥_{1}∥g∥_{1}.

and

∥(fdσ) ∗ (gdσ)∥_{∞} ≲ θ^{−1}∥f∥_{∞}∥g∥_{∞}.

The first estimate is straightforward; since f is an L^{1} function on the circle, fdσ is an L^{1} measure on R^{2}, and the convolution of two L^{1} functions is in L^{1} by Young’s inequality (or Fubini’s theorem).

Let’s look at the second estimate. Suppose f, g are supported on θ-arcs I, J, and let dσ_{I}, dσ_{J} be the restriction of surface measure to these arcs. Since we of course have the pointwise estimates

fdσ ≤ ∥f∥_{∞}dσ_{I}, gdσ ≤ ∥g∥_{∞}dσ_{J},

it suffices to show that

∥dσ_{I} ∗ dσ_{J}∥_{∞} ≲ θ^{−1}.

Let’s show this heuristically first. The measures dσ_{I} and dσ_{J} have total masses about θ respectively. So their convolution has mass about θ^{2}. On the other hand, the convolution is supported on the set-theoretic sum of these two arcs, which is comparable in size to a θ × θ^{2} rectangle. Assuming that the convolution spreads evenly, one thus expects a mean size of θ^{−1} as desired.

To do this more precisely, we may approximate dσ_{I} by \frac{1}{2ε}χ_{I_{ε}}, and similarly for dσ_{J}, where ε > 0 is a small number and I_{ε} is the ε neighbourhood of I. By the definition of induced Lebesgue measure, dσ_{I} is the weak limit of such measures. So it suffices to verify that

∥\frac{1}{2ε}χ_{I_{ε}} ∗ dσ_{J}∥_{∞} ≲ θ^{−1}

for all sufficiently small θ. But this just comes from the geometric observation that any translate of J intersects I_{ε} in an arc of length at most εθ^{−1}.

Now we have to piece all this together, and prove (2). We can’t just interpolate between (3) and (4) because of the support restriction on (4). So we have to split up the left-hand side of (2) into pieces in order to exploit (4).

We shall use something known as a Whitney decomposition. For every n ≥ 0, divide the circle into 2^{n} equal arcs, so that each arc at stage n has exactly two children at stage n + 1. We denote the set of all arcs at stage n by A_{n}. Define two arcs I, J ∈ A_{n} from the same stage n > 1 to be close if they are not adjacent, but their parents are adjacent, and write this relation as I ∼ J.

For every x ≠ y on the unit circle there is exactly one pair of arcs I, J containing x and y respectively such that I ∼ J. This implies that

{ \atop χ_{Ω}dσ}{ \atop χ_{Ω}dσ} = ∑_{I∼J} { \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}} = ∑_{n>1} ∑_{I,J∈A_{n}:I∼J} { \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}}.

This decomposition is phenomenon which is special to the bilinear perspective. I don’t know of any satisfactory way to duplicate this in a linear setting. Note that for each I there are only O(1) arcs J which are close to I.

To deal with the n summation, we just use the triangle inequality:

∥{ \atop χ_{Ω}dσ}{ \atop χ_{Ω}dσ}∥_{q/2} ≲ ∑_{n>1} ∥ ∑_{I,J∈A_{n}:I∼J} { \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}}∥_{q/2}.

Let’s try to estimate the q/2 norm by the L^{∞} norm and the L^{2} norm. For the L^{∞} norm we’ll just use the triangle inequality again, followed by (3), to get

∥ ∑_{I,J∈A_{n}:I∼J} { \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}}∥_{∞} ≲ ∑_{I,J∈A_{n}:I∼J} |Ω ∩ I||Ω ∩ J|.

This is a nice estimate - at least there’s no Fourier transforms on the right-hand side, but it would be better if we could have more tractable expressions such as Ω or 2^{−n} on the right-hand side. Fortunately, very crude estimates will do the trick. Clearly we may estimate |Ω ∩ J| by 2^{−n}. Since there are only O(1) J’s for each I, we thus have

∥ ∑_{I,J∈A_{n}:I∼J} { \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}}∥_{∞} ≲ ∑_{I∈A_{n}} |Ω ∩ I|2^{−n} = 2^{−n}|Ω|.

Alternatively, we may simply lift the restriction I ∼ J on the summation, and obtain the estimate

∥ ∑_{I,J∈A_{n}:I∼J} { \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}}∥_{∞} ≲ ( ∑_{I∈A_{n}} |Ω ∩ I|)( ∑_{J∈A_{n}} |Ω ∩ J|) = |Ω|^{2}.

Combining the two estimates we thus obtain

∥ ∑_{I,J∈A_{n}:I∼J} { \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}}∥_{∞} ≲ |Ω| min(|Ω|, 2^{−n}). (5)

Now for the L^{2} estimate. This time, the triangle inequality is a bad idea; L^{2} norms are perfect for exploiting orthogonality, and the triangle inequality just doesn’t do that at all. Everything relies on the following key observation of Fefferman: As I ∼ J vary, the functions { \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}} have Fourier transform supports which are essentially disjoint - so that these functions are essentially orthogonal. This is just done by computing the set theoretic sums of I and J and computing. Because of this (almost) orthogonality ^{1}, we thus have

∥ ∑_{I,J∈A_{n}:I∼J} { \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}}∥_{2} ≲ ( ∑_{I,J∈A_{n}:I∼J} ∥{ \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}}∥_{2}^{2})^{1/2}.

Now for (4):

∥ ∑_{I,J∈A_{n}:I∼J} { \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}}∥_{2} ≲ 2^{n/2}( ∑_{I,J∈A_{n}:I∼J} |Ω ∩ I||Ω ∩ J|)^{1/2}.

By the same arguments as before, we thus have

∥ ∑_{I,J∈A_{n}:I∼J} { \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}}∥_{2} ≲ 2^{n/2}(|Ω| min(2^{−n}, |Ω|))^{1/2}.

Combining this with (5) by Hölder’s inequality, we thus have

∥ ∑_{I,J∈A_{n}:I∼J} { \atop χ_{Ω}dσ_{I}} { \atop χ_{Ω}dσ_{J}}∥_{q/2} ≲ 2^{2n/q}(|Ω| min(2^{−n}, |Ω|))^{1−2/q}.

Summing over n, we obtain

∥{ \atop χ_{Ω}dσ}{ \atop χ_{Ω}dσ}∥_{q/2} ≲ ∑_{n>1} 2^{2n/q}(|Ω| min(2^{−n}, |Ω|))^{1−2/q}.

The right hand side can be computed (after tideous arithmetic) to be |Ω|^{1−2/q} = |Ω|^{2/p}, as desired. (Split the sum into the cases 2^{−n} ≥ |Ω| and 2^{−n} < |Ω|). This concludes the proof of the restriction conjecture in two dimensions.

A quite similar argument can be used to prove the Bochner-Riesz conjecture in two dimensions, but we shall not do so here.

As you can see, one of the key innovations here is the bilinear approach. Unfortuanately, one cannot apply the above argument directly to higher dimensions unless q/2 ≥ 2 - but as one can check, these cases are already covered by Tomas-Stein anyway. However, the bilinear philosophy has been useful in higher dimensions, and in fact all the best results on the restriction conjecture and related problems has come from precisely such an approach. Despite this, I won’t dwell much on this important technical tool, because it does tend to obscure other important ideas. (The original proof of Fefferman and Córdoba did not take such an explicitly bilinear approach, and was more elegant; but it was less obvious whether any of the ideas could be extended to other dimensions and exponents.)

1One can obtain perfect orthogonality by only considering every tenth (say) pair (I, J), and then adding up the ten smaller sums by the triangle inequality.

2. Geometric consequences of the restriction conjecture

We have seen that a purely geometrical fact - Besicovitch’s construction of a set that solved the Kakeya needle problem - was used to show the unboundedness of the disc multiplier. Thus we see a connection between these Fourier analysis conjectures and geometric packing problems. This connection to geometry is in fact present to virtually all oscillatory integral problems in two and more dimensions.

In this section we show that the restriction conjecture is similarly connected to a tube-packing problem, known as the Kakeya maximal function conjecture. Then we shall see that this geometric problem has a particularly simple consequence, known as the Kakeya set conjecture.

For the sake of argument, let us assume that the restriction conjecture for the sphere is true in R^{n}, or in other words that we have

∥f̂dσ∥_{q} ≲ ∥f∥_{p}

for all q > 2n/(n − 1) and q ≥ (n + 1)p′/(n − 1). In particular, we have

∥f̂dσ∥_{2n/(n−1)+ε} ≲ ∥f∥_{2n/(n−1)}

for all ε > 0; all constants depend on ε of course. In other words, any bounded measure on the sphere has a Fourier transform which is in L^{q} for every q > 2n/(n− 1).

The Fourier transform of f̂dσ is spread out over all space. Let’s localize matters to a large ball:

∥f̂dσ∥_{L^{2n/(n−1)+ε}(B(0,R))} ≲ ∥f∥_{2n/(n−1)}.

If we don’t care about growth of constants in R, then we can prove this easily. In fact, we have

∥f̂dσ∥_{L^{p}(B(0,R))} ≲ R^{n/p}∥f̂dσ∥_{∞} ≲ R^{n/p}∥f∥_{1} ≲ R^{n/p}∥f∥_{2n/(n−1)}.

So by Hölder’s inequality, we have

∥f̂dσ∥_{L^{2n/(n−1)}(B(0,R))} ≲ R^{ε}∥f∥_{2n/(n−1)}. (6)

(This pushing around of epsilons is not very hard to do, and can even be reversed; see [3]).

Now let’s test this inequality against some examples. The simplest example is the Knapp example: let f_{ω} be the characteristic function of a 1/\sqrt{R}-cap on the sphere, centered at a point ω in S^{n−1}. The Fourier transform of this is

{ \atop f_{ω}dσ}(x) = ∫_{|θ−ω|<1/\sqrt{R}} e^{2πix·θ} dθ.

We can break up x into the component (x · ω)ω parallel to ω, and the component x − (x · ω)ω perpendicular to ω. Let T_{ω}^{0} denote the tube of all x where the parallel component has magnitude less than R/100, and the perpendicular component has magnitude less than \sqrt{R}/100. Now we observe a simple fact: if x ∈ T_{ω}^{0}, then |x · θ − x · ω| ≤ 1/10. This implies that

|{ \atop f_{ω}dσ}(x)| ∼ | ∫_{|θ−ω|<1/\sqrt{R}} e^{2πix·ω} dθ| ∼ R^{−(n−1)/2}

for all x ∈ T_{ω}^{0}.

To summarize: we have a function f_{ω}, which has size 1 on a 1/\sqrt{R}-cap, such that { \atop f_{ω}dσ} has size R^{−(n−1)/2} on a tube T_{ω}^{0}, which incidentally has volume about R^{(n+1)/2}.

If you plug this example into (6), you don’t get anything much - the two sides more or less agree. But wait and watch.

Firstly, we may multiply f_{ω} by a phase in order to translate { \atop f_{ω}dσ} as far as we please. Thus, for any translate T_{ω} of T_{ω}^{0}, we may find a function f_{T} which has size 1 on the 1/\sqrt{R}-cap centered around ω, such that { \atop f_{T} dσ} has size R^{−(n−1)/2} on T_{ω}.

Again, we aren’t contradicting (6) yet. Let’s keep going.

So far we’re only playing with one very small cap on the sphere. The sphere S^{n−1} actually is covered by a large number (about R^{n−1}) of such caps, which can be made disjoint. Let Ω = {ω} be a collection of centers of such caps - this is basically a 1/\sqrt{R}-separated subset of S^{n−1}. Ω could potentially have cardinality R^{n−1}, but could also be much smaller. For each of these ω, let’s pick a tube T_{ω} of the form above, i.e. a \sqrt{R} ×R tube oriented in the direction ω - but let’s keep T_{ω} in B(0, R). Let f_{T_{ω}} be the function described above.

Now we test (6) with the function

f = ∑_{ω∈Ω} ε_{ω}f_{ω}

where, as usual, ε_{ω} = ±1 are iid signs. As before, the randomization is used so that we don’t have to think about any subtle issues of cancellation, and rely instead on Khinchin’s inequality.

The function f has a wildly oscillating phase, but its magnitude is pretty constant, being equal to 1 on a #Ω caps of size R^{−(n−1)/2}. So

∥f∥_{2n/(n−1)} ∼ (R^{−(n−1)/2}#Ω)^{(n−1)/2n}.

Now let’s work out the size of ∥f̂dω∥_{L^{2n/(n−1)}(B(0,R))}.

By Khinchin’s inequality, we have that

E(∥f̂dω∥_{L^{2n/(n−1)}(B(0,R))}^{2n/(n−1)}) ∼ ∥(∑_{ω∈Ω} |{ \atop f_{T_{ω}}dσ}|^{2})^{1/2}∥_{L^{2n/(n−1)}(B(0,R))}^{2n/(n−1)}.

From our estimates on { \atop f_{T_{ω}}dσ}, we thus have

E(∥f̂dω∥_{L^{2n/(n−1)}(B(0,R))}^{2n/(n−1)}) ≳ ∥R^{−(n−1)/2}(∑_{ω∈Ω} χ_{T_{ω}}^{2})^{1/2}∥_{2n/(n−1)}^{2n/(n−1)}.

Combining this with (6), we thus obtain

∥R^{−(n−1)/2}(∑_{ω∈Ω} χ_{T_{ω}}^{2})^{1/2}∥_{2n/(n−1)}^{2n/(n−1)} ≲ R^{ε}R^{−(n−1)/2}#Ω,

which simplifies after some algebra to

∥ ∑_{ω∈Ω} χ_{T_{ω}}∥_{n/(n−1)} ≲ R^{ε}(∑_{ω∈Ω} |T_{ω}|)^{(n−1)/n}.

(Recall that |T_{ω}| ∼ R^{(n+1)/2}.) Since this statement is scale invariant, we thus have (setting δ = 1/\sqrt{R})

Conjecture 2.1 (Kakeya maximal operator conjecture). Let T_{ω} be any collection of tubes of eccentricity 1/δ and equal sizes, whose orientations are δ-separated. Then we have

∥ ∑_{ω∈Ω} χ_{T_{ω}}∥_{n/(n−1)} ≲ δ^{−ε}(∑_{ω∈Ω} |T_{ω}|)^{(n−1)/n}. (7)

I’ll explain later why this conjecture is related to maximal operators. Traditionally we normalize the tubes T_{ω} to be δ × 1, but any other normalization will work.

To get a sense of perspective, note that both sides of (7) are equal (except for the δ^{−ε} factor) if the T_{ω} are disjoint.

Let’s investigate a consequence of this conjecture. Let us measure t he overlap of the tubes T_{ω} by the quantity α, defined by

| ⋃_{ω∈Ω} T_{ω}| = α ∑_{ω∈Ω} |T_{ω}|;

clearly 0 < α ≤ 1. From Hölder’s inequality we have

∑_{ω∈Ω} |T_{ω}| = ∥ ∑_{ω∈Ω} χ_{T_{ω}}∥_{1} ≤ ∥ ∑_{ω∈Ω} χ_{T_{ω}}∥_{n/(n−1)}| ⋃_{ω∈Ω} T_{ω}|^{1/n}.

Combining all this with (7) we obtain

δ^{ε} ≤ α.

In other words, α is essentially 1 up to extremely small powers of δ. Actually, at this point it is convenient to introduce the notation that A ⪅ B if A ≤ C_{ε}δ^{−ε}B for any ε > 0, and A ≈ B if A ⪅ B and B ⪅ A. We thus have

α ≈ 1. (8)

In other words, one consequence of the Kakeya maximal operator conjecture is that any collection of direction-separated tubes of the same size are essentially disjoint.

As a consequence, we can say the following statement concerning Besicovitch sets. A Besicovitch set is defined to be a subset of R^{n} which has a unit line segment in every direction; it solves the n-dimensional analogue of the Kakeya needle problem. Besicovitch’s construction (adapted to higher dimensions) shows that such sets can have measure zero, but it’s not known what dimension these sets have to be.

Conjecture 2.2 (Kakeya set conjecture). All Besicovitch sets have Hausdorff and Minkowski dimension n.

Just to briefly refresh the definition of Hausdorff and Minkowski dimension, at least for compact sets. The Minkowski dimension is simpler: we say that a set E in R^{n} has Minkowski dimension d if

lim_{δ→0} log_{δ} |E_{δ}| = n − d,

where E_{δ} is the δ-neighbourhood of E. The Hausdorff dimension is trickier. Define the Hausdorff d-measure H_{d}(E) of E to be the lim infimum of all quantities of the form ∑_{B} r(B)^{d}, where {B} is a collection of balls covering E and r(B) is the radius of B, and the limit is taken as sup_{B} r(B) → 0. We say that E has Hausdorff dimension d if H_{d+ε}(E) = 0 and H_{d−ε}(E) = ∞ for all ε > 0. The Hausdorff dimension always exists, while the Minkowski dimension does not necessarily exist; however, if it does, it is always at least as large as the Hausdorff dimension.

Proposition 2.3. The Kakeya maximal operator conjecture implies the Kakeya set conjecture.

Proof This is easiest to show for the Minkowski dimension. Let E be a Besicovitch set. Then E_{δ} contains a δ × 1 tube in every direction. In particular, it contains δ^{−(n−1)} tubes which are δ-separated in direction. From (8) we thus see that |E_{δ}| ⪆ 1, hence the claim.

Let’s do the Hausdorff one now. We have to show that H_{n−ε_{0}}(E) is infinite for any ε_{0} > 0. In other words, if we cover E by balls B of radius at most 2^{−j} (say), then we have to show that

∑_{B} r(B)^{n−ε_{0}} > c(j)

where c(j) → ∞ as j → ∞.

Fix j ≫ 0, ε_{0} > 0, and fix such a covering of E by balls. Suppose for contradiction that

∑_{B} r(B)^{n−ε_{0}} ≲ 1.

These balls may have wildly varying sizes. However, by making the balls bigger we may easily assume that the radius of each ball is a negative power of 2. We may also assume that any ball intersects at most O(1) balls of the same size, otherwise there is too much redundancy and we can reduce the left-hand side of the above equation.

There is a canonical measure associated to a Besicovitch set as follows. For each direction ω on the sphere, we have a unit line segment l_{ω} in the set, with an associated Lebesgue measure H^{1}(l_{ω}). Now if we average all these measures together, we get a measure μ = ∫_{S^{n−1}} H^{1}(l_{ω}) dω on the Besicovitch set E, with total mass 1. In particular, we have

∑_{B} μ(B) ≥ μ(⋃_{B} B) = 1.

This implies that there must exist a k ≥ j such that

∑_{B:r(B)=2^{−k}} μ(B) ≥ k^{−2}

since ∑_{k>j} k^{−2} < 1 for j large.

Fix such a k, and set δ = 2^{−k}. Let’s discard all balls B of size other than 2^{−k}; the balls no longer cover all of E, but that is irrelevant now. For each direction ω, let T_{ω} be the δ × 1 tube centered around l_{ω}. Observe that

H^{1}(l_{ω} ∩ B) ≲ δ^{1−n} ∫_{B} χ_{T_{ω}}(x) dx.

Thus we have

μ(B) ≲ δ^{1−n} ∫_{B}(∫_{S^{n−1}} χ_{T_{ω}} dω).

Combining this with the choice of k, and the bounded overlap of the balls B, we see that

δ^{1−n} ∫_{⋃_{B} B}(∫_{S^{n−1}} χ_{T_{ω}} dω) ⪆ 1

since k^{−2} ≈ 1. By an averaging argument, one can then find a δ-separated subset of directions Ω of cardinality about δ^{1−n} such that

∫_{⋃_{B} B} ∑_{ω∈Ω} χ_{T_{ω}} ⪆ 1.

By Hölder’s inequality we have

∫_{⋃_{B} B} ∑_{ω∈Ω} χ_{T_{ω}} ≤ ∥ ∑_{ω∈Ω} χ_{T_{ω}}∥_{n/(n−1)}| ⋃_{B} B|^{1/n}.

By our assumption (7), we therefore have

1 ⪅ (∑_{ω∈Ω} |T_{ω}|)^{n/(n−1)}| ⋃_{B} B|^{1/n}.

We can easily compute ∑_{ω∈Ω} |T_{ω}| ∼ 1, so that

| ⋃_{B} B| ≈ 1.

Since the balls B are essentially disjoint, this implies that the number of balls is ≈ δ^{−n}. This implies that

∑_{B:r(B)=δ} r(B)^{n−ε_{0}} ≈ δ^{−n}δ^{n−ε_{0}} = δ^{−ε_{0}} ≫ 1,

a contradiction.

To summarize, we have the following hierachy of conjectures: The restriction conjecture implies the Kakeya maximal operator conjecture, which implies the Hausdorff Kakeya set conjecture, which implies the Minkowski Kakeya set conjecture. (One can add many other related conjectures to this hierachy; for instance, the Bochner-Riesz conjecture implies the restriction conjecture. There are also interrelationships between the cone, spherical, and parabolic conjectures. See [3].)

3. Summary

The material I’ve covered in the course so far basically represents the main progress on the problem in the 70’s. In the 80’s there was a lull in activity, which remained until 1991, when Bourgain observed that these implications could be partially reversed. Specifically, progress on the Kakeya conjecture would give progress on the restriction conjecture (although it is unknown whether a full solution of Kakeya would give a full solution of restriction). Bourgain then went on to prove some new Kakeya estimates, and consequently some new restriction estimates beyond Tomas-Stein, in higher dimensions. This breakthrough spurred a fair amount of activity which has intensified throughout this decade.

Roughly speaking, the restriction conjecture seems to be a mix of some oscillation estimates and some geometric, Kakeya-style estimates. Understanding Kakeya helps in understanding restriction, but it is only one piece of the puzzle. There are many other pieces that we’re just beginning to learn - for instance, we’re beginning to understand the importance of the bilinear viewpoint - but we’re still missing part of the necessary machinery.

In the next installment of notes, I’ll describe some partial progress on the Kakeya conjecture, starting with the classical estimate of Drury, and then the better estimates of Córdoba (in two dimensions) and Wolff (in higher dimensions). Then we’ll see how these Kakeya estimates are used to prove new restriction theorems. After that, I’ll discuss some very recent developments of Bourgain, Wolff, and others (including myself), which are edging us closer to a full resolution of these problems.

References

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

2. W. Beckner, A. Carbery, S. Semmes, F. Soria, A note on restriction of the Fourier transform to spheres., Bull. London Math. Soc. 21 (1989), no. 4, 394–398.

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

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

5. T. Tao, A. Vargas, A bilinear approach to cone multipliers I. Restriction Estimates, to appear, Geometric and Functional Analysis.

6. T. Wolff, A sharp bilinear cone restriction estimate, preprint.

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

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