Local smoothing estimates, Bochner-Riesz estimates, and Kakeya estimates - a survey

Terence Tao (UCLA)

Abstract

In the last ten years there has been much progress and inter-connections regarding several conjectures in harmonic analysis, including the local smoothing, Bochner-Riesz, restriction, and Kakeya conjectures. We survey these results and connections in this talk.

The four basic problems:

• Wave equation estimates. Let u(x, t) solve the wave equation

□u(x, t) = F (x, t)
u(x, 0) = f(x)
u_{t}(x, 0) = g(x)

in R^{n+1}, where F, f, g are given data and □ = \frac{∂^{2}}{∂t^{2}} − Δ is the wave operator. We ask the question on how various space-time norms of u (L^{p}, Sobolev, mixed space-time norms, etc.) are controlled by various space-time norms of F and spatial norms of f and g (e.g. the energy norm ∥f∥_{H^{1}} + ∥g∥_{2}).

• Bochner-Riesz estimates. Let m(ξ) be a symbol on R^{n} which is singular on a surface such as a sphere, e.g.

m_{δ}(ξ) = (1 − |ξ|^{2})_{+}^{δ}.

We ask the question of what L^{p} estimates are available for the corresponding multiplier operator T defined by

{ \atop T f(ξ)} = m(ξ)f̂(ξ).

• The restriction problem. Let f be an L^{p} function on R^{n}. The Fourier transform f̂ is not continuous in general (unless p = 1), and hence cannot be evaluated at a single point; nevertheless, it is often possible to meaningfully restrict f̂ to a surface S providing that S has enough curvature. For which p and S is this possible?

• The Kakeya problem. Define a Besicovitch set (or Kakeya set) to be a subset of R^{n} which contains a unit line segment in every dimension. There are constructions of such sets with arbitrarily small (or zero) measure, but is it possible to create such a set with dimension strictly less than n?

• I’ve deliberately been vague in phrasing all four problems, because each problem comes with many variants. All four of these problems are still not completely settled, but there has been much progress recently.

• Although these problems appear quite dissimilar, they are quite inter-related. Oversimplifying a bit, the relationship is as follows:

5

Wave equation estimates

• The solution to the problem

□u(x, t) = F (x, t)
u(x, 0) = f(x)
u_{t}(x, 0) = g(x)

is given by

u(t) = cos(t\sqrt{−Δ})f + \frac{sin(t\sqrt{−Δ})}{\sqrt{−Δ}}g

+ ∫_{0}^{t} \frac{sin((t − s)\sqrt{−Δ})}{\sqrt{−Δ}}F (s) ds

so the problem of wave equation estimates can be (morally) reduced to asking how various norms of the space-time function

e^{it\sqrt{−Δ}}f(x)

are controlled by norms of a spatial function f(x).

• Since the wave evolution operator e^{it\sqrt{−Δ}} is unitary we have the energy estimate

∥e^{it\sqrt{−Δ}}f∥_{L_{x}^{2}} = ∥f∥_{2}

for all times t.

• The kernel E_{t} of this operator can be computed explicitly. In general it is only a distribution, not a function; however if one integrates enough times then it does become a function, and decays in time:

∥(1 + \sqrt{−Δ})^{−\frac{n+1}{2}−ε}E_{t}∥_{∞} ≲ t^{−\frac{n−1}{2}}

for all t ≠ 0. This gives

∥e^{it\sqrt{−Δ}}f∥_{L_{x}^{∞}} ≲ t^{−\frac{n−1}{2}}∥(1 + \sqrt{−Δ})^{\frac{n+1}{2}+ε}f∥_{1}.

• One can interpolate between these two estimates to obtain intermediate estimates such as

∥e^{it\sqrt{−Δ}}f∥_{L_{x}^{p}} ≲ ∥(1 + \sqrt{−Δ})^{(n+1)(\frac{1}{2}−\frac{1}{p})+ε}f∥_{p′}

for all 2 ≤ p ≤ ∞ and t ∼ 1. These estimates are due to Peral, and are essentially sharp, as shown by the so-called focussing example, in which f is supported on an annulus |x| = 1 + O(δ), and t is chosen to be t = 1 + O(δ). (This makes e^{it\sqrt{−Δ}}f quite large on the ball |x| = O(δ)).

• In the early 80s, it was realized that these fixedtime estimates, despite being sharp, were not precise enough to many applications (in particular, controlling low regularity solutions to non-linear wave equations). The main problem is that they require too much regularity on the initial data f, especially when p was far away from 2. To recover this “loss of derivatives”, time-averaging was introduced. In other words, one began to estimate quantities such as

(∫_{1}^{2} ∥e^{it\sqrt{−Δ}}f∥_{L_{x}^{p}}^{q} dt)^{1/q}

rather than try to obtain fixed-time estimates uniformly in t. These types of estimates (known collectively as local smoothing estimates) are better suited for applications (because one is usually more interested in the cumulative effect of a wave e^{it\sqrt{−Δ}}f than in uniform fixed-time estimates on that wave), and they are not seriously threatened by the focussing example.

• The first such estimates were obtained by Strichartz and later authors (Ginibre-Velo, Lindblad-Sogge, Kapitanski, Keel-T.), who showed that the above quantity is controlled by the L^{2} Sobolev norm ∥f∥_{H^{s}} of f for 2 ≤ q ≤ ∞ and p < ∞ as long as the scaling condition

\frac{n}{p} + \frac{1}{q} = \frac{n}{2} − s

and Knapp condition

\frac{(n − 1)/2}{p} + \frac{1}{q} = \frac{(n − 1)/2}{2}.

For instance, for n = 3 we have

∥e^{it\sqrt{−Δ}}f∥_{L_{x,t}^{4}} ≲ ∥f∥_{H^{1/2}},

while fixed-time estimates only give

∥e^{it\sqrt{−Δ}}f∥_{L_{x}^{4}} ≲ ∥f∥_{L_{1}^{4/3}}.

The argument is based on the T T^{∗} method, and uses the energy and decay estimates as key ingredients.

• The Knapp condition is dictated, not by the focussing example, but the so-called Knapp example or travelling wave example in which f is supported on a disk

{(x_{1}, x′) ∈ R × R^{n−1} : |x_{1}| ≲ δ^{2}, |x′| ≲ δ},

for some 0 < δ ≪ 1, while e^{it\sqrt{−Δ}}f is large on a translated disk

{(x_{1}, x′) ∈ R × R^{n−1} : |x_{1} − t| ≲ δ^{2}, |x′| ≲ δ}.

for times |t| ≲ 1. (For |t| ≳ 1 the wave begins to disperse).

• Strichartz estimates are very useful for studying nonlinear wave equations at low regularities, but are still not the perfect estimate. Part of the problem is that the right-hand side of a Strichartz estimate is always restricted to be in an L^{2} Sobolev space, while the lefthand side usually has different exponents than L^{2}. Some L^{p} → L^{q} estimates were then developed, for instance by Harmse and Oberlin, but they were not particularly satisfactory. The ideal estimate would be an L^{p} → L^{p} estimate, not lose any derivatives, and be a sharp example for both the focussing and Knapp examples.

• Sogge observed that there is essentially only one possible candidate for this “perfect estimate”, which is

∥e^{it\sqrt{−Δ}}f∥_{L_{x,t}^{2n/(n−1)}(R^{n}×[1,2])} ≲ ∥f∥_{2n/(n−1)}.

Such an estimate would recover all other known estimates for the wave equation by interpolation (mostly with quite trivial estimates such as the energy estimate).

• Unfortunately, this estimate is false - but only by a little bit. The best estimate we can hope for is

∥e^{it\sqrt{−Δ}}f∥_{L_{x,t}^{2n/(n−1)}(R^{n}×[1,2])} ≲ ∥f∥_{L_{ε}^{2n/(n−1)}}

for all ε > 0. This is known as Sogge’s local smoothing conjecture.

• This estimate is very difficult, even in two dimensions. When n = 2 Mockenhaupt obtained the estimate for all ε > 1/8; this was then improved slightly by Bourgain, Vargas, T., and Wolff, and the best result is now ε > \frac{5}{44}, obtained using Wolff’s recent cone restriction theorem.

• There are a couple reasons why one must lose an epsilon number of derivatives, but a particularly revealing one is due to Wolff (who modified an argument of Fefferman that we will see later). The construction is based around a Besicovitch set in R^{n} of measure zero; recall that a Besicovitch set contains a unit line segment in each direction. If one discretizes this set, looking only at a δ-separated set of directions, and thickening the corresponding line segments by δ, one obtains a collection of 1 × δ tubes T_{1}, …, T_{M} (with M ∼ δ^{1−n}) whose total volume goes to zero as δ → 0 (Besicovitch’s original construction gives a volume of about \frac{log log(1/δ)}{log(1/δ)}). Thus, these tubes have substantial overlap.

• Let T̃_{i} denote the tube T_{i} shifted backwards along its long direction w_{i} by a unit length, i.e. T̃_{i} = T_{i} − w_{i}. By performing Besicovitch’s construction carefully, one can arrange matters so that the shifted tubes T̃_{i} are disjoint, even though the original tubes T_{i} are overlapping quite heavily.

• Now we define f to be

f(x) = ∑_{i=1}^{δ^{n−1}} e^{iδ^{−2}x·w_{i}}χT̃_{i}(x).

The functions e^{iδ^{−2}x·w_{i}}χT̃_{i} are wave trains (about δ^{−2} Knapp examples cobbled together, actually). A computation shows that, essentially,

e^{it\sqrt{−Δ}}f(x) = ∑_{i=1}^{δ^{n−1}} e^{iδ^{−2}(x−tw_{i})·w_{i}}χ_{T̃_{i}+tw_{i}}(x).

Thus for t ∈ [1, 2] the wave trains have essentially moved from T̃_{i} to T_{i}. Since the T̃_{i} are disjoint but the T_{i} overlap heavily, it is easy to show that the L^{p} norm must increase significantly from t = 0 to t ∈ [1, 2] for all p > 2, which implies that the local smoothing conjecture is false without an additional epsilon of derivatives.

Bochner-Riesz estimates

• For any function m(ξ), define the multiplier T_{m} by

T̂_{m}f(ξ) = m(ξ)f̂(ξ).

• If the only singularity of m is at the origin, then the L^{p} mapping properties of T_{m} are well understood. Similarly if m is singular on a subspace. But things are not so well understood when m is singular on a curved surface such as the sphere.

• A model example of such operators are the Bochner-Riesz means S^{δ}, defined for δ ≥ 2 by

Ŝ^{δ}f(ξ) = (1 − |ξ|^{2})_{+}^{δ}f̂(ξ).

Originally, these means arose in the study of the L^{p} convergence properties of spherical Fourier summation. The higher δ is, the smoother the multiplier and more likely it is to be well-behaved.

• These means are always bounded on L^{2}, but not always on L^{p}. For instance, when δ = 0 these means are unbounded for all p ≠ 2 - this is an old result of Fefferman, and exploits the Besicovitch set construction as in Wolff’s argument.

• Herz observed that the convolution kernel of S^{δ} decayed like |x|^{−(n+1)/2−δ}, and deduced that these means could only be an L^{p} multiplier when

n/p, n/p′ > \frac{n + 1}{2} − δ.

• The Bochner-Riesz conjecture states that the above two necessary conditions are in fact sufficient for the L^{p} boundedness of S^{δ}. The critical value of p is p = 2n/(n − 1), in which case the conjecture mecomes

∥S^{ε}f∥_{2n/(n−1)} ≲ ∥f∥_{2n/(n−1)}.

This conjecture has been verified in two dimensions but is open in higher dimensions. For instance, in three dimensions the critical value of p is p = 3, but the conjecture has only been verified for p > 26/7 (Vargas, T.).

• Somewhat surprisingly, this conjecture is easily implied by the local smoothing conjecture. The point is that one can write

S^{ε} = (1 + Δ)_{+}^{ε} = ∫ a(t)e^{it\sqrt{−Δ}} dt

for some a(t). One can compute that a(t) decays like (1 + |t|)^{−1−ε}, and so by Hölder we have

|S^{ε}f(x)| ≲ (∫ (1 + |t|)^{−1−ε}|e^{it\sqrt{−Δ}}f|^{2n/(n−1)} dt)^{\frac{n−1}{2n}}.

Taking L^{2n/(n−1)} norms of both sides, and using rescaled versions of the local smoothing conjecture, one obtains the Bochner-Riesz conjecture.

Restriction estimates

• The classical restriction problem asks when we can meaningfully restrict the Fourier transform of an arbitrary L^{p} function to a compact surface S. The Fourier transform of an L^{1} function is continuous by Riemann-Lebesgue, and thus can always be restricted to a surface, but the Fourier transform of an arbitrary L^{2} function is another arbitrary L^{2} function, and so cannot be restricted in general.

• The question can be rephrased as whether the esti-

mate

∥f̂∥_{L^{q}(S)} ≲ ∥f∥_{L^{p}(R^{n})}

holds for test functions f and some exponent q. By duality, this is equivalent to the estimate

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

holding for all measures gdσ on S, where dσ is surface measure.

• These questions are also related to PDE; expressions such as ĝdσ represent solutions to the wave equation if S is a cone, the Schrödinger equation if S is a paraboloid, etc. Henceforth we set S to be the sphere.

• By setting g = 1 and noting that d̂σ decays like |x|^{−(n−1)/2}, we obtain the necessary condition p′ > 2n/(n − 1). There is also a Knapp example in this context; if one sets g to be supported on a spherical δ-cap, then ĝdσ is concentrated on a δ^{−1} × δ^{−2} tube, and one can obtain the condition p′ ≥ \frac{n+1}{n−1}q. The Restriction conjecture states that these necessary conditions are sufficient.

• For the sphere, this conjecture is only settled in two dimensions; for the cone, the analogous conjecture has recently been settled up to four dimensions by Wolff. Again, in three dimensions the critical exponent is p′ = 3, but the conjecture has only been verified for p′ > 26/7 (Vargas, T.).

• L^{2} restriction estimates such as

∥f̂∥_{L^{2}(S)} ≲ ∥f∥_{L^{p}(R^{n})}

were proven by Tomas and Stein using T T^{∗} methods (and are related to the Strichartz estimates mentioned earlier). By Plancherel’s theorem and a polar co-ordinates argument one can then obtain L^{2} local Bochner-Riesz estimates such as

∥S^{δ}f∥_{L^{2}(R^{n})} ≲ ∥f∥_{L^{p}(R^{n})}

which can then be converted into L^{p} estimates by a Hölder inequality argument.

• Conversely, Bochner-Riesz estimates imply restriction estimates. If f is compactly supported, then f̂ is smooth, and one can compute S^{δ}f quite explicitly using the method of stationary phase. In fact, one obtains

S^{δ}f(x) ≈ e^{i|x|}|x|^{−\frac{n+1}{2}−δ}f̂(\frac{x}{|x|})

for |x| large. A rigorous version of this “folk calculation” of Fefferman has been used to show that the Bochner-Riesz conjecture implies the restriction conjecture.

• One strategy to prove restriction estimates is to square both sides and bilinearize the estimate as

∥ĝdσĥdσ∥_{L^{p′/2}(R^{n})} ≲ ∥g∥_{L^{q′}(S)}∥h∥_{L^{q′}(S)}

and then to decompose g and h with respect to the angular separation between their supports. This technique was first developed in the 70s in the case p′ = 4 (thus converting linear L^{4} estimates into bilinear L^{2} estimates), but it is only in the last few years that this technique has been effectively used for other values of p. All the current best results in this field, including Wolff’s restriction theorem, are obtained by bilinear techniques.

• The reason why bilinear techniques seem to be effective is that the Knapp example, which is the primary obstruction to proving restriction estimates, disappear in the bilinear setting if one assumes that the supports of g and h have a large angular separation. (The case when g and h have small angular separation can then be handled by a rescaling argument).

Kakeya estimates

• Recall that a Besicovitch set is a set which contains a unit line segment in every direction. The classical construction of Besicovitch shows that such sets can have zero measure, but it is an open problem (except in two dimensions) whether they can have Minkowski or Hausdorff dimension strictly less than n. The Kakeya conjecture asserts that this is impossible (i.e. all Besicovitch sets have full dimension).

• There is a discretized version of this conjecture which is stronger. Namely, let T be a collection of δ × 1 tubes in R^{n} such that the directions of any two tubes are δ-separated. Then the Kakeya maximal function conjecture asserts that

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

for all p ≥ n/(n − 1). The quantity on the righthand side is natural, as it occurs when the tubes T all intersect at a common point. Roughly speaking, this conjecture asserts that this type of intersection is pretty much the worst way that tubes which are separated in angle can intersect.

• An easy argument shows that if the above estimate is true for some p, then all Besicovitch sets have Hausdorff and Minkowski dimension at least p′. In three dimensions, this can quickly be used to show that Besicovitch sets have dimension at least 2, basically by exploiting the fact that any two lines (tubes) can intersect in at most one point (ball) and performing some elementary combinatorics. This was later improved by Bourgain and later by Wolff to 5/2, using some more combinatorics and some geometrical facts (e.g. any two intersecting lines determine a unique plane). This is still the best bound on the maximal function and Hausdorff dimension problems, but the bound on the easier Minkowski problem has recently been improved to 5/2 + 10^{−10}. (Katz, Laba, T.)

• In higher dimensions there is a promising recent approach of Bourgain and extended by Katz, T., which treats line segments as arithmetic progressions and invokes some sophisticated results from additive combinatorics, including some recent work of Gowers. This purely arithmetical approach has yielded a lower bound of \frac{6n}{11} + \frac{5}{11} for the Hausdorff dimension and \frac{4n}{7} + \frac{3}{7} for the Minkowski dimension just by using fairly elementary combinatorics; in contrast, the geometrical arguments of Wolff give a lower bound of \frac{n}{2} + 1.

• We have seen already that these sets are connected with the previous conjectures. Indeed, by pursuing these arguments a bit further one can show that any of the local smoothing, Bochner-Riesz, or restriction conjectures imply the Kakeya conjectures listed here.

• Conversely, given a good Kakeya estimate it is possible to obtain a non-trivial restriction estimate (either linear or bilinear), or Bochner-Riesz estimate. Very briefly, the idea is as follows.

• To obtain a restriction estimate we need to control the L^{p} norm of a quantity f̂dσ. A model case is the Knapp example, in which f is concentrated on a δcap centered at ω (say) and f̂dσ is concentrated on a δ^{−1} × δ^{−2} tube T oriented in the direction ω, and looks roughly like e^{iω·x}χ_{T}(x).

• By a partition of both space and frequency, we can decompose f̂dσ as a linear combination of Knapp examples:

f̂dσ(x) ≈ ∑_{T} c_{T}e^{iω·x}χ_{T}(x).

We can usually control some l^{r} norm of the c_{T} by an L^{q} norm of f.

• To estimate this quantity, we use some sort of orthogonality (or standard restriction estimates) to control the oscillatory sum by its square function:

∥ ∑_{T} c_{T}e^{iω·x}χ_{T}(x)∥_{p} ≲ ∥(∑_{T} |c_{T}e^{iω·x}χ_{T}(x)|^{2})^{1/2}∥_{p};

in practice this is not easy to do, and often we must lose a power of δ or something similar at this stage.

• The square function simplifies to the non-oscillatory

expression

∥ ∑_{T} |c_{T}|^{2}χ_{T}∥_{p/2}^{1/2},

and this can be handled by Kakeya estimates.

• A similar strategy also has allowed recent progress on wave equation estimates. Again, the idea is to break the wave up into wave packets (Knapp examples), use restriction estimates (this time for the cone) to replace the sum of wave packets by its square function, and use Kakeya estimates (now for light rays, rather than arbitrary tubes) to estimate this function. This is where the present best estimates on all of the oscillatory estimates come from.