AN INFORMAL PROOF OF WOLFF’S RESTRICTION THEOREM

TERENCE TAO

Abstract. We give an informal proof of Wolff’s recent bilinear restriction theorem [6].

1. Introduction

Let n ≥ 2 be an integer. We say that a function ϕ on R^{n+1} is a red wave if its space-time Fourier transform ϕ̂ is an L^{2} measure on the set

Σ^{red} = \{(ξ, |ξ|) : ∠(ξ, e_{1}) ≤ \frac{π}{8} , 1 ≤ |ξ| ≤ 2\}

where e_{1} is a basis vector. Similarly, we say that ψ is a blue wave if ψ̂ is an L^{2} measure on

Σ^{blue} = \{(ξ, −|ξ|) : ∠(ξ, e_{1}) ≤ \frac{π}{8} , 1 ≤ |ξ| ≤ 2\}.

Red and blue waves both solve the free wave equation, but propagate along different sets of characteristics.

We define the energy of ϕ, ψ by

E(ϕ) = ∥ϕ(t)∥_{2}^{2}, E(ψ) = ∥ψ(t)∥_{2}^{2}  (1)

where t ∈ R is arbitrary. This definition is independent of the choice of t, and is equivalent to more standard definitions of energy thanks to the frequency localization.

In [6] Wolff proved

Theorem 1.1. [6] Let 2 ≥ p > \frac{n+3}{n+1}. For all red waves ϕ and blue waves ψ, we have

∥ϕψ∥_{p} ≲ E(ϕ)^{1/2}E(ψ)^{1/2}.  (2)

The endpoint p = \frac{n+3}{n+1} was recently obtained in [4], and is sharp. The purpose of this note is to give an informal overview of the arguments and estimates used to prove the non-endpoint case (2). A rigorous version of the arguments below appears in [5].

The idea is to prove the localized version

∥ϕψ∥_{L^{p}(Q)} ≲ E(ϕ)^{1/2}E(ψ)^{1/2}  (3)

of (2) for all cubes Q of side-length R ≫ 1, where the constant is either independent of R or grows very slowly in R. In [6] Wolff achieves the latter, and uses some standard arguments in [1], [3] to remove the growth in R; in [5] I achieve the former.

1991 Mathematics Subject Classification. 42B15, 35L05.

To estimate the left-hand side of (3) we shall interpolate between L^{1} and L^{2} estimates. By integrating (1) in time we see that

∥ϕ∥_{L^{2}(Q)} ≲ R^{1/2}E(ϕ)^{1/2}  (4)

and similarly for ψ, hence

∥ϕψ∥_{L^{1}(Q)} ≲ RE(ϕ)^{1/2}E(ψ)^{1/2}.

Now suppose we could prove

∥ϕψ∥_{L^{2}(Q)} ≲ R^{−(n−1)/4}E(ϕ)^{1/2}E(ψ)^{1/2}.  (5)

Then by interpolation we would obtain (3) since p > (n + 3)/(n + 1).

Now, (5) cannot possibly hold as stated, since the left-hand side grows in R and the right-hand side decays in R. However, there is a substitute, which is roughly like

∥ϕψ∥_{L^{2}(Q\Q′)} ≲ R^{−(n−1)/4}E(ϕ)^{1/2}E(ψ)^{1/2}  (6)

where Q′ is a finite union of cubes of side-length ≪ R, such that |Q′| ≪ |Q|. This estimate does not suffer the defect of being obviously false as (5) does, and it turns out that one can still salvage (3) from this estimate and the L^{1} estimate by a rather technical induction argument.

Now lets see how to prove (6). What we’ll do is try to prove (5) first, and then realize halfway through the argument that we have to throw out a fraction of Q in order to make the estimate work.

We follow Bourgain and partition Q into \sqrt{R}-cubes q. We square (5) as

∑_{q} ∥ϕψ∥_{L^{2}(q)}^{2} ≲ R^{−(n−1)/2}E(ϕ)E(ψ).  (7)

We now follow the standard procedure first established by Fefferman and Cordoba for the circle, and extended to the cone by Bourgain, Wolff, Mockenhaupt, Vargas and others. Divide the set Σ^{red} into angular sectors Γ of width 1/\sqrt{R}, and partition ϕ = ∑_{Γ} ϕ_{Γ} accordingly. By orthogonality in frequency we have

E(ϕ) ∼ ∑_{Γ} E(ϕ_{Γ}).

The Fourier support of ϕ_{Γ} is contained in a 1 × 1/R × 1/\sqrt{R} × … × 1/\sqrt{R} box, which means by the uncertainty principle that ϕ_{Γ} is roughly constant on dual 1 × R × \sqrt{R} × … × \sqrt{R} boxes oriented in the (null) direction ω_{Γ} of Γ. It would then be logical to decompose ϕ_{Γ} by a partition of unity subordinate to these boxes; however these boxes are too eccentric to be tractable, and we shall instead use the slightly larger R × \sqrt{R} tubes instead.

More precisely, for each Γ, we let T be a finitely overlapping cover of R^{n+1} by R × \sqrt{R} tubes oriented in the direction ω_{Γ}, and we divide ϕ_{Γ} = ∑_{T} ϕ_{T} where each ϕ_{T} is concentrated on T and still has Fourier support on Γ (or perhaps a slight thickening of Γ). For this discussion I’ll pretend that ϕ_{T} physically vanishes outside T and vanishes in frequency outside Γ; this is not quite achievable rigorously, but there are substitutes which are effectively as good. By spatial orthogonality we have

E(ϕ_{Γ}) ∼ ∑_{T:T//ω_{Γ}} E(ϕ_{T}).

To summarize, we have a large number of R × \sqrt{R} tubes T, all oriented in directions normal to Σ^{red}, and a decomposition ϕ = ∑_{T} ϕ_{T} into red waves supported on T and having frequency support in a 1/\sqrt{R}-sector, and with the Bessel inequality

∑_{T} E(ϕ_{T}) ≲ E(ϕ).  (8)

By our construction every single tube T appears in the decomposition ϕ = ∑_{T} ϕ_{T}; but in practice only some of these tubes will contain a significant amount of energy. To visualize the decomposition we recommend imagining that only a relatively small collection of tubes T appear, and each tube carries a roughly equal amount of energy. (This follows the pigeonholing philosophy adopted in [6]).

We now write the left-hand side of (7) as

∑_{q} ∥ ∑_{T} ϕ_{T} ψ∥_{L^{2}(q)}^{2}.

By the triangle inequality, this is bounded by

∑_{q}(∑_{T} ∥ϕ_{T} ψ∥_{L^{2}(q)})^{2}.  (9)

We thus look for ways to estimate ∥ϕ_{T} ψ∥_{L^{2}(q)}.

There is a standard argument based on Plancherel and Cauchy-Schwarz which allows one to prove the estimate

∥ϕψ∥_{2} ≲ E(ϕ)^{1/2}E(ψ)^{1/2}

for all red waves ϕ and blue waves ψ. The key point is that Σ^{red} and Σ^{blue} are always transverse.

We could apply that estimate here, but it is quite inefficient because (a) it doesn’t take advantage of the frequency localization of ϕ_{T}, and (b) it doesn’t take advantage of the spatial localization to q. To exploit (a), we observe that we can improve the above standard estimate to

∥ϕ_{T} ψ∥_{2} ≲ R^{−(n−1)/4}E(ϕ_{T})^{1/2}E(ψ)^{1/2}

simply by repeating the Plancherel and Cauchy-Schwarz argument. The key new ingredient is that if ϕ_{T} is supported on Γ, then the intersection of Γ with any translate of Σ^{blue} is always an n − 1-dimensional set of measure O(R^{−(n−1)/2}).

Suppose q is supported in the slab R^{n} × I for some interval I of length \sqrt{R}. By integrating (1) over I, we may rewrite the above estimate as

∥ϕ_{T} ψ∥_{L^{2}(q)} ≲ R^{−(n+1)/4}∥ϕ_{T} ∥_{L^{2}(R^{n}×I)}∥ψ∥_{L^{2}(R^{n}×I)}.

However, ϕ_{T} and ψ satisfy the free wave equation, and thus should obey finite speed of propagation. This means that in the time interval I, the value of ϕ_{T} and ψ at q should be unaffected by the values of ϕ_{T} and ψ outside of Cq. One can make this heuristic precise, and improve the above estimate to

∥ϕ_{T} ψ∥_{L^{2}(q)} ≲ R^{−(n+1)/4}∥ϕ_{T} ∥_{L^{2}(Cq)}∥ψ∥_{L^{2}(Cq)} + R^{−N}E(ϕ_{T})^{1/2}E(ψ)^{1/2}.

Ignoring the error term, and inserting this estimate back into (9), we obtain

∑_{q}(∑_{T} R^{−(n+1)/4}∥ϕ_{T} ∥_{L^{2}(Cq)}∥ψ∥_{L^{2}(Cq)})^{2}.

To show (7), it thus suffices to show that

∑_{q} ∥ψ∥_{L^{2}(Cq)}^{2}(∑_{T} ∥ϕ_{T} ∥_{L^{2}(Cq)})^{2} ≲ RE(ϕ)E(ψ).

Since we’re pretending ϕ_{T} to be supported on T, we may rewrite the left-hand side heuristically as

∑_{q} ∥ψ∥_{L^{2}(Cq)}^{2}( ∑_{T:Cq⊂CT} ∥ϕ_{T} ∥_{L^{2}(Cq)})^{2}.

We now perform the following inspired Cauchy-Schwarz^{1}:

( ∑_{T:Cq⊂CT} ∥ϕ_{T} ∥_{L^{2}(Cq)})^{2} ≤( ∑_{T:q⊂CT} \frac{∥ϕ_{T} ∥_{L^{2}(Cq)}^{2}}{∥ψ∥_{L^{2}(CT)}^{2}} )

( ∑_{T:Cq⊂CT} ∥ψ∥_{L^{2}(CT)}^{2})

Now pretend for the moment that we had the energy estimate

∑_{T:Cq⊂CT} ∥ψ∥_{L^{2}(CT)}^{2} ≲ R^{1/2}E(ψ)  (10)

for each q.

Then after combining all the above estimates and simplifying, we reduce to showing that

∑_{q} ∥ψ∥_{L^{2}(Cq)}^{2} ∑_{T:Cq⊂CT} \frac{∥ϕ_{T} ∥_{L^{2}(Cq)}^{2}}{∥ψ∥_{L^{2}(CT)}^{2}} ≲ R^{1/2}E(ϕ).

Interchange the q and T summations, we estimate the left-hand side by

∑_{T}( sup_{Cq⊂CT} ∥ϕ_{T} ∥_{L^{2}(Cq)}^{2}) ∑_{q:Cq⊂CT} \frac{∥ψ∥_{L^{2}(Cq)}^{2}}{∥ψ∥_{L^{2}(CT)}^{2}} .

The inner sum is clearly O(1). By (4), the supremum is O(R^{1/2}E(ϕ_{T})). The claim then follows from (8).

It remains to show (10). Fix q. Let’s first prove the weaker estimate

∑_{T:Cq⊂CT} ∥ψ∥_{L^{2}(T′)}^{2} ≲ R^{1/2}E(ψ)

where T′ is the portion of CT which is at a distance ∼ R from q. We make the important geometrical observation that the T′ are finitely overlapping, and occupy the \sqrt{R}-neighbourhood of a red light cone

C = {(x, t) : (x − x_{0}, t − t_{0}) is normal to Σ^{red}}.

It thus suffices to show that

∥ψ∥_{L^{2}(C)} ≲ E(ϕ)^{1/2}

since the claim then follows by averaging over time translations by O(\sqrt{R}).

If C were a spacelike hypersurface, then this would just follow from standard energy estimates. Of course, C is a null hypersurface rather than a spacelike one, however the point is that it is only null in red directions, whereas ψ propagates along blue directions. Because of this, one can rescue the above energy estimates by some basic Fourier integral operator theory. (In Wolff’s paper, this observation is phrased in terms of red tubes and blue tubes, and observing that any given blue tube can only hit at most one of the red half-tubes T′ mentioned above).

1This trick is implicitly in Wolff [6], but as everything is pigeonholed in that paper it just takes the form of an algebraic manipulation.

Okay, so we’ve proved (10) for the distant half of things, when T is replaced by T′. What about the other half? Unfortunately, (10) is false for this half; the closer one gets to q, the more the tubes T overlap each other, and (10) begins to blow up. In other words, we have a failure of (10) - and hence of (5) when the tubes T form a “bush”, concentrating at a cube q. However, it turns out that these tubes can only form significant bushes at a very small number of such cubes q, where by “significant” I mean that virtually all the significant tubes T are involved in this bush. This is basically due to the fact that any two tubes with a reasonable angular separation can only intersect in at most one cube q. Now if one removes the portions of Q which are too close (distance ≪ R) to one of these bad cubes q, one can recover (10) and thus obtain (6).

References

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

[2] G. Mockenhaupt, A note on the cone multiplier, Proc. AMS 117 (1993), 145–152.

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

[4] T. Tao, Endpoint bilinear restriction theorems for the cone, and some null form estimates, to appear.

[5] T. Tao, Non-endpoint bilinear restriction theorems for the cone, expository notes.

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

Department of Mathematics, UCLA, Los Angeles, CA 90024

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