A PROOF OF WOLFF’S RESTRICTION THEOREM

TERENCE TAO

Abstract. We re-prove Wolff’s recent bilinear restriction theorem [6].

1. Introduction

Let n ≥ 2 be an integer. We say that a function ϕ : R^{n+1} → H is a red wave if it takes values in a finite dimensional complex Hilbert space, and 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 : R^{n+1} → H′ is a blue wave if H′ is a finite dimensional complex Hilbert space and 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. The vector valued formulation will be convenient for technical reasons.

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.

The main result of this paper is the following bilinear estimate.

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)

Here we adopt the convention that ϕψ : R^{n+1} → H ⊗H′ denotes the tensor product of ϕ and ψ.

The endpoint p = \frac{n+3}{n+1} was recently obtained in [5], and is sharp. The arguments here are a simplified version of those in that paper.

1991 Mathematics Subject Classification. 42B15, 35L05.

2. Notation

We let C denote various large numbers that vary from line to line, and let A ≲ B or A = O(B) denote the estimate A ≤ CB where C depends only on n and on the parameter N if this appears in the equation. Similarly we use A ≪ B to denote A ≤ C^{−1}B. Generally speaking, we shall use the ≲ notation to control error terms, but will need the more precise ≤ notation for the main terms due to the inductive nature of the argument.

Note that our constants C are independent of the dimension of the spaces H, H′. This will be important for the induction argument.

We shall always treat R^{n+1} as endowed with the Euclidean metric and never with the Minkowski metric, so that terms such as length |x|, angle ∠(x, y), etc. retain their usual meaning in R^{n+1}.

If ϕ(x, t) is a function of both space and time, we use ϕ(t) to denote the spatial function ϕ(t)(x) = ϕ(x, t).

We use the hat notation for the spatial Fourier transform

f̂(ξ) = ∫_{R_{n}} e^{−2πix·ξ}f(x) dx

as well as the spacetime Fourier transform

ϕ̂(ξ, τ) = ∫_{R_{n+1}} e^{−2πi(x·ξ+tτ)}ϕ(x, t) dxdt,

with the meaning being clear from context. We define the frequency support supp(f̂) of a function f to be the support of the Fourier transform f̂.

We define Q(x_{Q}, t_{Q}; r_{Q}) to be the n + 1-dimensional cube in R^{n+1} centered at (x_{Q}, t_{Q}) with side-length r_{Q} and with sides parallel to the axes. We call the interval [t_{Q} − r_{Q}/2, t_{Q} + r_{Q}/2] the lifespan of Q. If Q = Q(x_{Q}, t_{Q}; r_{Q}) is a cube and c > 0, we use cQ to denote the cube cQ = Q(x_{Q}, t_{Q}; cr_{Q}).

Let \underline{Σ} = \underline{Σ}_{n} denote the spatial region

\underline{Σ} = \{ξ : \frac{1}{2} ≤ |ξ| ≤ 4, ∠(ξ, e_{1}) ≤ π/4\}.

If (x_{0}, t_{0}) ∈ R^{n+1}, define the red cone with vertex (x_{0}, t_{0}) to be the set

C^{red}(x_{0}, t_{0}) = \{(x_{0} + rω, t_{0} − r) : r ∈ R, ω ∈ S^{n−1} ∩ \underline{Σ}\},

and the blue cone at this vertex to be

C^{blue}(x_{0}, t_{0}) = \{(x_{0} + rω, t_{0} + r) : r ∈ R, ω ∈ S^{n−1} ∩ \underline{Σ}\}.

For any r ≫ 1, we define C^{red}(x_{0}, t_{0}; r), C^{blue}(x_{0}, t_{0}; r) to be the r-neighbourhoods of C^{red}(x_{0}, t_{0}), C^{blue}(x_{0}, t_{0}) respectively.

We now make precise the heuristic that red waves propagate along red cones and blue waves propagate along blue cones. Fix a(ξ) to be a bump function supported on \underline{Σ} which equals 1 on the spatial projection of Σ^{red} and Σ^{blue}. We define the evolution operator U(t) by

{ \atop U(t)f}(ξ) = a(ξ)e^{2πit|ξ|}f̂(ξ)

and the kernel K_{t} by

K_{t}(x) = ∫ a(ξ)e^{2πi(x·ξ+t|ξ|)} dξ.

We have the propagation law

ϕ(t) = U(t)ϕ(0) = ϕ(0) ∗ K_{t}  (3)

for all red waves ϕ and all times t_{1}, t_{2}. A stationary phase computation gives the pointwise estimate

|K_{t}(x)| ≲ (1 + dist((x, t), C^{red}(0, 0))^{−N}  (4)

for arbitrarily large N. Applying (3) with t = 0 and then using Young’s inequality and (4) we obtain ∥ϕ(0)∥_{∞} ≲ E(ϕ)^{1/2}; by time translation invariance and time reversal symmetry we thus have

∥ϕ∥_{∞} ≲ E(ϕ)^{1/2}, ∥ψ∥_{∞} ≲ E(ψ)^{1/2}.  (5)

Throughout the paper, we use η_{0} to denote a non-negative Schwarz function on R^{n} with total mass 1 and whose Fourier transform is supported on the unit disk; such a function can be constructed for instance by η_{0} = φ̂ ∗ φ̂, where φ is a real even C^{∞} function supported near the origin such that φ(0) = 1. For any r > 0 we define η_{r} by

η_{r}(x) = r^{−n}η_{0}(\frac{x}{r}).  (6)

If ϕ is a red wave, we define the margin of ϕ to be the quantity

dist(supp(ϕ̂), ∂Σ^{red})

where ∂Σ^{red} is the boundary of Σ^{red} on the light cone, and the angular dispersion of ϕ to be the quantity

diam\{\frac{ξ}{|ξ|} : (ξ, τ) ∈ supp(ϕ̂)\}.

We define the same notions for blue waves ψ by time-reversal symmetry.

3. Beginning of proof

We can now begin the proof of Theorem 1.1. Henceforth p shall be fixed.

The estimate (2) is global in spacetime. Our strategy shall be to replace this global estimate by an estimate localized to a cube Q, and use induction to send r_{Q} → ∞. More precisely, we define

Definition 3.1. For each R ≫ 1, define A(R) to be the best constant for which the inequality

∥ϕψ∥_{L^{p}(Q)} ≤ A(R)E(ϕ)^{1/2}E(ψ)^{1/2}  (7)

holds for all red and blue waves ϕ, ψ (including vector-valued waves) with margin at least 1/100 − R^{−1/N}, and all cubes Q of side-length R.

From trivial estimates we see that A(R) is finite for each R. The margin requirements are technicalities which are needed for the induction on R to work properly, as many of our decompositions will decrease the margin of ϕ and ψ slightly. However, it is clear by a finite partition of space and frequency, and some mild Lorentz transforms that Theorem 1.1 will obtain if we can show that A(R) is uniformly bounded in R.

We shall in fact show Proposition 3.2. For any R ≫ 1 and any 0 < c ≪ 1, we have

A(R) ≤ (1 + Cc)A(R/2) + Cc^{−C}R^{−ε}.  (8)

where ε > 0 is a constant depending only on n and p.

By choosing c = R^{−ε/2C} we thus see that

A(R) + 1 ≤ (1 + CR^{−ε/2C})(A(R/2) + 1);

telescoping this, we obtain the desired boundedness of A(R).

It remains to show Proposition 3.2. We first develop a number of preliminaries.

4. Energy estimates on light cones of opposite color

By integrating (1) along the life-span of a cube we see that

∥ϕ∥_{L^{2}(Q)} ≤ r_{Q}^{1/2}E(ϕ)^{1/2}, ∥ψ∥_{L^{2}(Q)} ≤ r_{Q}^{1/2}E(ψ)^{1/2}  (9)

for any cube Q. In particular, by Hölder we have

∥ϕψ∥_{L^{1}(Q)} ≲ r_{Q}E(ϕ)^{1/2}E(ψ)^{1/2}.  (10)

The purpose of this section is to obtain the following improvement to (9).

Lemma 4.1. Let ϕ be a red wave. Then for any (x_{0}, t_{0}) ∈ R^{n+1} and R ≫ 1 we have

∥ϕ∥_{L^{2}(C^{blue}(x_{0},t_{0};R))} ≲ R^{1/2}E(ϕ)^{1/2}.

Proof By translation invariance we may take (x_{0}, t_{0}) = 0.

The blue cone is a null surface, and so standard energy estimates will not prove this estimate. However this can be salvaged since ϕ is red, so that the characteristics of ϕ will be transverse to the blue cone.

We turn to the details. By (3), it suffices to show that

(∫ ∥χ_{E_{t}}U(t)f∥_{L^{2}(R^{n})}^{2} dt)^{1/2} ≲ R^{1/2}∥f∥_{L^{2}(R^{n})}

for all f ∈ L^{2}(R^{n}), where E_{t} is the slice E_{t} = {x : (x, t) ∈ C^{blue}(0, 0; R)}. We shall apply the T T^{∗} method. By duality the above estimate is equivalent to

∥ ∫ U(t)^{∗}(χ_{E_{t}}F (t)) dt∥_{L^{2}(R^{n})} ≲ R^{1/2}∥F ∥_{L^{2}(R^{n+1})}.

We square this as

∫ ∫ ⟨χ_{E_{s}}U(s)U(t)^{∗}(χ_{E_{t}}F (t)), F (s)⟩ dtds ≲ R∥F ∥_{2}^{2}

where ⟨f, g⟩ = ∫R^{n} f(x)\overline{g(x)} dx is the usual complex inner product. By Cauchy-Schwarz and Young’s inequality it suffices to show that

|⟨χ_{E_{s}}U(s)U(t)^{∗}(χ_{E_{t}}F (t)), F (s)⟩| ≲ (1 + |s − t|/R)^{−N}∥F (t)∥_{2}∥F (s)∥_{2}

for all s, t and some sufficiently large N.

When |s − t| ≲ R this follows from Cauchy-Schwarz and the L^{2} boundedness of the operator χ_{E_{s}}U(s)U(t)^{∗}χ_{E_{t}}, so we may assume that |s−t| ≫ R. The convolution operator U(s)U(t)^{∗} behaves essentially like U(s − t), and satisfies a similar kernel estimate to (4), namely

|K(x)| ≲ (1 + dist((x, s − t), C^{red}(0, 0))^{−N}

for all N. The claim then follows from the transversality of C^{red}(0, 0) and C^{blue}(x_{0}, t_{0}) and crude estimates.

By time reversal symmetry a similar estimate holds with the roles of red and blue reversed.

5. Bilinear L^{2} estimates in the low dispersion case.

A key ingredient in the proof of Theorem 1.1 is

Lemma 5.1. [4, 6] Let ϕ be a red wave, and let ψ be a blue wave. If ϕ has angular dispersion O(1/r) for some r ≫ 1, then

∥ϕψ∥_{L^{2}(R^{n+1})} ≲ r^{−(n−1)/2}E(ϕ)^{1/2}E(ψ)^{1/2}.  (11)

In particular, with no dispersion hypothesis on ϕ we have the well-known estimate

∥ϕψ∥_{2} ≲ E(ϕ)^{1/2}E(ψ)^{1/2}.  (12)

Proof From hypothesis, the frequency support of ϕ is contained in a a sector Γ of width O(1/r). By Plancherel’s theorem it thus suffices to show that

∥fdσ_{1} ∗ gdσ_{2}∥_{2} ≲ r^{−(n−1)/2}∥f∥_{L^{2}(Γ)}∥g∥_{L^{2}(Σ^{blue})}

for all f and g, where dσ_{1} and dσ_{2} denote surface measure on Γ and Σ^{blue} respectively. By Young’s inequality we have

∥fdσ_{1} ∗ gdσ_{2}∥_{1} ≲ ∥f∥_{1}∥g∥_{1}

so it suffices by interpolation to show that

∥fdσ_{1} ∗ gdσ_{2}∥_{∞} ≲ r^{−(n−1)}∥f∥_{∞}∥g∥_{∞},

which by positivity reduces to showing that

∥dσ_{1} ∗ dσ_{2}∥_{∞} ≲ r^{−(n−1)}.  (13)

But this follows from the observation that the intersection of Γ and x − Σ^{blue} is always transverse and has n − 1-dimensional measure at most O(r^{−(n−1)}) for any x ∈ R^{n+1}.

This estimate is global in spacetime, however for our applications it will be more convenient to work with a localized form of Lemma 5.1.

Lemma 5.2. Let ϕ be a red wave, angular dispersion O(1/r), and margin at least 1/200, and let ψ be a blue wave with margin at least 1/200. Then we have

∥ϕψ∥_{L^{2}(Q)} ≲ r^{−(n+1)/2}∥ϕ∥_{L^{2}(CQ)}∥ψ∥_{L^{2}(CQ)} + r^{−N}E(ϕ)^{1/2}E(ψ)^{1/2} (14)

for all cubes Q of side-length r and all N.

Proof Let S ⊂ R^{n} denote the n-cube

S = {x ∈ R^{n} : (x, t) ∈ CQ for some t}.

It suffices to show that

∥ϕψ∥_{L^{2}(Q)} ≲ r^{−(n−1)/2}∥ϕ(t_{0})∥_{L^{2}(S)}∥ψ(t_{0})∥_{L^{2}(S)} + r^{−N}E(ϕ)^{1/2}E(ψ)^{1/2} (15)

for all t in the life-span of S, since (14) follows by averaging this over t and applying Cauchy-Schwarz.

It remains to prove (15). By time translation invariance we may assume that t_{0} = 0.

Define the wave ϕ̃ by defining

ϕ̃(0) = ϕ(0)(χ_{\frac{1}{2} S} ∗ η_{200})

and

ϕ̃(t) = U(t)ϕ̃(0).

One easily verifies that ϕ̃ is a red wave with E(ϕ̃) ≲ E(ϕ). By construction we see that ϕ(0) − ϕ̃(0) is extremely small on \frac{1}{4}S, indeed we have

∥ϕ(0) − ϕ̃(0)∥_{L^{∞}( \frac{1}{4} S)} ≲ r^{−N}

for arbitrarily large N. By (3) and (4) we thus have (if the constant C was chosen sufficiently large)

∥ϕ − ϕ̃∥_{L^{∞}(Q)} ≲ r^{−N}.

Combining this with (9) and the triangle inequality we see that

∥ϕψ∥_{L^{2}(Q)} ≲ ∥ϕ̃ψ∥_{L^{2}(Q)} + r^{−N}E(ϕ)^{1/2}E(ψ)^{1/2}

for all N. Now define

ψ̃(0) = ψ(0)(χ_{\frac{1}{2} S} ∗ η_{200})

and

ψ̃(t) = U(−t)ψ̃(0).

By similar arguments to before we have

∥ϕψ∥_{L^{2}(Q)} ≲ ∥ϕ̃ψ̃∥_{L^{2}(Q)} + r^{C−N}E(ϕ)^{1/2}E(ψ)^{1/2}.

The claim then follows by observing that

E(ϕ̃) ≲ ∥ϕ(0)∥_{L^{2}(S)} + r^{−N}E(ϕ)^{1/2}

and similarly for ψ̃.

6. Wave packets

In this section we derive another basic tool, namely the ability to break up a red wave ϕ into “wave packets” which have low angular dispersion and which are concentrated on tubes.

Let R, Q, c be as in Proposition 3.2. Set r = 2^{−J}R, where J is an integer chosen so that r ∼ \sqrt{R}.

Let E be a maximal 1/r-separated subset of S^{n−1} ∩ \underline{Σ}, and let L denote the lattice L = c^{−2}rZ^{n}. We define a red tube to be any set T = T (ω_{T} , x_{T}) of the form

T = {(x, t) ∈ R^{n+1} : |x − (x_{T} + ω_{T} t)| ≤ r}

where x_{T} ∈ L and ω_{T} ∈ E. We let T denote the set of all red tubes.

If T is a red tube, we define the cutoff function χ̃_{T} on R^{n+1} by

χ̃_{T} (x, t) = (1 + \frac{|x − x_{T} − ω_{T} t|}{r})^{−100n}.  (16)

Lemma 6.1. Suppose that ϕ is a red wave with margin at least θ ≫ r^{−1}. Then there exists a decomposition

ϕ = ∑_{T ∈T} ϕ_{T} .  (17)

into red waves ϕ_{T} of margin at least θ − Cr^{−1} and angular dispersion O(r^{−1}) satisfying the following properties.

• If Q′ ranges over some index set, and a_{Q′,T} are non-negative numbers such that ∑_{Q′} a_{Q′,T} = 1 for each T, then we have the Bessel inequality

∑_{Q′} E( ∑_{T ∈T} a_{Q′,T} ϕ_{T}) ≤ (1 + Cc)E(ϕ).  (18)

• If T ∈ T is a tube with dist(T, Q) ≫ R, then

∥ϕ_{T} ∥_{L^{∞}(Q)} ≲ dist(T, Q)^{C−N}E(ϕ)^{1/2}.  (19)

• We have the estimate

∑_{T ∈T} sup_{q∈Q_{J} (Q)} χ̃_{T} (x_{q}, t_{q})^{−3}∥ϕ_{T} ∥_{L^{2}(Cq)}^{2} ≲ c^{−C}rE(ϕ).  (20)

Proof By translation invariance we may assume Q is centered at the origin.

Partition

S^{n−1} ∩ \underline{Σ} = ⋃_{ω∈E} A_{ω}

where A_{ω} consists of those points in S^{n−1} ∩ \underline{Σ} which are closer to ω than any other element of E. Thus A_{ω} is in the O(1/r)-neighbourhood of ω.

Let G ⊂ SO(n) denote the set of all rotations in R^{n} which differ from the identity by O(1/r). Let dΩ be a smooth compactly supported probability measure on the interior of G.

For each Ω ∈ G and ω ∈ E, we define the Fourier projection operators P_{Ω,ω} for test functions f on R^{n} by

{ \atop P_{Ω,ω}f}(ξ) = χ_{Ω(A_{ω})}(ξ/|ξ|)f̂(ξ).

Note that

∑_{ω∈E} P_{Ω,ω}ϕ(0) = ϕ(0)

for all Ω ∈ G and red waves ϕ of frequency 1.

Write

η^{x_{0}}(x) = η_{0}(\frac{c^{2}}{r} (x − x_{0}))

for all x_{0} ∈ L, where η_{0} is as in (6). From the Poisson summation formula we have

1 = ∑_{x_{0}∈L} η^{x_{0}}.

We define the functions ϕ_{T} at time 0 by

ϕ_{T} (x, 0) = η^{x_{T}} (x) ∫ (P_{Ω,ω_{T}} ϕ(0))(x) dΩ  (21)

and at other times by

ϕ_{T} (t) = U(t)ϕ_{T} (0).

One may verify that ϕ_{T} is a red wave the correct margin and angular dispersion properties, and that (17) holds.

We now prove (18). We expand the left-hand side as

(∑_{Q′} ∥ ∫ ∑_{ω∈E} ∑_{x_{0}∈L} a_{Q′,T (ω,x_{0})}η^{x_{0}}P_{Ω,ω}(ϕ(0)) dΩ∥_{2}^{2})^{1/2}.

By Minkowski’s inequality this is less than or equal to

∫ (∑_{Q′} ∥ ∑_{ω} ∑_{x_{0}} a_{Q′,T (ω,x_{0})}η^{x_{0}}(x)P_{Ω,ω}(ϕ(0))∥_{2}^{2})^{1/2} dΩ.  (22)

Define the set Y ⊂ S^{n−1} by

Y = ⋃_{ω∈E}{α ∈ A_{ω} : dist(α, S^{n−1}\A_{ω}) > Cc^{2}/r},

and define P_{Ω(Y)} to be the multiplier

{ \atop P_{Ω(Y)}f}(ξ) = χ_{Ω(Y)}(\frac{ξ}{|ξ|})f̂(ξ).

By the triangle inequality, (22) is less than the sum of

∫ (∑_{Q′} ∥ ∑_{ω} ∑_{x_{0}} a_{Q′,T (ω,x_{0})}η^{x_{T}} P_{Ω,ω}P_{Ω(Y)}ϕ(0)∥_{2}^{2})^{1/2} dΩ.  (23)

and

∫ (∑_{q_{0}} ∥ ∑_{ω} ∑_{x_{0}} a_{Q′,T (ω,x_{0})}η^{x_{0}}(x)P_{Ω,ω}(1 − P_{Ω(Y)})ϕ(0)∥_{2}^{2})^{1/2} dΩ.  (24)

Consider the quantity (23). The contributions of each ω are orthogonal as ω varies, so we can rewrite (23) as

∫ (∑_{Q′} ∑_{ω} ∥ ∑_{x_{0}} a_{Q′,T (ω,x_{0})}η^{x_{0}}(x)P_{Ω,ω}P_{Ω(Y)}ϕ(0)∥_{2}^{2})^{1/2} dΩ,

which can be re-arranged as

∫ (∑_{ω} ∫ |(P_{Ω,ω}P_{Ω(Y)}ϕ(0))(x)|^{2} ∑_{Q′}(∑_{x_{0}} a_{Q′,T (ω,x_{0})}η^{x_{0}}(x))^{2} dx)^{1/2} dΩ.

Clearly this is less than

∫ (∑_{ω} ∫ |(P_{Ω,ω}P_{Ω(Y)}ϕ(0))(x)|^{2}(∑_{Q′} ∑_{x_{0}} a_{Q′,T (ω,x_{0})}η^{x_{0}}(x))^{2} dx)^{1/2} dΩ.

Summing in Q′ and then in x_{0}, this simplifies to

∫ (∑_{ω} ∫ |(P_{Ω,ω}P_{Ω(Y)}ϕ(0))(x)|^{2} dx)^{1/2} dΩ.

By the orthogonality of the P_{Ω,ω}, we thus have

(23) ≤ ∫ ∥P_{Ω(Y)}ϕ(0)∥_{2} dΩ ≤ E(ϕ)^{1/2}.

Now consider (24). Repeating the above argument, but noting that we must use almost orthogonality instead of orthogonality, we obtain

(24) ≲ ∫ ∥(1 − P_{Ω(Y)})ϕ(0)∥_{2} dΩ.

By Cauchy-Schwarz we thus have

(24) ≲ (∫ ∥(1 − P_{Ω(Y)})ϕ(0)∥_{2}^{2} dΩ)^{1/2};

by Plancherel we thus have

(24) ≲ (∫ |ϕ̂(ξ, 0)|^{2}(∫ χ_{Ω(Y)^{c}}(\frac{ξ}{|ξ|}) dΩ) dξ)^{1/2}.

The inner integral can be seen to be O(c^{2}) for all ξ, so by Plancherel again we have

(24) ≲ cE(ϕ)^{1/2}.

Combining our estimates for (23) and (24) we obtain (18).

We now prove (19). From the definition of ϕ_{T} it is easy to verify that (19) holds at time t = 0 with Q replaced by CQ. The claim then follows from (3) and (4).

Now we prove (20). We shall need the auxiliary estimate

Lemma 6.2. If T is a red tube, and ϕ is a red wave with frequency support in the sector {(ξ, |ξ|) : |ξ| ∼ 1, ∠(ξ, ω_{T}) ≲ C/r}, then we have

∥ϕ(t_{0})∥_{L^{2}(T ∩R^{n}×{t_{0}})} ≲ ∥χ̃_{T}^{3} (0)ϕ(0)∥_{2}  (25)

for all t_{0} = O(R).

Proof We have the identity

ϕ(x, t_{0}) = ∫ e^{2πi(x·ξ+t_{0}|ξ|)}φ_{T} (ξ)ϕ̂(0)(ξ) dξ

where φ is a bump function adapted to the sector \{ξ ∈ \underline{Σ} : ∠(ξ, ω_{T}) ≲ 1/r\}. By standard stationary phase estimates and the observation that t_{0} = O(r^{2}), we thus have

ϕ(t_{0}) = ϕ(0) ∗ K

where the kernel K satisfies the estimates

|K(x)| ≲ r^{1−n}(1 + |x − t_{0}ω_{T} |/r)^{−5N}(1 + |(x − t_{0}ω_{T}) · ω_{T} |)^{−5N}.

The claim (25) then follows from a dyadic decomposition of K around the point t_{0}ω_{T}, followed by Young’s inequality.

By the above Lemma, the left-hand side of (20) is majorized by

r ∑_{T} sup_{q} χ̃_{T} (x_{q}, t_{q})^{−3}∥χ̃_{T_{q}}(0)^{3}ϕ_{T} (0)∥_{2}^{2},

where for each T, q, T_{q} is a translate of T which intersects q. From (21) and the rapid decay of η_{0}, this is majorized by

c^{−C}r ∑_{T} sup_{q} χ̃_{T} (x_{q}, t_{q})^{−3}∥χ̃_{T_{q}}(0)^{3}χ̃_{T} (0)^{4} ∫ |P_{Ω,ω_{T}} ϕ(0)| dΩ∥_{2}^{2}.

From (16) we have the elementary inequality

χ̃_{T_{q}}(x, 0)χ̃_{T} (x, 0) ≲ χ̃_{T} (x_{q}, t_{q})

so we may bound the previous by

c^{−C}r ∑_{T} ∥χ̃_{T} (0) ∫ |P_{Ω,ω_{T}} ϕ(0)| dΩ∥_{2}^{2}.

Summing in x_{T}, we reduce to showing that

∑_{ω∈E} ∥ ∫ |P_{Ω,ω}ϕ(0)| dΩ∥_{2}^{2} ≲ E(ϕ).

But this follows from Minkowski’s inequality followed by Plancherel’s theorem.

7. The key decomposition

Let R, c be as in Proposition 3.2. By combining the results of the previous three sections together, we can now derive the main proposition used to derive Proposition 3.2.

If Q is a cube of side-length R, and j ≥ 0 is an integer, we may partition Q into 2^{(n+1)j} cubes of side-length 2^{−j}R; we use Q_{j}(Q) to denote the collection of these cubes.

Proposition 7.1. Let R ≫ 1, 0 < c ≪ 1 and θ ≫ R^{−1/2}. Let Q be a spacetime cube of side-length R, ϕ be a red wave of margin at least θ, and ψ be a blue wave.

Then there exist red waves ϕ^{(Q′)} for all Q′ ∈ Q_{2}(Q) with margin at least θ − CR^{−1/2} such that the following properties hold.

• We have the reconstruction formula

ϕ = ∑_{Q′∈Q_{2}(Q)} Φ^{(Q′)}.  (26)

and the Bessel inequality

∑_{Q′∈Q_{2}(Q)} E(ϕ^{(Q′)}) ≤ (1 + Cc)E(ϕ)  (27)

• We have

∥ϕ^{(Q′)}ψ∥_{L^{p}(Q\(1+c)Q′)} ≲ c^{−C}R^{−ε}E(ϕ)^{1/2}E(ψ)^{1/2}  (28)

for all Q′ ∈ Q_{2}(Q).

Roughly speaking, ϕ^{(Q′)} is the portion of ϕ which concentrates in Q′. Proof We shall define the functions ϕ^{(Q′)} for each Q′ ∈ Q_{2}(Q) by

ϕ^{(Q′)} = ∑_{T ∈T} \frac{∥ψχ̃_{T} ∥_{L^{2}(Q′)}^{2}}{∥ψχ̃_{T} ∥_{L^{2}(Q)}^{2}} ϕ_{T} .  (29)

Note that (26) follows from (17), and (27) follows from (18).

We now prove (28). Fix Q′. From (10) we have

∥ϕ^{(Q′)}ψ∥_{L^{1}(Q\(1+c)Q′)} ≲ c^{−C}RE(ϕ)^{1/2}E(ψ)^{1/2}

so by interpolation and the hypothesis p > \frac{n+3}{n+1} it suffices to show that

∥ϕ^{(Q′)}ψ∥_{L^{2}(Q\(1+c)Q′)} ≲ c^{−C}R^{−(n−1)/4}E(ϕ)^{1/2}E(ψ)^{1/2}.  (30)

We may assume that c ≫ R^{−1/N}, since the claim follows trivially from (12) and otherwise.

We shall use a (heavily disguised) version of the arguments in Wolff [6]. If q ∈ Q_{J}(Q) intersects Q\(1+c)Q′, then dist(q, Q′) ≳ cR. By squaring (30), we thus reduce to showing that

∑_{q∈Q_{J} (Q):dist(q,Q′)≳cR} ∥ϕ^{(Q′)}ψ∥_{L^{2}(q)}^{2} ≲ c^{−C}r^{−(n−1)}E(ϕ)E(ψ).  (31)

Consider a single summand from (31). From (29) and the triangle inequality we have

∥ϕ^{(Q′)}ψ∥_{L^{2}(q)} ≤ ∑_{T ∈T} \frac{∥ψχ̃_{T} ∥_{L^{2}(Q′)}^{2}}{∥ψχ̃_{T} ∥_{L^{2}(Q)}^{2}} ∥ϕ_{T} ψ∥_{L^{2}(q)}.  (32)

Consider the tubes T which do not intersect CQ. By (19) and (9), their total contribution to (32) is O(R^{C−N}E(ϕ)^{1/2}E(ψ)^{1/2}). As for the tubes T which do intersect CQ, we may apply Lemma 5.2 to obtain

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

Since E(ϕ_{T}) ≲ E(ϕ) by (18) and we are only summing O(R^{C}) tubes, the total contribution of the error term to (32) is O(R^{C−N}E(ϕ)^{1/2}E(ψ)^{1/2}). Combining all these estimates we obtain

∥ϕ^{(Q′)}ψ∥_{L^{2}(q)} ≲ ∥ψ∥_{L^{2}(Cq)} ∑_{T ∈T} \frac{∥ψχ̃_{T} ∥_{L^{2}(Q′)}^{2}}{∥ψχ̃_{T} ∥_{L^{2}(Q)}^{2}} ∥ϕ_{T} ∥_{L^{2}(Cq)} + R^{C−N}E(ϕ)^{1/2}E(ψ)^{1/2}.

Inserting this back into (31), we see that it suffices to show that

∑_{q∈Q_{J} (Q):dist(q,Q′)≳cR} ∥ψ∥_{L^{2}(Cq)}^{2}( ∑_{T ∈T} \frac{∥ψχ̃_{T} ∥_{L^{2}(Q′)}^{2}}{∥ψχ̃_{T} ∥_{L^{2}(Q)}^{2}} ∥ϕ_{T} ∥_{L^{2}(Cq)})^{2} ≲ c^{−C}r^{2}E(ϕ)E(ψ). (33)

Using the trivial estimate

\frac{∥ψχ̃_{T} ∥_{L^{2}(Q′)}^{2}}{∥ψχ̃_{T} ∥_{L^{2}(Q)}^{2}} ≤ \frac{∥ψχ̃_{T} ∥_{L^{2}(Q′)}}{∥ψχ̃_{T} ∥_{L^{2}(Q)}}

followed by Cauchy-Schwarz, we have

( ∑_{T ∈T} \frac{∥ψχ̃_{T} ∥_{L^{2}(Q′)}^{2}}{∥ψχ̃_{T} ∥_{L^{2}(Q)}^{2}} ∥ϕ_{T} ∥_{L^{2}(Cq)})^{2} ≤ ( ∑_{T ∈T} \frac{∥ϕ_{T} ∥_{L^{2}(Cq)}^{2}}{∥ψχ̃_{T} ∥_{L^{2}(Q)}^{2}χ̃_{T} (x_{q}, t_{q})})

( ∑_{T ∈T} ∥ψχ̃_{T} ∥_{L^{2}(Q′)}^{2}χ̃_{T} (x_{q}, t_{q})).

Observe that

∑_{T ∈T} ∥ψχ̃_{T} ∥_{L^{2}(Q′)}^{2}χ̃_{T} (x_{q}, t_{q}) ≲ ∥ψχ∥_{2}^{2}

where

χ = ( ∑_{T ∈T} χ̃_{T} (x_{q}, t_{q})χ̃_{T}^{2} )^{1/2}χ_{Q′}.

Since dist(q, Q′) ≳ cR, we see from elementary geometry that

χ(x, t) ≲ c^{−C}(1 + \frac{dist((x, t), C^{red}(x_{q}, t_{q}))}{r})^{−10n}.

From Lemma 4.1 and the triangle inequality we thus have

∥ψχ∥_{2}^{2} ≲ c^{−C}rE(ψ).

Inserting all these estimates back into (33), we see that it will suffice to show that

∑_{q∈Q_{J} (Q)} ∥ψ∥_{L^{2}(Cq)}^{2} ∑_{T ∈T} \frac{∥ϕ_{T} ∥_{L^{2}(Cq)}^{2}}{∥ψχ̃_{T} ∥_{L^{2}(Q)}^{2}χ̃_{T} (x_{q}, t_{q})} ≲ c^{−C}rE(ϕ).  (34)

We re-arrange the left-hand side as

∑_{T ∈T} ∑_{q} χ̃_{T} (x_{q}, t_{q})^{−3}∥ϕ_{T} ∥_{L^{2}(Cq)}^{2} \frac{∥ψ∥_{L^{2}(Cq)}^{2}χ̃_{T} (x_{q}, t_{q})^{2}}{∥ψχ̃_{T} ∥_{L^{2}(Q)}^{2}}

and estimate this by

∑_{T ∈T} (sup_{q} χ̃_{T} (x_{q}, t_{q})^{−3}∥ϕ_{T} ∥_{L^{2}(Cq)}^{2}) ∑_{q∈Q_{J} (Q)} \frac{∥ψ∥_{L^{2}(Cq)}^{2}χ̃_{T} (x_{q}, t_{q})^{2}}{∥ψχ̃_{T} ∥_{L^{2}(Q)}^{2}}

Since the inner sum is O(1), the desired estimate (34) then follows from (20).

By time reversal, one can formulate an analogue of Proposition 7.1 for blue waves, decomposing a blue wave ψ with appropriate margin requirements as ψ = ∑_{Q′∈Q(Q)} ψ^{(Q′)} and satisfying the analogues of (27) and (28).

8. Proof of Proposition 3.2

Armed with Proposition 7.1 we can now prove Proposition 3.2.

Fix R ≫ 1, 0 < c ≪ 1, and let Q_{0} be a cube of side-length R. Let ϕ, ψ be red and blue waves with normalized energy E(ϕ) = E(ψ) = 1 and margin at least 1/100 − R^{−1/N}. We have to show that

∥ϕψ∥_{L^{p}(Q_{0})} ≤ (1 + Cc)A(R/2) + Cc^{−N}R^{−ε}.  (35)

We will actually show that

∥ϕψ∥_{L^{p}(⋃_{Q′∈Q_{2}(Q))}(1−c)Q′)} ≤ A(R/2) + Cc^{−N}R^{−ε}  (36)

for all cubes Q of side-length 2R. If one raises to the p^{th} power, and then averages over translations of Q by O(R), one obtains (35).

It remains to show (36). Fix Q. By Proposition 7.1 we may decompose ϕ = ∑_{Q′∈Q_{2}(Q)} ϕ^{(Q′)}, where the ϕ^{(Q′)} are red waves with margin at least 1/100 − R^{−1/N} − CR^{−1/2} such that (27) and (28) hold. By the triangle inequality we have

∥ϕψ∥_{L^{p}(⋃_{Q′∈Q_{2}(Q))}(1−c)Q′)} ≤ ∑_{Q′∈Q_{2}(Q)} ∥ϕ^{(Q′)}ψ∥_{L^{p}((1−c)Q′)}

+ ∑_{Q′,Q′′∈Q_{2}(Q):Q′≠Q′′} ∥ϕ^{(Q′)}ψ∥_{L^{p}((1−c)Q′′)}.

By (28), we thus have

∥ϕψ∥_{L^{p}(⋃_{Q′∈Q_{2}(Q))}(1−c)Q′)} ≤ ∑_{Q′∈Q_{2}(Q)} ∥ϕ^{(Q′)}ψ∥_{L^{p}((1−c)Q′)} + Cc^{−N}R^{−ε}. (37)

Now let Φ denote the vector-valued red wave

Φ = (ϕ^{(Q′)})_{Q′∈Q_{2}(Q)},

and apply the time-reversed analogue of Proposition 7.1 with ϕ replaced by ψ and ψ replaced by Φ, we may decompose ψ = ∑_{Q′∈Q_{2}(Q)} ψ^{(Q′)} where the ψ^{(Q′)} are blue waves with margin at least 1/100 − R^{−1/N} − CR^{−1/2} satisfying

∑_{Q′∈Q_{2}(Q)} E(ψ^{(Q′)}) ≤ E(ψ) = 1  (38)

and

∥Φψ^{(Q′′)}∥_{L^{p}(Q\(1+c)Q′′)} ≲ c^{−C}R^{−ε}E(Φ)^{1/2}E(ψ)^{1/2} ≲ c^{−C}R^{−ε}

for all Q′′ ∈ Q_{2}(Q). Since |ϕ^{(Q′)}| ≤ |Φ| for all Q′ ∈ Q_{2}(Q), we may use the triangle inequality as before, and improve (37) to

∥ϕψ∥_{L^{p}(⋃_{Q′∈Q_{2}(Q))}(1−c)Q′)} ≤ ∑_{Q′∈Q_{2}(Q)} ∥ϕ^{(Q′)}ψ^{(Q′)}∥_{L^{p}(Q′)} + Cc^{−N}R^{−ε}.

If R is large enough, the waves ϕ^{(Q′)} and ψ^{(Q′)} have margin at least 1/100 − (R/2)^{−1/N}, and so we can apply (7) to obtain

∥ϕψ∥_{L^{p}(⋃_{Q′∈Q_{2}(Q))}(1−c)Q′)} ≤ A(R/2) ∑_{Q′∈Q_{2}(Q)} E(ϕ^{(Q′)})^{1/2}E(ψ^{(Q′)})^{1/2} + Cc^{−N}R^{−ε}.

The claim (36) then follows from Cauchy-Schwarz, (27), and (38).

References

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

[2] D. Foschi, S. Klainerman, Homogeneous L^{2} bilinear estimates for wave equations, to appear, Amer. Math. J.

[3] S. Klainerman, M. Machedon, Remark on Strichartz-type inequalities. With appendices by Jean Bourgain and Daniel Tataru. Internat. Math. Res. Notices 5 (1996), 201–220.

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

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

[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