Abstract. This paper is a continuation of [35], in which new bilinear estimates for surfaces in R^{3} were proven. We give a concrete improvement to the square function estimate of Mockenhaupt [20]. We apply these estimates to give new progress on several open problems concerning the wave and Schrödinger equation in R^{2+1}, and convolution with curves in R^{3}.

A BILINEAR APPROACH TO CONE MULTIPLIERS II. APPLICATIONS.

TERENCE TAO AND ANA VARGAS

1. Introduction

Let S_{1}, S_{2} two smooth compact hypersurfaces with boundary in R^{3}, with Lebesgue measure dσ_{1} and dσ_{2} respectively. If 0 < q ≤ ∞, we say that the bilinear adjoint restriction estimate R_{S_{1},S_{2}}^{∗}(2 × 2 → q) holds if one has

∥∏_{t=1}^{2}{ \atop f_{t}dσ_{t}}∥_{L^{q}(R^{3})} ≲ ∏_{t=1}^{2}∥f_{t}∥_{2}.

for all test functions f_{1}, f_{2} supported on S_{1}, S_{2} respectively.

The question of finding, for specified S_{1}, S_{2}, the range of q for which R_{S_{1},S_{2}}^{∗}(2 × 2 → q) holds is open. The estimate is trivial for q = ∞, and if S_{1} = S_{2} is a subset of a hyperplane then no estimate is possible for q < ∞. On the other hand, if S_{1} and S_{2} are transverse in the sense that the normals to S_{1} lie in a set which is separated from the normals to S_{2}, then we have the estimate for q = 2 (and hence for q ≥ 2 by interpolation). Indeed, by Plancherel’s theorem R_{S_{1},S_{2}}^{∗}(2 × 2 → 2) is equivalent to

∥f_{1}dσ_{1} ∗ f_{2}dσ_{2}∥_{2} ≲ ∥f_{1}∥_{2}∥f_{2}∥_{2},

and the result follows by interpolating between the trivial estimate

∥f_{1}dσ_{1} ∗ f_{2}dσ_{2}∥_{1} ≲ ∥f_{1}∥_{1}∥f_{2}∥_{1},

and the estimate

∥f_{1}dσ_{1} ∗ f_{2}dσ_{2}∥_{∞} ≲ ∥f_{1}∥_{∞}∥f_{2}∥_{∞}

which follows from transversality. If S_{1}, S_{2} are not transverse but have nonvanishing Gaussian curvature then one can also obtain the estimate R_{S_{1},S_{2}}^{∗}(2×2 → 2) from the Tomas-Stein theorem and Hölder’s inequality.

If one assumes only that S_{1} and S_{2} are transverse, then the above argument is sharp, as can be seen by taking S_{1} and S_{2} to be subsets of transverse hyperplanes. The argument is still sharp for certain curved sets S_{1}, S_{2}; for instance if S_{1} and S_{2} are unit-seperated subsets of the cylinder

{(ξ_{1}, ξ_{2}, ξ_{3}) : |ξ_{1}|^{2} + |ξ_{2}|^{2} = 1},

1991 Mathematics Subject Classification. 42B10, 42B25.

then by taking f_{1}, f_{2} to be bump functions one can make { \atop f_{t}dσ_{t}} comparable to R^{−1/2} on (disjoint) R × R × 1 slabs for some arbitrarily large R, as a simple stationary phase computation shows. By multiplying f_{1} or f_{2} by a suitable phase one can make the two slabs co-incide, and so the estimate R_{S_{1},S_{2}}^{∗}(2 × 2 → q) is impossible for q < 2.

However, it was observed by Bourgain [3] that the estimate R_{S_{1},S_{2}}^{∗}(2 × 2 → q) can hold for some q < 2 if the surfaces S_{1} and S_{2} have slightly more curvature than the cylinder example; in particular, if S_{1} and S_{2} are separated subsets of a small portion of the light cone then R_{S_{1},S_{2}}^{∗}(2 × 2 → 2 − τ) holds for some small τ > 0. Indeed, examples suggest that one should have R_{S_{1},S_{2}}^{∗}(2 × 2 → q) for all q ≥ 5/3 in this case (see [34, 35]; this conjecture is due to Machedon and Klainerman).

Estimates of this form were studied in [35], the prequel to this paper. We state the two relevant results of that paper here. The first result pertains to pairs of surfaces which resemble projectively separated subsets of the light cone, or of transverse subsets of non-parallel cylinders.

Definition 1.1. Suppose that S_{1} and S_{2} are compact surfaces with boundary in R^{3}. If ξ ∈ S_{t} , t = 1, 2, we use n_{t}(ξ) ∈ S^{2}/± to denote the unit normal to S_{t} at ξ. We say that the pair S_{1} and S_{2} are of disjoint conic type if the following statements hold:

• (Transversality) For all ξ_{t} ∈ S_{t}, t = 1, 2 we have n_{t}(ξ_{t}) ∈ N_{t}, where N_{1} and N_{2} are small disjoint caps in S^{2}/± which are seperated by a distance comparable to 1.

• (Null direction) The map dn_{t} : T_{ξ_{t}}S_{t} → T_{ξ_{t}}S_{t} has eigenvalue 0 with multiplicity one in the direction w_{t}(ξ_{t}) ∈ S^{2}/±. We also assume that the remaining eigenvalue has magnitude ∼ 1.

• (Tranversality of null directions) For all ξ_{t} ∈ S_{t}, t = 1, 2 we have w_{t}(ξ_{t}) ∈ W_{t}, where W_{1} and W_{2} are small disjoint caps in S^{2}/± which are seperated by a distance comparable to 1. Furthermore, the maximal angular seperation between W_{1} and N_{2}, or W_{2} and N_{1}, is strictly less than π/2.

Theorem 1.2. [35] If S_{1} and S_{2} are of disjoint conic type, then R_{S_{1},S_{2}}^{∗}(2 × 2 → q) for all q > 2 − \frac{8}{121}.

If one assumes more curvature on S_{1}, S_{2} (for instance of S_{1}, S_{2} are separated subsets of the sphere or paraboloid) then better results are known. More precisely:

Theorem 1.3. [35] Let S be a graph of an elli ptic phase function (in the sense of [34] or [18, 19]), and let S_{1}, S_{2} be unit-seperated subsets of S. Then

R_{S_{1},S_{2}}^{∗}(2 × 2 → q) (1)

for all q > 2 − \frac{2}{17}.

The purpose of this paper is to give applications of these results to outstanding conjectures in harmonic analysis and wave equations. We will use the philosophy of [34] to allow us to pass efficiently from the bilinear estimates stated above to the linear estimates which appear in applications.

We have the following results. Firstly, in Section 2 we show pointwise convergence of solutions of the free Schrödinger equation to their initial data for low regularities. More precisely, we have pointwise convergence for all data in H^{s}, s > 15/32. This improves on previous work by [2], [18].

In Section 3 we show that Theorem 1.2 implies some new null form estimates for the L_{x,t}^{q} norms of products of solutions to the wave equation in R^{2+1}. This gives some progress on a recent conjecture of Foschiand Klainerman [12]. As an application of these null form estimates we improve upon the local smoothing estimate proven by Schlag and Sogge [28].

In [3], the estimates of the type considered in Theorem 1.2 were used to obtain an improvement to Mockenhaupt’s square function estimate [20]. By combining this argument with the bilinear philosophy in [34] we are able to obtain a quantitative improvement, reducing the N^{1/4} factor in Mockenhaupt’s estimate to 1/4 − 1/238. We also prove the corresponding amount of progress on the Bochner-Riesz multiplier problem, Sogge’s local smoothing conjecture, and the L^{p} smoothing conjecture for the convolution operator with the helix.

Figure 1. Logical layout of this paper.

Although there are many common themes in these results, particularly the bilinear philosophy and the use of the estimates R^{∗}(2×2 → q), some of the sections can be read independently, although all sections utilize some basic lemmas in harmonic analysis which we place in an Appendix. We display the logical dependencies of the results of this paper in Figure 1.

2. Application to the Schrodinger maximal function

Consider the solution e^{itΔ}f of the free Schrödinger equation with initial datum f. We are interested in proving a.e. pointwise convergence e^{itΔ}f → f(x) when t → 0, under certain assumptions on the (weak) regularity of f. We follow the line started by Carleson [5] who proved convergence for f in the Sobolev space H^{1/4}(R). This is a one dimensional result. For n ≥ 3, the best result known was proven independently by Sjölin [29] and Vega [36], who showed convergence for f ∈ H^{s}(R^{n}) whenever s > 1/2. For functions f in R^{2}, due to a better knoweledge of the restriction properties of the paraboloid in R^{3}, convergence has been proven under weaker assumptions. Bourgain [2], [4] proved that there is s < 1/2 for which the result holds. This theorem was refined by Moyua, Vargas and Vega [18], and the argument there could be improved using the results of Tao, Vargas and Vega in [34] and those in [35]. However we provide a simpler argument in this section based on the bilinear philosophy which gives slightly better exponents.

Theorem 2.1. Suppose that q is such that R_{S_{1},S_{2}}^{∗}(2 × 2 → q) whenever S_{1}, S_{2} are unit-separated subsets of the standard paraboloid in R^{3}. Then, for all s > 1 − 1/q and all f ∈ H^{s}(R^{2}),

lim_{t→0} e^{itΔ}f(x) = f(x) a.e. x ∈ R^{2}.

Combining this with (1), we thus obtain pointwise convergence of solutions to the free Schrödinger equation to their initial data, if the data is in H^{s} for s > 15/32 = .46875. This improves upon the results in [18], which required

s > \frac{164 + \sqrt{2}}{339} = .4879475 …

As usual, the convergence will be a consequence of an estimate for a maximal function.

Theorem 2.2. Under the same assumptions^{1},

∥ sup_{t} | ∫ f̂(ξ)e^{2πi(<x,ξ>+t|ξ|^{2})} dξ|∥_{L^{2q}(R^{2})} ≲ ∥f∥_{H^{s}(R^{2})}.

Note that this estimate is somewhat stronger than what is needed for pointwise convergence. By the usual arguments of Littlewood-Paley decomposition and Plancherel’s theorem, it suffices to prove the following proposition.

Proposition 2.3. Under the same assumptions,

∥ sup_{t} | ∫ f̂(ξ)e^{2πi(<x,ξ>+t|ξ|^{2})} dξ|∥_{2q} ≤ N^{1−1/q}∥f∥_{2}. (2)

for all N > 0 and all f with supp f ⊂ B(0, N)\B(0, N/2), Proof

1All estimates in this paper will be a priori.

We rescale to make the support of our function independent of N:

∥ sup_{t} | ∫ f̂(ξ)e^{2πi(<x,ξ>+t|ξ|^{2})} dξ|∥_{2q}

= N^{2−1/q}∥ sup_{t} | ∫_{B(0,1)} f̂(Nξ)e^{2πi(<x,ξ>+t|ξ|^{2})} dξ|∥_{2q}.

By Plancherel’s theorem, it thus suffices to show that we havE

∥ sup_{t} |ĝ dσ|∥_{2q} ≤ C∥g∥_{2}

for all functions g supported in a compact subset S of the paraboloid. We square this as

∥ sup_{t} |ĝ dσĝ dσ|∥_{q} ≤ C∥g∥_{2}∥g∥_{2} (3)

For each j ≥ 0 we break up S into dyadic “squares” τ_{k}^{j}, and write τ_{k}^{j} ∼ τ_{k′}^{j} if τ_{k}^{j},τ_{k′}^{j} are not adjacent but have adjacent parents. For each j ≥ 0 write g = ∑ g_{k}^{j} where g_{k}^{j}(ξ) = gχ_{τ_{k}^{j}}(ξ). Then,

ĝ dσĝ dσ = ∑_{j} ∑_{k,k′:τ_{k}^{j}∼τ_{k′}^{j}} { \atop g_{k}^{j} dσ}{ \atop g_{k′}^{j} dσ}.

From the triangle inequality, the left hand side of (3) is thus majorized by

∑_{j} ∑_{k,k′:τ_{k}^{j}∼τ_{k′}^{j}} ∥ sup_{t} |{ \atop g_{k}^{j} dσ}(·, t){ \atop g_{k′}^{j} dσ}(·, t)|∥_{q}. (4)

We fix j. From the geometry of τ_{k}^{j} we see then that the temporal Fourier transform

of { \atop g_{k}^{j} dσ}(x, ·) is supported in an interval of length 2^{−j}. The same statement holds

for { \atop g_{k}^{j} dσ}(x, ·){ \atop g_{k′}^{j} dσ}(x, ·). Hence, we can use Lemma 7.3 to bound

sup_{t} |{ \atop g_{k}^{j} dσ}(x, t){ \atop g_{k′}^{j} dσ}(x, t)| ≲ (2^{−j})^{1/q}∥{ \atop g_{k}^{j} dσ}(x, ·){ \atop g_{k′}^{j} dσ}(x, ·)∥_{L^{q}(dt)}.

Therefore, we estimate (4) by

(∑_{j} ∑_{k,k′:τ_{k}^{j}∼τ_{k′}^{j}} 2^{−j/q}∥{ \atop g_{k}^{j} dσ}{ \atop g_{k′}^{j} dσ}∥_{L^{q}(dtdx)})^{1/2}.

By R_{S_{1},S_{2}}^{∗}(2 × 2 → q), an affine transformation, and parabolic rescaling (cf. [34], Proposition 2.6) the last expression is not bigger than

C(∑_{j} ∑_{k,k′:τ_{k}^{j}∼τ_{k′}^{j}} 2^{−j/q}2^{−2j}2^{4j/q}∥g_{k}^{j}∥_{2}∥g_{k′}^{j}∥_{2})^{1/2}.

Finally, we sum on k and j and take square root. Note that 5/3 < q implies 3/q − 2 < 0.

The best estimate of the form R^{∗}(2 × 2 → q) that one can hope for is q = 5/3. Thus this argument can at best give convergence in H^{s} for s > 2/5. However, the strongest counterexample known to date (see [10]) only gives s ≥ 1/4 as a necessary condition, which suggests that the above argument is not sharp. The use of Lemma 7.3 in the above seems particularly weak.

We remark that the above argument extends without difficulty to the more general dispersive equations studied in [19].

3. Null form estimates for L^{p}, p < 2

In this section we consider estimates on one-sided solutions ϕ^{±} = e^{±i\sqrt{−Δ}t}f to the free wave equation in R^{2+1}, in terms of Sobolev norms of the initial data f. (Estimates for general solutions to the free wave equation can be obtained by the usual decomposition ϕ = ϕ^{+} + ϕ^{−} into one-sided solutions).

Strichartz’ estimate [31] in 2 + 1 dimensions states that

∥ϕ^{±}∥_{L_{x,t}^{6}} ≲ ∥f∥Ḣ^{1/2}.

We may bilinearize this in the usual manner as

∥ϕ^{±}ψ^{±}∥_{L_{x,t}^{3}} ≲ ∥f∥Ḣ^{1/2}∥g∥Ḣ^{1/2},

where ϕ^{±} = e^{±i\sqrt{−Δ}t}f, ψ^{±} = e^{±i\sqrt{−Δ}t}g, and the two signs ± need not agree.

Without any multiplier weights on the quantity ϕ^{±}ψ^{±}, this is the best estimate available in pure Lebesgue norms L_{x,t}^{q}. (If one allows mixed norms such as L_{t}^{q}L_{x}^{r} then other estimates are possible; see e.g. [13], [12] and the references therein.)

However, if one allows weights on the right-hand side then there are many further estimates available. More precisely, if we let D_{0}, D_{+} and D_{−} be the Fourier multipliers corresponding to the functions^{2} |ξ|, |ξ|+|τ|, ||ξ| − |τ|| respectively, then we may consider estimates of the form

∥D_{0}^{β_{0}}D_{+}^{β_{+}}D_{−}^{β_{−}}(ϕ^{±}ψ^{±})∥_{L_{x,t}^{q}} ≲ ∥f∥Ḣ^{α_{1}} ∥g∥Ḣ^{α_{2}} (5)

for various real numbers q, β_{0}, β_{+}, β_{−}, α_{1}, α_{2}. In the remainder of the section we will always assume that α_{1} ≥ α_{2}, and that the sign on ϕ is positive; the remaining cases will follow from symmetry. We will distinguish two cases (++) and (+−) of (5), depending on the sign on ψ^{±}. Henceforth we fix the sign ± to be the same at every occurence.

When q = 2 such estimates are important in the study of semi-linear wave equations with quadratic non-linearities; such equations arise naturally in the study of wave maps. (See [refs?]) Estimates of this type with q ≠ 2, β_{+} = β_{−} = 0 and β negative have also been recently utilized in [17]. However for q < 2 no estimates were known in R^{2+1}.

The study of these estimates and generalizations was systematically taken up in [12] (see also [14, 15, 16], and earlier work by [1]), and the following necessary conditions were found:

2In this section and the next, we use the variables (ξ, τ) = (ξ_{1}, ξ_{2}, τ) to parameterize frequency space R^{2+1}.

Proposition 3.1. [12] With the above notation and assumptions, the inequalities

(6)  2 − \frac{3}{q} + β_{0} + β_{+} + β_{−} = α_{1} + α_{2}
q ≥ \frac{5}{3}
(7)  β_{−} ≥ \frac{3}{2q} − \frac{1}{2}
(8)  α_{1} ≤ β_{−} + 2 − \frac{3}{q}
α_{1} ≤ β_{−} + \frac{5}{4} − \frac{3}{2q}
α_{1} ≤ β_{−} + 1

are necessary in order for (5) to hold. In the (++) case we have the additional necessary conditions

β_{0} ≥ \frac{3}{q} − 2
(9)  β_{0} ≥ \frac{5}{q} − 3

and in the (+−) case we have the additional necessary condition

α_{1} + α_{2} ≥ \frac{5}{q} − 2. (10)

The condition (6) comes from scaling considerations, while the remaining conditions come from considering examples of f_{1}, f_{2} with various frequency supports. Condition (7) is also related to Lorentz invariance considerations (see [12]). It was conjectured in [12] that (5) is true whenever all of the above necessary conditions hold, unless at least two of the inequalities hold with strict equality. This conjecture has been verified for q = 2 but is largely open otherwise.

We will concern ourselves with the case \frac{5}{3} ≤ q < 2, for which no estimates of the type (5) were previously known to exist in 2 + 1 dimensions. In this case many of the above conditions are redundant, and one only requires (6), (7), (8), (9) and (10).

Our results are as follows.

Theorem 3.2. Let \frac{5}{3} ≤ q ≤ 2 be suchthat R_{S_{1},S_{2}}^{∗}(2 × 2 → q) holds for all pairs S_{1}, S_{2} of surfaces of disjoint conic type. Suppose that (6) holds, (7) is satisfied with strict inequality, and

α_{1} < β_{−} + 2 − \frac{3}{q} − 4(\frac{1}{q} − \frac{1}{2}). (11)

If

α_{1} + α_{2} > \frac{5}{q} − 2 + 4(\frac{1}{q} − \frac{1}{2}) (12)

then the (+−) form of (5) holds. If

β_{0} > \frac{5}{q} − 3 + 2(\frac{1}{q} − \frac{1}{2}) (13)

then the (++) form of (5) holds.

In particular, by Theorem 1.2 the above result holds for all 2 − \frac{8}{121} < q ≤ 2. The factors of (\frac{1}{q} − \frac{1}{2}) represent some inefficiencies in our arguments; however in the endpoint case q = \frac{5}{3} we have been able to replace (11), (13) by their sharp counterparts (8), (9) (except for endpoints).

Analogues of the above theorem also hold for the null forms

Q_{0}(ϕ^{+}, ψ^{±}) = ∂_{t}ϕ^{+}∂_{t}ψ^{±} + ∇_{x}ϕ^{+} · ∇_{x}ψ^{±}
Q_{ij}(ϕ^{+}, ψ^{±}) = ∂_{x_{i}}ϕ^{+}∂_{x_{j}}ψ^{±} − ∂_{x_{j}}ϕ^{+}∂_{x_{i}}ψ^{±}
Q_{0j}(ϕ^{+}, ψ^{±}) = ∂_{t}ϕ^{+}∂_{x_{j}}ψ^{±} − ∂_{x_{j}}ϕ^{+}∂_{t}ψ^{±};

From an analysis of the multiplier (see e.g. [12]), one expects these null-forms to behave like

Q_{0}(ϕ^{+}, ψ^{−}) ∼ D_{+}D_{−}(ϕ^{+}ψ^{−})
Q_{0j}(ϕ^{+}, ψ^{−}) ∼ D_{+}^{1/2}D_{−}^{1/2}(D_{0}^{1/2}ϕ^{−}D_{0}^{1/2}ψ^{−})
Q_{ij}(ϕ^{+}, ψ^{−}) ∼ D_{+}^{1/2}D_{−}^{1/2}(D_{0}^{1/2}ϕ^{+}D_{0}^{1/2}ψ^{−})

in the (+−) case and

Q_{0}(ϕ^{+}, ψ^{+}) ∼ D_{+}D_{−}(ϕ^{+}ψ^{+})
Q_{0j}(ϕ^{+}, ψ^{+}) ∼ D_{+}^{1/2}D_{−}^{1/2}(D_{0}^{1/2}ϕ^{+}D_{0}^{1/2}ψ^{+})
Q_{ij}(ϕ^{+}, ψ^{+}) ∼ D_{0}D_{+}^{−1/2}D_{−}^{1/2}(D_{0}^{1/2}ϕ^{+}D_{0}^{1/2}ψ^{+})

in the (++) case. A modification of the arguments below show that our results extend to these null forms as if the above heuristics were exact.

3.3. Preliminaries: Some Minkowski geometry. In this section we briefly state some elementary facts of Minkowski geometry that we shall need, together with some notation.

If Q is a subset of R^{2}, we define the lifts S^{+}(Q), S^{−}(Q) of Q to the upper and lower light-cones respectively by

S^{±}(Q) = {(ξ, ±|ξ|) : ξ ∈ Q}.

We give these light cones two measures: the induced measure dξ which is the pushforward of Lebesgue measure on R^{2} under S^{±}, and the invariant measure \frac{dξ}{|ξ|}, which is left unchanged by the action of the unimodular Lorentz group SO(2, 1). Note that dξ is also comparable to the measure induced from the Euclidean metric on R^{2+1}.

The group generated by the Lorentz group, the isotropic dilations, is the group of orientation-preserving conformal linear transformations Conf^{+}(R^{2+1}) of Minkowski space. This group has the following 2-transitivity property: if A, A′ ∈ S^{+}(R^{2}), B, B′ ∈ S^{±}(R^{2}) are such that A, B are linearly independent and A′, B′ are linearly independent, then there exists a conformal linear transformation L such that L(A) = A′ and L(B) = B′. This fact will often allow us to place the Fourier transforms of f and g in specified locations in frequency space.

In order to handle the D_{−} weight, we will need the identity

|ξ + η| − ||ξ| − |η|| = \frac{|ξ + η|^{2} − (|ξ| − |η|)^{2}}{|ξ + η| + ||ξ| − |η||} ∼ \frac{|ξ||η|∠(ξ, −η)^{2}}{|ξ + η|} (14)

for the (+−) case,and

|ξ| + |η| − |ξ + η| = \frac{(|ξ| + |η|)^{2} − |ξ + η|^{2}}{|ξ| + |η| + |ξ + η|} ∼ \frac{|ξ||η|∠(ξ, η)^{2}}{|ξ| + |η|} (15)

for the (++) case; here ∠(ξ, η) denotes the angle subtended by ξ and η with respect to the origin.

At certain stages in the argument it will be convenient to view the light cone in null co-ordinates (a_{1}, a_{2}, a_{3}) defined by

(a_{1}, a_{2}, a_{3}) = (\frac{τ + ξ_{1}}{2} , ξ_{2}, \frac{τ − ξ_{1}}{2} ).

The double light cone |τ| = |ξ| in these co-ordinates becomes

4a_{1}a_{3} = a_{2}^{2} (16)

with the upper and lower light cone corresponding to a_{1}, a_{3} ≥ 0 and a_{1}, a_{3} ≤ 0 respectively. Note that for any constants A, B, C > 0 such that AC = B^{2}, the map

(a_{1}, a_{2}, a_{3}) → (Aa_{1}, Ba_{2}, Ca_{3})

preserves the light cone and is hence a conformal linear transformation. If two sets Q^{1}, Q^{2} are very close in one of the co-ordinates (especially the a_{2} co-ordinate), we will frequently apply the above type of transformation to make the sets Q^{1}, Q^{2} unit-separated in this co-ordinate.

3.4. Preliminaries: The quantity C^{±}. In this section we recast the desired estimate (5) in terms of a quantity C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, D), which we will shortly define. We then collect some basic properties of C^{±} to use in the proof of Theorem 3.2. Henceforth q < 2 will be fixed to be such that R_{S_{1},S_{2}}^{∗}(2 × 2 → q) for all surfaces S_{1}, S_{2} of disjoint conic type.

Definition 3.5. If α_{1}, α_{2} ∈ R, Q^{1}, Q^{2} are subsets of R^{2} and D is a Fourier multiplier on R^{2+1} given by m(ξ, τ), we define C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, D) to be the best constant C (possibly infinite) such that

∥D(ϕ^{+}ψ^{±})∥_{L_{x,t}^{q}} ≤ C∥f∥Ḣ^{α_{1}} ∥g∥Ḣ^{α_{2}}

for all f, g whose Fourier transforms are supported in Q^{1}, Q^{2} respectively.

Thus (5) can be written as

C_{α_{1},α_{2}}^{±}(R^{2}, R^{2}, D_{0}^{β_{0}}D_{+}^{β_{+}}D_{−}^{β_{−}}) ≲ 1. (17)

On the other hand, from Plancherel’s theorem we see that C_{0,0}^{±}(Q^{1}, Q^{2}, 1) is essentially the best constant C that appears in the estimate

∥{ \atop fdσ_{1}}{ \atop gdσ_{2}}∥_{q} ≤ C∥f∥_{2}∥g∥_{2} (18)

for all f and g supported on the surfaces S^{+}(Q^{1}), S^{±}(Q^{2}) respectively, where dσ_{1}, dσ_{2} are induced measure dξ as defined in Section 3.3. From the hypothesis R^{∗}(2 × 2 → q) we thus have

Proposition 3.6. If Q^{1}, Q^{2} are such that the surfaces S^{+}(Q^{1}), S^{±}(Q^{2}) are of disjoint conic type, then

C_{0,0}^{±}(Q^{1}, Q^{2}, 1) ≲ 1

where the constant depends only on the constants in the disjoint conic type condition.

Our strategy shall be to use the familiar techniques of dyadic decomposition and rescaling to reduce (5) to something treatable by Proposition 3.6. Due to the geometry of the cone some of our decompositions and rescalings will be nonisotropic, and some of our techniques will not be totally efficient.

We now formalize abstractly the decomposition and rescaling properties that we shall need. We first observe the trivial relationship

C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, D_{1} + D_{2}) ≤ C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, D_{1}) + C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, D_{2}). (19)

Also, if P is a multiplier which is bounded on L^{q} (in particular, if P is in one of the standard multiplier classes) then we have

C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, P D) ≤ C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, D). (20)

We also observe a scaling property, which can be deduced from dimensional analysis.

Lemma 3.7. If j is an integer, and the notation is as in Definition 3.5, then

C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, D) = 2^{( \frac{3}{q} +α_{1}+α_{2}−2)j}C_{α_{1},α_{2}}^{±}(2^{j}Q^{1}, 2^{j}Q^{2}, D_{2^{j}}),

where D_{j} is given by the multiplier

m_{2^{j}}(ξ, τ) = m(2^{−j}ξ, 2^{−j}τ).

In the case when α_{1} = α_{2} = 1/2 there is an additional scaling because the Ḣ^{1/2} norm is invariant under Lorentz transformations of ϕ^{+}, ψ^{±}. Indeed, from the identity

ϕ̂^{+}(ξ, τ) = Cf̂(ξ)δ(τ − |ξ|) dξ,

where we view the spacetime Fourier transform of ϕ^{+} as a measure, we can write the Ḣ^{1/2} norm of f in an invariant way as

∥f∥Ḣ^{1/2} = C∥\frac{ϕ̂^{+}}{dμ} ∥_{L^{2}(dμ)}

where dμ is the invariant measure dξ/|ξ| on the upper light cone and \frac{_{ϕ^{+}}}{dμ} denotes the Radon-Nikodym derivative. A similar argument works for ψ^{±}.

Since Lorentz transformations are unimodular, they preserve L^{q} and we obtain the following convenient fact.

Lemma 3.8. If L ∈ SO(2, 1) is a Lorentz transformation, Q^{1}, Q^{2} are subsets of R^{2} and D is a Fourier multiplier given by m(ξ, τ), then

C_{1/2,1/2}^{±}(Q^{1}, Q^{2}, D) = C_{1/2,1/2}^{±}(Q_{L}^{1}, Q_{L}^{2}, D_{L}),

where Q_{L}^{1}, Q_{L}^{2} are given by

S^{+}(Q_{L}^{1}) = L(S^{+}(Q^{1})), S^{±}(Q_{L}^{2}) = L(S^{±}(Q^{2})),

and D_{L} is given by the multiplier m ∘ L^{−1}.

Combining these two lemmas we have

Corollary 3.9. If L is an element of Conf^{+}(R^{2+1}), Q^{1}, Q^{2} are subsets of R^{2} and D is a Fourier multiplier given by m(ξ, τ), then

C_{1/2,1/2}^{±}(Q^{1}, Q^{2}, D) = | det L|^{\frac{1}{q} − \frac{1}{3}} C_{1/2,1/2}^{±}(Q_{L}^{1}, Q_{L}^{2}, D_{L}),

where Q_{L}^{1}, Q_{L}^{2}, D_{L} are as in Lemma 3.8.

We now discuss the problem of controlling C^{±}(Q^{1}, Q^{2}, D) by a suitable partition of Q^{1}, Q^{2}. We begin with a variant of Schur’s test.

Lemma 3.10. Suppose the notation is as in Definition 3.5, and suppose that we are given partitions Q^{1} = ⋃_{k} Q_{k}^{1}, Q^{2} = ⋃_{l} Q_{l}^{2} into essentially disjoint sets. Suppose that A > 0 is such that

sup_{k} ∑_{l} C_{α_{1},α_{2}}^{±}(Q_{k}^{1}, Q_{l}^{2}, D) < A

and

sup_{l} ∑_{k} C_{α_{1},α_{2}}^{±}(Q_{k}^{1}, Q_{l}^{2}, D) < A.

Then

C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, D) ≲ A.

Proof Fix f, g with Fourier transforms supported on Q^{1}, Q^{2} respectively, and decompose f = ∑_{k} f_{k}, g = ∑_{l} g_{l}, ϕ^{+} = ∑_{k} ϕ_{k}^{+}, ψ^{+} = ∑_{l} ψ_{l}^{+} accordingly. We need to estimate

∥ ∑_{k} ∑_{l} D(ϕ_{k}^{+}ψ_{l}^{+})∥_{q}. (21)

By the triangle inequality and Definition 3.5, this is majorized by

∑_{k,l} C_{α_{1},α_{2}}^{±}(Q_{k}^{1}, Q_{l}^{2}, D)∥f_{k}∥Ḣ^{α_{1}} ∥g_{l}∥Ḣ^{α_{2}}

But by the hypothesis on k, l, and Schur’s test, we may majorize this by

A(∑_{k} ∥f_{k}∥2 ; Ḣ^{α_{1}} )^{1/2}(∑_{l} ∥g_{l}∥2 ; Ḣ^{α_{2}} )^{1/2}

and the claim follows from Definition 3.5 and Plancherel’s theorem.

With the notation as above, we observe that the space-time Fourier transform of D(ϕ^{+}ψ^{±}) is supported on the set

supp(m) ∩ (S^{+}(Q^{1}) + S^{±}(Q^{2})) = supp(m) ∩ {(ξ + η, |ξ| ± |η|) : ξ ∈ Q^{1}, η ∈ Q^{2}}. (22)

Thus, if this set is empty, then C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, D) = 0. This gives an immediate corollary of Lemma 3.10:

Corollary 3.11. Suppose the notation is as in Lemma 3.10. Write Q_{k}^{1} ∼ Q_{l}^{2} if there exists ξ ∈ Q_{k}^{1}, η ∈ Q_{l}^{2} such that

m(ξ + η, |ξ| ± |η|) ≠ 0.

Suppose further that for each k (resp. l) there are at most O(1) l (resp. k) such that Q_{k}^{1} ∼ Q_{l}^{2}. Then

C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, D) ≲ sup_{k,l:Q_{k}^{1}∼Q_{l}^{2}} C_{α_{1},α_{2}}^{±}(Q_{k}^{1}, Q_{l}^{2}, D).

Corollary 3.11 is quite efficient, but it only works when the relation Q_{k}^{1} ∼ Q_{l}^{2} is only sparsely satisfied. In the general case we have Lemma 3.10, but this gives poor estimates. We can improve on this Lemma by exploiting quasi-orthogonality, although we have not been able to obtain the optimal exponents in this manner. More precisely, we have

Lemma 3.12. Suppose the notation is as in Lemma 3.10. Suppose also that for distinct (k, l) the sets

S^{+}(Q_{k}^{1}) + S^{±}(Q_{l}^{2})

lie in essentially disjoint rectangles in R^{2+1}. Then we have

C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, D) ≲ (KL)^{\frac{1}{q} − \frac{1}{2}} sup_{k,l} C_{α_{1},α_{2}}^{±}(Q_{k}^{1}, Q_{l}^{2}, D)

where K, L are the cardinalities of the index sets of k, l respectively.

Proof We repeat the proof of Lemma 3.10. This time, however, we exploit the hypothesis, which implies by (22) that the functions D(ϕ_{k}^{+}ψ_{l}^{+}) have Fourier transforms supported in essentially disjoint rectangles. Thus we use Lemma 7.1 instead of the triangle inequality, to majorize (21) by

(∑_{k} ∑_{l} ∥D(ϕ_{k}^{+}ψ_{l}^{+})∥_{q}^{q})^{1/q}.

By Hölder’s inequality this is majorized by

(KL)^{\frac{1}{q} − \frac{1}{2}} (∑_{k} ∑_{l} ∥D(ϕ_{k}^{+}ψ_{l}^{+})∥_{q}^{2})^{1/2}.

By Definition 3.5 this is majorized by

(KL)^{\frac{1}{2} − \frac{1}{q}} sup_{k,l} C_{α_{1},α_{2}}^{±}(Q_{k}^{1}, Q_{l}^{2}, D)(∑_{k} ∑_{l} ∥f_{k}∥2 ; Ḣ^{α_{1}} ∥g_{l}∥2 ; Ḣ^{α_{2}} )^{1/2}.

The claim then follows from Definition 3.5 and Plancherel’s theorem.

To close this section we give an easy consequence of Plancherel’s theorem.

Lemma 3.13. Let the notation be as in Definition 3.5. If Q^{1}, Q^{2} are supported on the annuli {ξ : |ξ| ∼ 2^{j}}, {η : |η| ∼ 2^{k}}, then

C_{α_{1},α_{2}}^{±}(Q^{1}, Q^{2}, D) ∼ 2^{−α_{1}j}2^{−α_{2}k}C_{0,0}^{±}(Q^{1}, Q^{2}, D).

3.14. Proof of Theorem 3.2. The first part of the proof consists of repeated applications of the above decomposition and rescaling lemmas until all the multipliers are essentially constant, and we are in the one of three model cases for some j ≥ 0:

Figure 2. Case I_{j}, viewed from the ξ_{2} and τ axes. The supports of ϕ̂ and ψ̂ are on almost diametrically opposite sides of the same light cone, so that ϕ̂ψ is close to the τ axis.

• Case I_{j}: We are in the (++) case, and f̂, ĝ are supported on the sets

Q_{I_{j}}^{1} = {ξ_{1} = 1 + O(2^{−j}), ξ_{2} ∼ 2^{−j}}, Q_{I_{j}}^{2} = {η_{1} = −1 + O(2^{−j}), η_{2} ∼ 2^{−j}} (23)

respectively. In this case D_{0}, D_{+}, D_{−} are comparable to 2^{−j}, 1, and 1 respectively. This case is good when β_{0} is sufficiently large.

• Case II_{j}: We are in the (+−) case, and f̂, ĝ are supported on the sets

Q_{II_{j}}^{1} = {ξ : ξ_{1} = 1 + O(2^{−j}), ξ_{2} ∼ 2^{−j}}, Q_{II_{j}}^{2} = {η : η_{1} = −1 + O(2^{−j}), η_{2} ∼ 2^{−j}} (24)

respectively. In this case D_{0}, D_{+}, D_{−} are comparable to 2^{−j}, 2^{−j}, and 2^{−2j} respectively. This case is good when α_{1} − β_{−} is sufficiently small.

• Case III_{j}: We are in either the (++) or (+−) case, and and f̂, ĝ are supported on the sets

Q_{III_{j}}^{1} = {ξ : ξ_{1} ∼ 1, |ξ_{2}| ≪ 1}, Q_{III_{j}}^{2} = {η : η_{1} ∼ ∓2^{j}, |η_{2}| ≪ 2^{j}} (25)

respectively. In this case D_{0}, D_{+}, D_{−} are comparable to 2^{j}, 2^{j}, and 1 respectively. This case is good when α_{1} + α_{2} is sufficiently large.

The above estimates on D_{−} may be verified by (14), (15). We illustrate the frequency supports of ϕ, ψ, and ϕψ schematically in Figures 2, 3, 4. Note that the perspectives are different in each case.

Figure 3. Case II_{j}, viewed from the ξ_{1} and τ axes. The supports of ϕ̂ and ψ̂ are on almost diametrically opposite sides of opposing light cones, so that ϕ̂ψ is close to the origin.

Our techniques require breaking the functions f_{1}, f_{2} into many dyadic pieces, and summing them in fairly crude ways; this causes a logarithmic loss in many of our estimates. However, since we are assuming that the inequalities (7), (11), (12), (13) are satisfied with strict inequality, this loss is of no importance. It is quite likely that a more careful treatment would allow one to recover many of the endpoints. We remark that these reductions are valid for all q and all dimensions n (with (7) generalized to β_{−} ≥ \frac{n+1}{2q} − \frac{n−1}{2}).

We then treat each of the model cases using further decompositions, Lemma 3.12, rescaling, and the hypothesis R^{∗}(2 × 2 → q). Our techniques here are not optimal. One possibility for improvement would be to use Littlewood-Paley theory for equally spaced projections in frequency space (cf. Lemma 7.2) and develop vector-valued analogues of the estimate R^{∗}(2 × 2 → q).

We begin the reduction to these cases. By Definition 3.5 it suffices to prove the estimate (17). We partition R^{2} into the dyadic annuli A_{k} = {ξ : |ξ| ∼ 2^{k}} for Figure 4. Case III_{j} in the (++) case, viewed in cross-section with the ξ_{1}, τ plane and from the τ axis. The supports of ϕ̂ and ψ̂ are at widely separated angles and scales.

integer k. By Lemma 3.10, (17) will follow from the estimate

C_{α_{1},α_{2}}^{±}(A_{m}, A_{k}, D_{0}^{β_{0}}D_{+}^{β_{+}}D_{−}^{β_{−}}) ≲ 2^{−ε|m−k|}

uniformly in m, k for some ε > 0. From the assumption α_{1} > α_{2}, symmetry, and Lemma 3.13 it suffices to verify the case when k ≥ m. Accordingly we write k = m + j for some j ≥ 0. From Lemma 3.7 and the scaling condition (6) we may take m to equal 0, so we need only prove

C_{α_{1},α_{2}}^{±}(A_{0}, A_{j}, D_{0}^{β_{0}}D_{+}^{β_{+}}D_{−}^{β_{−}}) ≲ 2^{−εj}.

We now split the multiplier D_{0}^{β_{0}}D_{+}^{β_{+}}D_{−}^{β_{−}} into several pieces. In the (++) case we decompose the identity as I = P_{+} + (I − P_{+}), where P_{+}, (I − P_{+}) are given by multipliers supported on |ξ| ≪ |τ|, |ξ| ≳ |τ| respectively. In the (+−) case we decompose the identity as I = P_{−} + (I − P_{−}), where P_{−}, (I − P_{−}) are given by multipliers supported on |ξ| + |τ| ≪ 2^{j}, |ξ| + |τ| ∼ 2^{j} respectively.

By (19), we reduce to showing that

C_{α_{1},α_{2}}^{+}(A_{0}, A_{j}, P_{+}D_{0}^{β_{0}}D_{+}^{β_{+}}D_{−}^{β_{−}}) ≲ 2^{−εj}. (26)

C_{α_{1},α_{2}}^{−}(A_{0}, A_{j}, P_{−}D_{0}^{β_{0}}D_{+}^{β_{+}}D_{−}^{β_{−}}) ≲ 2^{−εj}. (27)

C_{α_{1},α_{2}}^{±}(A_{0}, A_{j}, (I − P_{±})D_{0}^{β_{0}}D_{+}^{β_{+}}D_{−}^{β_{−}}) ≲ 2^{−εj}. (28)

These three estimates will reduce to the cases I_{j}, II_{j}, III_{j} respectively (possibly after redefining j).

3.15. The contribution of P_{+}. The argument for this section is closely related to the one in [17].

Let f, g have Fourier transforms supported on A_{0}, A_{j} respectively. From (22) and the definition of P_{+}, we see that P_{+}D_{0}^{β_{0}}D_{+}^{β_{+}}D_{−}^{β_{−}}(ϕ^{+}ψ^{+}) vanishes unless j ≤ C. By enlarging the annuli A_{k} if necessary we may assume that j = 0. The indices α_{1}, α_{2} are now irrelevant by Lemma 3.13, and will be discarded.

The region given by (22) is essentially a ball of unit radius, on which D_{−}^{β_{−}} and D_{+}^{β_{+}} are standard multipliers which can then be discarded by (20). Thus (26) reduces to

C^{+}(A_{0}, A_{0}, P_{+}D_{0}^{β_{0}}) ≲ 1.

We now perform another decomposition. For all j ≥ C, let Δ_{j}^{0} be given by a multiplier adapted to the cylindrical region {(ξ, τ) : |ξ| ∼ 2^{−j}}. Then on the region given by (22) we essentially have

P_{+}D_{0}^{β_{0}} = ∑_{j≥C} 2^{−β_{0}j}Δ_{j}^{0},

so by (19) it suffices to show that

C^{+}(A_{0}, A_{0}, Δ_{j}^{0}) ≲ 2^{(β_{0}−ε)j}.

Fix j, and divide A_{0} into essentially disjoint squares Q of width 2^{−j}. With the notation of Corollary 3.11 with D = Δ_{j}^{0}, we observe that Q^{1} ∼ Q^{2} only holds when Q^{1} + Q^{2} ⊂ B(0, C2^{−j}). Thus the criteria of Corollary 3.11 are satisfied, and we reduce to showing that

C^{+}(Q^{1}, Q^{2}, Δ_{j}^{0}) ≲ 2^{(β_{0}−ε)j}

uniformly for all Q^{1}, Q^{2} such that Q^{1} + Q^{2} ⊂ B(0, C2^{−j}).

Fix Q^{1}, Q^{2}. By a rotation we may place these cubes on the ξ_{1} axis, and by Lemma 3.7 we may make them a distance 1 fromthe origin, so that we are now in the case I_{j} mentioned at the start of this section. Discarding now the Δ_{j}^{0} projection using (20), we will be done if we can show that

C^{+}(Q_{I_{j}}^{1}, Q_{I_{j}}^{2}, 1) ≲ 2^{(β_{0}−ε)j}. (29)

Note that this is (except for the epsilon) the same estimate one would be faced with when specializing (5) to the case I_{j}. When j is small then this follows from Proposition 3.6 so we may assume that j ≫ 1.

We would like to apply Proposition 3.6 for all j, but unfortunately the surfaces S^{+}(Q_{I_{j}}^{1}), S^{+}(Q_{I_{j}}^{2}) are too flat to be of disjoint conic type uniformly in j (after rescaling), and we must decompose and rescale further. Partition Q_{I_{j}}^{1} (resp. Q_{I_{j}}^{2}) into about 2^{j} rectangles Q_{1} (resp. Q_{2}) each of length 2^{−2j} in the e_{1} direction and 2^{−j} in the e_{2} direction. From a Taylor expansion of |ξ| we see that the conditions of Lemma 3.12 are satisfied, so we have

C^{+}(Q_{I_{j}}^{1}, Q_{I_{j}}^{2}, 1) ≲ 2^{2j( \frac{1}{q} − \frac{1}{2} )} sup_{Q^{1},Q^{2}} C^{+}(Q^{1}, Q^{2}, 1). (30)

By Corollary 3.9 we may assume that

Q^{1} = {ξ_{1} = 1 + O(2^{−2j}), ξ_{2} ∼ 2^{−j}}, Q^{2} = {η_{1} = −1 + O(2^{−2j}), η_{2} ∼ 2^{−j}}.

By (18) we thus need to estimate the quantity

∥{ \atop fdσ_{1}}{ \atop gdσ_{2}}∥_{q}

for all f and g supported on S^{+}(Q^{1}), S^{+}(Q^{2}) respectively. In null co-ordinates these two surfaces can be written as

S^{+}(Q^{1}) = \{(a_{1}, a_{2}, a_{3}) : a_{1} = 1 + O(2^{−2j}), a_{2} ∼ 2^{−j}, a_{3} = \frac{a_{2}^{2}}{4a_{1}} \}

S^{+}(Q^{2}) = \{(a_{1}, a_{2}, a_{3}) : a_{3} = 1 + O(2^{−2j}), a_{2} ∼ 2^{−j}, a_{1} = \frac{a_{2}^{2}}{4a_{3}} \}

We now rescale these surfaces to be of unit size, applying the (non-conformal) scaling

L(a_{1}, a_{2}, a_{3}) = (2^{2j}a_{1}, 2^{j}a_{2}, 2^{2j}a_{3}).

The surfaces transform to

L(S^{+}(Q^{1})) = \{(a_{1}, a_{2}, a_{3}) : a_{1} = 2^{2j} + O(1), a_{2} ∼ 1, a_{3} = \frac{a_{2}^{2}}{2^{−2j+2}a_{1}} \}

L(S^{+}(Q^{2})) = \{(a_{1}, a_{2}, a_{3}) : a_{3} = 2^{2j} + O(1), a_{2} ∼ 1, a_{1} = \frac{a_{2}^{2}}{2^{−2j+2}a_{3}} \}.

Since 2^{−2j+2}a_{1}, 2^{−2j+2}a_{3} are almost constant, these surfaces are essentially unit parabolic cylinders oriented in the (1, 0, 0) and (0, 0, 1) directions. These two surfaces are of disjoint conic type, and so the hypothesis R^{∗}(2 × 2 → q) applies. Undoing the scaling, we obtain

2^{−5j/q}∥{ \atop fdσ_{1}}{ \atop gdσ_{1}}∥_{q} ≲ 2^{−3j/2}∥f∥_{2}2^{−3j/2}∥g∥_{2}.

Thus

C^{+}(Q^{1}, Q^{2}, 1) ≲ 2^{−3j}2^{5j/q}.

Combining this with (30) we obtain

C^{+}(Q_{I_{j}}^{1}, Q_{I_{j}}^{2}, 1) ≲ 2^{2j( \frac{1}{q} − \frac{1}{2} )}2^{−3j}2^{5j/q} (31)

and (29) follows from (13).

For q < 7/4 there is a better way to prove (29), simply by replacing Q_{I_{j}}^{1}, Q_{I_{j}}^{2} by the larger sets Q_{I_{0}}^{1}, Q_{I_{0}}^{2} and using Proposition 18. This proves (29) for β_{0} > 0 (and can also capture the endpoint β_{0} = 0, simply by refusing to decompose using Δ_{j}). If we could prove the optimal estimate R^{∗}(2 × 2 → 5/3) then one would obtain the optimal value of β_{0} given by (9).

3.16. The contribution of P_{−}. By arguing similarly to the previous section, we may assume that j = 0 and discard the α_{1}, α_{2} subscripts.

On the region given by (22), |ξ| + |τ| ∼ |ξ| and so D_{0}^{β_{0}} is equal to D_{+}^{β_{0}} times a harmless multiplier. Thus by (20) it suffices to show that

C^{−}(A_{0}, A_{0}, P_{−}D_{+}^{β_{0}+β_{+}}D_{−}^{β_{−}}) ≲ 1.

On the region given by (22), we essentially have the decomposition

P_{−}D_{+}^{β_{0}+β_{+}}D_{−}^{β_{−}} = ∑_{j≥C} ∑_{m≥0} 2^{−(β_{0}+β_{+})j}2^{−β_{−}(j+m)}Δ_{j}^{+}Δ_{j+m}^{−},

where Δ_{j}^{+}, Δ_{j}^{−} are given by multipliers which are bump functions adapted to |ξ| + |τ| ∼ 2^{−j}, | |ξ| − |τ| | ∼ 2^{−j} respectively.

By (19) it thus suffices to show that

C^{−}(A_{0}, A_{0}, Δ_{j}^{+}Δ_{j+m}^{−}) ≲ 2^{(β_{0}+β_{−}+β_{+}−ε)j}2^{(β_{−}−ε)m}

uniformly in j, m.

Fix j,m. Partition A_{0} into disjoint sectors Γ with radial width 1 and angular width C^{−1}2^{−j− \frac{m}{2}} . We use the notation of Corollary 3.11 with D = Δ_{j}^{+}Δ_{j+m}^{−}. From (14) we see that Γ^{1} ∼ Γ^{2} only occurs when the two sectors Γ^{1}, −Γ^{2} subtend an angle of ∼ 2^{−j− \frac{m}{2}} . By Corollary 3.11 it thus suffices to show that

C^{−}(Γ^{1}, Γ^{2}, Δ_{j}^{+}Δ_{j+m}^{−}) ≲ 2^{(β_{0}+β_{−}+β_{+}−ε)j}2^{(β_{−}−ε)m}

uniformly for Γ^{1} ∼ Γ^{2}. By a rotation we may assume that

Γ^{1} = \{ξ : ξ_{1} ∼ 1, ξ_{2} ∼ 2^{−j− \frac{m}{2}} \}, Γ^{2} = \{η : η_{1} ∼ −1, η_{2} ∼ 2^{−j− \frac{m}{2}} \}.

We now partition Γ^{1} and Γ^{2} further into disjoint rectangles Q^{1}, Q^{2} which have dimensions 2^{−j} ×2^{−j− \frac{m}{2}} . We observe that Q^{1} ∼ Q^{2} only when Q^{1}+Q^{2} ⊂ B(0, C2^{−j}). Thus by Corollary 3.11 again, we reduce to showing that

C^{−}(Q^{1}, Q^{2}, Δ_{j}^{+}Δ_{j+m}^{−}) ≲ 2^{(β_{0}+β_{−}+β_{+}−ε)j}2^{(β_{−}−ε)m}.

for Q^{1} ∼ Q^{2}; by Lemma 3.7 we may assume

Q^{1} = \{ξ : ξ_{1} = 1+O(2^{−j}), ξ_{2} ∼ 2^{−j− \frac{m}{2}} \}, Q^{2} = \{η : η_{1} = −1+O(2^{−j}), η_{2} ∼ 2^{−j− \frac{m}{2}} \}.

The projection Δ_{j+m}^{−} is given by a standard multiplier on the region given by (22), so we may discard this projection together with the Δ_{j}^{+} multiplier by (20). Since the α_{1}, α_{2} indices can be set arbitrarily by Lemma 3.13, we write our estimate as

C_{1/2,1/2}^{−}(Q^{1}, Q^{2}, 1) ≲ 2^{(β_{0}+β_{−}+β_{+}−ε)j}2^{(β_{−}−ε)m}.

We now eliminate the m parameter by applying the conformal linear transformation

L(ξ_{1}, ξ_{2}, τ) = (\frac{τ + ξ_{1}}{2} − 2^{m} \frac{τ − ξ_{1}}{2} , 2^{\frac{m}{2}} ξ_{2}, \frac{τ + ξ_{1}}{2} + 2^{m} \frac{τ − ξ_{1}}{2} ).

By Corollary 3.9 and a computation involving a Taylor expansion of |ξ|, |η|, we have

C_{1/2,1/2}^{−}(Q^{1}, Q^{2}, 1) = | det L|^{\frac{1}{q} − \frac{1}{3}} C_{1/2,1/2}^{−}(Q_{II_{j}}^{1}, Q_{II_{j}}^{2}, 1)

where Q_{II_{j}}^{1}, Q_{II_{j}}^{2} were defined in the model case II_{j}. We simplify this using | det L| = C2^{\frac{3m}{2}} and (6) as

C_{1/2,1/2}^{−}(Q_{II_{j}}^{1}, Q_{II_{j}}^{2}, 1) ≲ 2^{(α_{1}+α_{2}+ \frac{3}{q} −2−ε)j}2^{(β_{−}− \frac{3}{2q} + \frac{1}{2} −ε)m}.

By (7) it suffices to verify this for m = 0:

C_{1/2,1/2}^{−}(Q_{II_{j}}^{1}, Q_{II_{j}}^{2}, 1) ≲ 2^{(α_{1}+α_{2}+ \frac{3}{q} −2−ε)j}.

This is essentially what one would get by specializing (5) to the case II_{j}. As in the previous section we may assume that j ≫ 1.

We will treat this using a further conformal transformation

L̃(ξ_{1}, ξ_{2}, τ) = (\frac{τ + ξ_{1}}{2} − 2^{2j} \frac{τ − ξ_{1}}{2} , 2^{j}ξ_{2}, \frac{τ + ξ_{1}}{2} + 2^{2j} \frac{τ − ξ_{1}}{2} ).

By Corollary 3.9 we then have

C_{1/2,1/2}^{−}(Q_{II_{j}}^{1}, Q_{II_{j}}^{2}, 1) ≲ | det L̃|^{\frac{1}{q} − \frac{1}{3}} C_{1/2,1/2}^{−}(Q̃^{1}, Q̃^{2}, 1)

where Q̃^{1}, Q̃^{2} are the regions

Q̃^{1} = {ξ : ξ_{2} ∼ 1, |ξ| + ξ_{1} = 2 + O(2^{−j})}
Q̃^{2} = {η : η_{2} ∼ 1, |η| − η_{1} = 2 + O(2^{−j})}.

Since | det L̃| = C2^{3j}, we thus reduce to

C_{1/2,1/2}^{−}(Q̃^{1}, Q̃^{2}, 1) ≲ 2^{(α_{1}+α_{2}−1−ε)j}. (32)

These two sets can be each broken up into about 2^{j} squares R^{1}, R^{2} of sidelength 2^{−j}. Because these sets lie on angle-separated subsets of the cone, the hypotheses of Lemma 3.12 apply, and we have

C_{1/2,1/2}^{−}(Q̃^{1}, Q̃^{2}, 1) ≲ (2^{2j})^{\frac{1}{q} − \frac{1}{2}} sup_{R^{1},R^{2}} C_{1/2,1/2}^{−}(R^{1}, R^{2}, 1). (33)

By Corollary 3.9 we may choose

R^{1} = R^{2} = {ξ : ξ_{1} = 1 + O(2^{−j}), ξ_{2} ∼ 2^{−j}}.

By repeating the proof of (31) (with the (+−) sign instead of (++)) we have

C_{1/2,1/2}^{−}(R^{1}, R^{2}, 1) ≲ 2^{2j( \frac{1}{q} − \frac{1}{2} )}2^{−3j}2^{5j/q}.

Combining this with (33) and simplifying, we have

C_{1/2,1/2}^{−}(Q̃^{1}, Q̃^{2}, 1) ≲ 2^{( \frac{9}{q} −5)j},

and (32) follows from (12).

As in the previous section, there is an alternate argument which is superior for small q. By replacing Q̃^{1}, Q̃^{2} by their O(1)-neighbourhoods and using Proposition 3.6, we obtain

C_{1/2,1/2}^{−}(Q̃^{1}, Q̃^{2}, 1) ≲ 1,

which gives (32) if α_{1} + α_{2} > 1. This is superior to the previous argument when q < 9/5, and for q = 5/3 it gives an almost sharp result.

3.17. The contribution of I − P_{±}. Let f, g have Fourier transforms supported on A_{0}, A_{j} respectively. By considering the (++), (+−) cases separately, we see that one has

|ξ| ∼ |ξ| + |τ| ∼ 2^{j}

on the region given by (22). Thus by (20) and standard multiplier calculus, the multipliers D^{+}, D_{0} may be replaced by the constant 2^{j}. By Lemma 3.13 we thus reduce to

C_{1/2,1/2}^{±}(A_{0}, A_{j}, (I − P_{±})D_{−}^{β_{−}}) ≲ 2^{−(β_{0}+β_{+}+ \frac{1}{2} −α_{2}+ε)j},

which we write using (6) as

C_{1/2,1/2}^{±}(A_{0}, A_{j}, (I − P_{±})D_{−}^{β_{−}}) ≲ 2^{(β_{−}+ \frac{3}{2} − \frac{3}{q} −α_{1}−ε)j}.

From (14), (15) we see that the value of the multiplier for D_{−} cannot exceed 1. Thus we may write

D_{−}^{β_{−}} = ∑_{m≥0} 2^{−mβ_{−}}Δ_{m}^{−}

on the region given by (22), where Δ_{m}^{−} was defined in the previous section. Discarding the harmless I − P_{±} multiplier by (20) and using (19), it thus suffices to show that

C_{1/2,1/2}^{±}(A_{0}, A_{j}, Δ_{m}^{−}) ≲ 2^{(β_{−}−ε)m}2^{(β_{−}+ \frac{3}{2} − \frac{3}{q} −α_{1}−ε)j}

uniformly in m ≥ 0.

Fix j,m. Partition A_{0} (resp. A_{j}) into disjoint sectors Γ^{1} (resp. Γ^{2}) with radial width 1 (resp. 2^{j}) and angular width C^{−1}2^{− \frac{m}{2}} . We use the notation of Corollary 3.11 with D = Δ_{m−j}^{−}. From (14),(15) we see that Γ^{1} ∼ Γ^{2} only occurs when Γ^{1}, ±Γ^{2} differ in the angular variable by ∼ 2^{−m/2}. Thus by Corollary 3.11 it suffices to show that

C_{1/2,1/2}^{±}(Γ^{1}, Γ^{2}, Δ_{m}^{−}) ≲ 2^{(β_{−}−ε)m}2^{(β_{−}+ \frac{3}{2} − \frac{3}{q} −α_{1}−ε)j}

uniformly for Γ^{1} ∼ Γ^{2}. The operator Δ_{m}^{−} is given by a standard multiplier when restricted to the region given by (22), so we may discard it by (20). By a rotation we may assume that

Γ^{1} = \{ξ : ξ_{1} ∼ 1, |ξ_{2}| ≪ 2^{− \frac{m}{2}} \}, Γ^{2} = \{η : η_{1} ∼ 2^{j}, η_{2} ∼ 2^{j− \frac{m}{2}} \}.

We now apply Corollary 3.9 with the conformal linear transformation

L(ξ_{1}, ξ_{2}, τ) = (\frac{τ + ξ_{1}}{2} − 2^{m} \frac{τ − ξ_{1}}{2} , 2^{\frac{m}{2}} ξ_{2}, \frac{τ + ξ_{1}}{2} + 2^{m} \frac{τ − ξ_{1}}{2} )

to obtain (after applying another mild conformal transformation)

C_{1/2,1/2}^{±}(Γ^{1}, Γ^{2}, 1) ∼ | det L|^{\frac{1}{q} − \frac{1}{3}} C_{1/2,1/2}^{±}(Q_{III_{j}}^{1}, Q_{III_{j}}^{2}, 1)

where Q_{III_{j}}^{1}, Q_{III_{j}}^{2} were defined in the model case III_{j}. Since | det L| = C2^{3m/2}, we thus reduce to showing that

C_{1/2,1/2}^{±}(Q_{III_{j}}^{1}, Q_{III_{j}}^{2}, 1) ≲ 2^{(β_{−}− \frac{3}{2q} + \frac{1}{2} −ε)m}2^{(β_{−}+ \frac{3}{2} − \frac{3}{q} −α_{1}−ε)j}.

By (7) it suffices to consider the case m = 0. By Lemma 3.13 this becomes

C_{0,0}^{±}(Q_{III_{j}}^{1}, Q_{III_{j}}^{2}, 1) ≲ 2^{(β_{−}+2− \frac{3}{q} −α_{1}−ε)j}. (34)

This should be compared with condition (8). This estimate is essentially what one would get by specializing (5) to the case III_{j}. As in the previous sections we may assume j ≫ 1.

It would be convenient if the two sets Q_{III_{j}}^{1}, Q_{III_{j}}^{2} were of the same size. Accordingly, we partition Q_{III_{j}}^{2} into about 2^{2j} squares Q^{2} of sidelength 1, and partition Q_{III_{j}}^{1} into the singleton partition {Q_{III_{j}}^{1}}. Because of the angular separation between Q_{III_{j}}^{1} and Q_{III_{j}}^{2}, we see that the conditions of Lemma 3.12 are satisfied, and so we have

C_{0,0}^{±}(Q_{III_{j}}^{1}, Q_{III_{j}}^{2}, 1) ≲ 2^{2j( \frac{1}{q} − \frac{1}{2} )} sup_{Q^{2}} C_{0,0}^{±}(Q_{III_{j}}^{1}, Q^{2}, 1). (35)

By Corollary 3.9 and Lemma 3.13 we may move Q^{2} to

Q^{2} = {η : η_{1} = ∓2^{j} + O(1), η_{2} = O(1)}

without seriously affecting the position of Q_{III_{j}}^{1}.

We would like to now apply Proposition 3.6. Unfortunately, the surface S^{±}(Q^{2}) is too flat to be of disjoint conic type uniformly in j, and we must make another decomposition. Partition Q_{III_{j}}^{1} into about 2^{j} sets of the form

Q^{1} = {ξ ∈ Q_{III_{j}}^{1} : ξ_{1} + |ξ| = a + O(2^{−j})}

where a ∼ 1, and partition Q^{2} trivially as {Q^{2}}. From a Taylor expansion of |ξ| we see that the condition of Lemma 3.12 is satisfied, so we have

C_{0,0}^{±}(Q_{III_{j}}^{1}, Q^{2}, 1) ≲ 2^{j( \frac{1}{q} − \frac{1}{2} )} sup_{Q^{1}} C_{0,0}^{±}(Q^{1}, Q^{2}, 1). (36)

By Lemma 3.7 we may set a = 2 without seriously affecting the position of Q^{2}. From (34), (35), (36), we thus reduce to

C_{0,0}^{±}(Q^{1}, Q^{2}, 1) ≲ 2^{−3j( \frac{1}{q} − \frac{1}{2} )}2^{(β_{−}+2− \frac{3}{q} −α_{1}−ε)j}.

By (18) it therefore suffices to show

∥{ \atop fdσ_{1}}{ \atop gdσ_{2}}∥_{q} ≲ 2^{−3j( \frac{1}{q} − \frac{1}{2} )}2^{(β_{−}+2− \frac{3}{q} −α_{1}−ε)j}∥f∥_{2}∥g∥_{2} (37)

for f, g supported on S^{+}(Q^{1}), S^{±}(Q^{2}) respectively. In null co-ordinates the surfaces become

S^{+}(Q^{1}) = \{(a_{1}, a_{2}, a_{3}) : a_{1} = 1 + O(2^{−j}), a_{2} = O(1), a_{3} = \frac{a_{2}^{2}}{4a_{1}} \}

S^{±}(Q^{2}) = \{(a_{1}, a_{2}, a_{3}) : a_{3} ∼ ±2^{j} + O(1), a_{2} = O(1), a_{1} = \frac{a_{2}^{2}}{4a_{3}} \}

We now apply the non-conformal scaling

L(a_{1}, a_{2}, a_{3}) = (2^{j}a_{1}, a_{2}, a_{3}).

The surfaces

L(S^{+}(Q^{1})) = \{(a_{1}, a_{2}, a_{3}) : a_{1} = 2^{j} + O(1), a_{2} = O(1), a_{3} = \frac{a_{2}^{2}}{2^{−j+2}a_{1}} \}

L(S^{±}(Q^{2})) = \{(a_{1}, a_{2}, a_{3}) : a_{3} ∼ ±2^{j} + O(1), a_{2} = O(1), a_{1} = \frac{a_{2}^{2}}{2^{−j+2}a_{3}} \}

are essentially non-parallel parabolic cylinders and are of disjoint conic type uniformly in j.

Translating the estimate R^{∗}(2 × 2 → q) for these surfaces back to S^{+}(Q^{1}) and S^{±}(Q^{2}), we obtain

2^{−j/q}∥{ \atop fdσ_{1}}{ \atop gdσ_{2}}∥_{q} ≤ 2^{−j/2}∥f∥_{2}∥g∥_{2}.

Combining this with (11) we obtain (37) as desired.

4. Application to Sogge’s local smoothing conjecture

We continue the study of one-sided solutions ϕ^{±} = e^{±i\sqrt{−Δ}t}f to the free wave equation in R^{2+1}, which we initiated in Section 3.

Figure 5. Local smoothing estimates LS(p → q, α).

We consider local smoothing estimates of the form

∥ϕ^{±}∥_{L^{q}([1,2]×R^{2})} ≲ ∥f∥_{L_{α}^{p}},

where 1 ≤ p, q ≤ ∞ and α ∈ R. We call such an estimate LS(p → q, α). By time reversal symmetry it suffices to consider the solution ϕ^{+}.

There are several known necessary conditions for LS(p → q, α) to hold. From translation invariance considerations we have q ≥ p. From the focussing example in which f is a function of height 1 adapted to a δ-neighbourhood of the circle of radius r for some r ∈ [1, 2], and ϕ^{±} has size δ^{−1/2} on the δ-ball centered at (r, 0), we obtain the condition

\frac{3}{q} − \frac{1}{2} ≥ \frac{1}{p} − α. (38)

From the Knapp example, in which ϕ^{±}(x, t) for |t| ≲ 1 is essentially a bump of magnitude 1 supported on a δ × δ^{1/2} rectangle, we obtain the condition

\frac{3}{2q} ≥ \frac{3}{2p} − α. (39)

Finally there is the condition

\frac{1}{q} + \frac{1}{2} ≥ \frac{2}{p} − α

which comes from considering the case when f is a delta function.

We display these conditions, and the ranges for which they are dominant, in Figure 5. The endpoints A, H, I correspond to the estimates LS(1 → 1, 1/2 + ε), LS(∞ → ∞, 1/2 + ε), LS(1 → ∞, 3/2 + ε), with the epsilon loss being necessary. The endpoint B is the energy estimate LS(2 → 2, 0). These four estimates, and the ones obtained by interpolation between them, are the best that one can do from fixed-time estimates alone; any further estimate represents some gain in smoothness by averaging over time, hence the term “local smoothing”.

The optimal local smoothing estimate would be LS(4 → 4, ε), which is point D in Figure 5. This is the local smoothing conjecture of Sogge, which is still open, although Mockenhaupt, Seeger, and Sogge [21] have the partial result LS(4 → 4, 1/8 + ε). We improve upon this result in Section 6.

By interpolating the conjecture D with the known estimate I, we obtain

Conjecture 4.1. [28] We have LS((q/3)′ → q, 3/2 − 6/q + ε) for all q ≥ 4 and ε > 0.

This conjecture is true for q ≥ 6 thanks to Strichartz’ estimate LS(2 → 6, 1/2), which is the point G. In [28] Schlag and Sogge improved this to q ≥ 5, which is the point F. We do not know if the ε is necessary for q > 4, although the Besicovitch-Rado-Kinney construction and an example in Wolff [37] both show that it is necessary when q = 4.

The purpose of this section is to improve this result to

Theorem 4.2. Suppose that q_{0} < 2 is such that (5) is true in the (++) case for q = q_{0}, β_{0} = β_{+} = 0, β_{−} = \frac{3}{2q_{0}} − \frac{1}{2} + 2ε, and α_{1} = α_{2} = \frac{3}{4q′_{0}} + ε for arbitrarily small ε > 0. Then Conjecture 4.1 is true for all q ≥ q_{0} + 3.

From Theorem 1.2 and Theorem 3.2 we see that the hypotheses of this theorem hold for q_{0} = 2− \frac{8}{121} +ε. Thus Conjecture 4.1 is true for q > 5− \frac{8}{121}. Unfortunately, since (5) can only hold when q_{0} ≥ 5/3 we see that this theorem cannot prove the full conjecture. We discuss an alternative approach in Section 6. Proof Fix q_{0}; without loss of generality we may assume that q = 3 + q_{0}. Write p = (q/3)′ and α = 3/2 − 6/q + ε.

Roughly speaking, the idea is as follows. The hypothesis is a bilinear variant of the (false) estimate LS(2 → 2q, α̃), where α̃ is given by (38). Strictly speaking this estimate is false because of the Knapp condition (39), however the bilinear version (5) can still hold, and this turns out to be good enough for interpolation purposes (as long as the final estimate is on the critical line p = (q/3)′). Interpolating this estimate, which corresponds to C on Figure 5, with the point H, we obtain the desired result E. This is a variant of the argument in [28], which used the q = 2 estimate, although the version of H used in [28] is slightly different. The approach should also be compared with the sharp restriction theorems proven in [34, 35].

By Littlewood-Paley decomposition and giving an epsilon up in the α index, we may assume that f has frequency support on the annulus |ξ| ∼ N for some N; we may assume that N ≫ 1 since in the low frequency case ϕ^{±} is given by a smoothing operator applied to f.

We have to prove

∥ϕ^{+}∥_{L^{q}([1,2]×R^{2})} ≲ N^{α}∥f∥_{L^{p}}.

Since q > p we can use translation invariance and finite speed of propagation to replace [1, 2] × R^{2} by [1, 2] × B(0, 1). We square this as

∥ϕ^{+}ϕ^{+}∥_{L^{q/2}([1,2]×B(0,1))} ≲ N^{2α}∥f∥_{L^{p}}∥f∥_{L^{p}}.

For each j > 0, we divide up the circle into 2^{j} arcs τ_{k}^{j} of length 2^{−j}, and partition f = ∑_{k} f_{j,k}, ϕ^{+} = ∑_{k} ϕ_{j,k}^{+}, where f̂_{j,k} is supported in the sector {ξ : |ξ| ∼ N, ξ/|ξ| ∈ τ^{j,k}}. By the Whitney decomposition (cf. the proof of Proposition 2.3 or [34]) we have

ϕ^{+}ϕ^{+} = ∑_{j} ∑_{k∼k′} ϕ_{j,k}^{+}ϕ_{j,k′}^{+}

so it suffices to show that

∥ ∑_{k∼k′} ϕ_{j,k}^{+}ϕ_{j,k′}^{+}∥_{L^{q/2}([1,2]×B(0,1))} ≲ N^{2α}2^{−εj}∥f∥_{L^{p}}∥f∥_{L^{p}} (40)

uniformly in j for some ε > 0.

Fix j. We see that ϕ_{j,k}^{+}ϕ_{j,k′}^{+} are supported in essentially disjoint boxes as k ∼ k′ both vary. Since q/2 > 2, we may apply Lemma 7.1 and estimate the left-hand side of (40) by

( ∑_{k∼k′} ∥ϕ_{j,k}^{+}ϕ_{j,k′}^{+}∥_{L^{q/2}([1,2]×B(0,1))}^{r})^{1/r}

where r = (q/2)′.

Suppose for the moment that we could prove the estimate

∥ϕ_{j,k}^{+}ϕ_{j,k′}^{+}∥_{L^{q/2}([1,2]×B(0,1))} ≲ N^{2α}2^{−εj}∥f_{j,k}∥_{L^{p}}∥f_{j,k′}∥_{L^{p}} (41)

for all k ∼ k′. Then the left-hand side of (40) would be bounded by

N^{2α}2^{−εj} ( ∑_{k∼k′}(∥f_{j,k}∥_{L^{p}}∥f_{j,k′}∥_{L^{p}})^{r})^{1/r} .

Since q > 4 and p = (q/3)′, we have r ≥ p/2, so the above expression is majorized by

N^{2α}2^{−εj} ( ∑_{k∼k′}(∥f_{j,k}∥_{L^{p}}∥f_{j,k′}∥_{L^{p}})^{p/2})^{2/p} .

By Cauchy-Schwarz and the fact that there are only finitely many k′ associated to each k, we may bound this by

N^{2α}2^{−εj} (∑_{k} ∥f_{j,k}∥_{L^{p}}^{p})^{1/p} (∑_{k′} ∥g_{j,k′}∥_{L^{p}}^{p})^{1/p} ,

which is (40).

Thus it suffices to prove (41). By a rotation it suffices to show that

∥ϕ^{+}ψ^{+}∥_{L^{q/2}([1,2]×B(0,1))} ≲ N^{2α}2^{−εj}∥f∥_{p}∥g∥_{p} (42)

whenever f, g have frequency support on the sectors

{ξ : ξ_{1} ∼ N, |ξ_{2}| ≪ 2^{−j}N},
{η : η_{1} ∼ N, η_{2} ∼ 2^{−j}N}

respectively; here the notation is as in Section 3.

We first show this in the easy case when 2^{j} > \sqrt{N}. By Hölder’s inequality it suffices to show that

∥ϕ^{+}∥_{L^{q}([1,2]×B(0,1))} ≲ N^{α}2^{−εj}∥f∥_{p}.

We use the integral representation

ϕ^{+}(t, x) = ∫ e^{2πix·ξ}e^{2πit|ξ|}f̂(ξ) dξ.

From the support of f and the restrictions t ∼ 1, 2^{j} > \sqrt{N} we have the Taylor approximation

t|ξ| = tξ_{1} + O(1),

where the O(1) error is smooth. Discarding this error^{3} it thus suffices to show that

∥ ∫ e^{2πix·ξ}e^{2πitξ_{1}} f̂(ξ) dξ∥_{L^{q}([1,2]×B(0,1))} ≲ N^{α}2^{−εj}∥f∥_{p}.

But by a change of variables and the Fourier inversion formula this becomes

∥f∥_{L^{q}(B(0,C))} ≲ N^{α}2^{−εj}∥f∥_{p}.

But this follows (with ε = α/2) from Lemma 7.3, the frequency support of f, and the definitions of α, p, q.

We now turn to the case when 2^{j} ≲ \sqrt{N}. It suffices to show

∥ϕ^{+}ψ^{+}∥_{L^{q/2}([1,2]×B(0,1))} ≲ (2^{−2j}N)^{1− \frac{1}{p} − \frac{3}{q} +ε}N^{\frac{2}{p} − \frac{3}{q} +ε}∥f∥_{p}∥g∥_{p}, (43)

for arbitrarily small ε, since (42) follows from the definitions of p, α, q, the hypothesis 2^{j} ≲ \sqrt{N}, and some algebra. Since (1/p, 1/q) is on the line segment between (1/2, 1/2q_{0}) and (1/∞, 1/∞), it suffices to prove (43) with (p, q) replaced by (2, 2q_{0}) and (∞, ∞) respectively. Substituting these pairs of exponents into (43) and discarding some epsilons we obtain (after some algebra)

3See [32], Lemma 2.1, [6], Lemma 2.10, or the references therein. A similar argument also appears in the proof of Proposition 5.2 in this paper.

∥ϕ^{+}ψ^{+}∥_{L^{q_{0}} ([1,2]×B(0,1))} ≲ N^{α_{1}}N^{α_{2}}(2^{−2j}N)^{−β_{−}}∥f∥_{2}∥g∥_{2}

and

∥ϕ^{+}ψ^{+}∥_{L^{∞}([1,2]×B(0,1))} ≲ 2^{−2j}N∥f∥_{∞}∥g∥_{∞}

with the same restrictions on f, g as before, where α_{1},α_{2}, β_{−} were defined in the statement of the Theorem.

The former estimate is a direct consequence of the hypothesis (5) with the specified values of q, β_{0}, β_{+}, β_{−}, α_{1}, α_{2}, since D_{−} is equal to a standard multiplier with height 2^{−2j}N on the frequency support of ϕ^{+}ψ^{+}. Thus it suffices to prove the latter estimate. By Hölder’s inequality, symmetry it suffices to show that

∥ϕ^{+}∥_{L^{∞}([1,2]×B(0,1))} ≲ N^{1/2}2^{−j}∥f∥_{∞}.

It suffices to show this for 2^{j} = \sqrt{N}, since in the case 2^{j} < \sqrt{N} can be recovered by decomposing f and using the triangle inequality. But by repeating the Taylor approximation argument used earlier this inequality reduces to ∥f∥_{L^{∞}(B(0,C))} ≲ ∥f∥_{∞}, which is of course trivial.

5. An improvement to Mockenhaupt’s square function estimate

Let N be a large dyadic number, and consider the neighbourhood S_{N^{2}} of the upper unit light cone

S_{N^{2}} = {(ξ, τ) : |ξ| ∼ 1, τ = |ξ| + O(1/N^{2})}.

Following [20], we divide this region into N regions

E_{m} = \{(re^{2πiθ}, τ) : r ∼ 1, θ = \frac{m}{N} + O(\frac{1}{N}), τ = r + O(1/N^{2})\}

for m = 0, …, N − 1; note that each E_{m} is essentially a 1 × \frac{1}{N} × \frac{1}{N^{2}} rectangle. Let ψ_{m} be a bump function essentially adapted to E_{m}. We will think of the index set 0, …, N − 1 as being arranged in a circle, so N − 1 is adjacent to 0.

In [20], the following square-function estimate was proven for arbitrary test functions f:

∥ ∑_{m} ψ̂_{m} ∗ f∥_{4} ≲ N^{1/4}∥(∑_{m} |ψ̂_{m} ∗ f|^{2})^{1/2}∥_{4}. (44)

This estimate has many applications; we will discuss some of these in Section 6. In [3], it was observed that the 1/4 exponent in (44) could be improved to 1/4−τ

for some τ > 0. The purpose of this section is to refine the argument in [3] and give an explicit value for τ. More precisely, we have

Theorem 5.1. If κ > 0 is such that R^{∗}(2 × 2 → 2 − κ) holds for unit-separated subsets of the upper unit light cone and τ < κ/(16 − 4κ), then

∥ ∑_{m} ψ̂_{m} ∗ f∥_{4} ≲ N^{1/4−τ} ∥(∑_{m} |ψ̂_{m} ∗ f|^{2})^{1/2}∥_{4}, (45)

In particular, we have (45) for all τ < 1/238. The best possible value of τ is of course τ = 1/4, but this estimate is well beyond the techniques of this paper. We remark that the best possible value of κ is 1/3.

Our approach combines the arguments in [3] with the bilinear philosophy of [34],

[35] and this paper. As in [34], we begin by reducing the above linear estimate to a bilinear estimate.

Lemma 5.2. Let 0 < τ < 1/4 be fixed. If (45) holds, then one has

∥(∑_{m} ψ̂_{m} ∗ f)(∑_{m′} ψ̂_{m′} ∗ g)∥_{2} ≲ N^{1/2−2τ} ∥(∑_{m} |ψ̂_{m} ∗ f|^{2})^{1/2}∥_{4}∥(∑_{m′} |ψ̂_{m′} ∗ g|^{2})^{1/2}∥_{4} (46)

for all f,g where the ranges of m and m′ are separated in the sense that dist(E_{m}, E_{m′}) ∼ 1 for all m, m′. Conversely, if (46) holds under this separation condition, then (45) holds.

Proof The former implication is an immediate consequence of Hölder’s inequality, so we consider the latter. Assume that (46) holds. To prove (45) it suffices by squaring to estimate the quantity

∥(∑_{m} ψ̂_{m} ∗ f)(∑_{m} ψ̂_{m} ∗ f)∥_{2};

it will be important that we do not conjugate the second factor. For each 1 ≤ 2^{j} ≤ N we break the interval [0, N) into 2^{j} dyadic subintervals τ_{j,k}, and write τ_{j,k} ∼ τ_{j,k′} if τ_{j,k}, τ_{j,k′} are not adjacent, but have adjacent parents, where we identify 0 with N for the purposes of determining adjacency. When 2^{j} = N we also write τ_{j,k} ∼ τ_{j,k′} if τ_{j,k}, τ_{j,k′} are adjacent or equal. We then have the identity (cf. the arguments in Proposition 2.3 and Theorem 4.2)

∑_{m} ψ̂_{m} ∗ f ∑_{m} ψ̂_{m} ∗ f = ∑_{1≤2^{j}≤N} ∑_{τ_{j,k}∼τ_{j,k′}}( ∑_{m∈τ_{j,k}} ψ̂_{m} ∗ f)( ∑_{m′∈τ_{j,k′}} ψ̂_{m′} ∗ f).

By the triangle inequality it suffices to consider the contribution of a single j, providing we obtain an exponential gain in j. More precisely, we will show

\begin{aligned}∥ ∑_{τ_{j,k}∼τ_{j,k′}} & ( ∑_{m∈τ_{j,k}} ψ̂_{m} ∗ f)( ∑_{m′∈τ_{j,k′}} ψ̂_{m′} ∗ f)∥_{2} \\ & ≲ (2^{−j}N)^{1/2−2τ} (∥(∑_{m} |ψ̂_{m} ∗ f|^{2})^{1/2}∥_{4})^{2} \end{aligned} (47)

We first consider the easy case when 2^{j} = N. In this case the only pairs m, m′ which appear are those such that |m − m′| ≲ 1, where the norm |m − m′| is taken in the obvious circular sense. By polarization the left-hand side of (47) is therefore majorized by

∥ ∑_{m} |ψ̂_{m} ∗ f|^{2}∥_{2} = (∥(∑_{m} |ψ̂_{m} ∗ f|^{2})^{1/2}∥_{4})^{2}

as desired.

It remains to consider the case when 2^{j} < N. Again, we observe from elementary geometry that the frequency supports of

( ∑_{m∈τ_{j,k}} ψ̂_{m} ∗ f)( ∑_{m′∈τ_{j,k′}} ψ̂_{m′} ∗ f)

are essentially disjoint as τ_{j,k}, τ_{j,k′} vary. Thus we may estimate (47) by

( ∑_{τ_{j,k}∼τ_{j,k′}} ∥( ∑_{m∈τ_{j,k}} ψ̂_{m} ∗ f)( ∑_{m′∈τ_{j,k′}} ψ̂_{m′} ∗ f)∥_{2}^{2})^{1/2}. (48)

We now claim that (46) implies that

\begin{matrix}∥ & ( ∑_{m∈τ_{j,k}} ψ̂_{m} ∗ f)( ∑_{m′∈τ_{j,k′}} ψ̂_{m′} ∗ g)∥_{2} \\ ≲ (2^{−j}N)^{1/2−2τ} ∥( ∑_{m∈τ_{j,k}} |ψ̂_{m} ∗ f|^{2})^{1/2}∥_{4}∥( ∑_{m′∈τ_{j,k′}} |ψ̂_{m′} ∗ g|^{2})^{1/2}∥_{4} \end{matrix} (49)

for all f,g. When j = 0 this is just (46). Now consider the case j > 0. By a rotation we may assume that the index sets τ_{j,k}, τ_{j,k′} are within C2^{−j}N of the origin, so that the sets E_{m}, E_{m′} are within C2^{−j} of the e_{1}, e_{3} plane. We now apply the conformal linear transformation

(ξ_{1}, ξ_{2}, τ) ↦ (\frac{τ + ξ_{1}}{2} − 2^{2j} \frac{τ − ξ_{1}}{2} , 2^{j}ξ_{2}, \frac{τ + ξ_{1}}{2} + 2^{2j} \frac{τ − ξ_{1}}{2} ).

One may verify that the norms on both sides of (49) scale the same way under this transformation. Furthermore, the sets E_{m}, E_{m′} map to very similar sets but with the role of N replaced by 2^{−j}N. The claim then follows from (46).

Applying (49) to (48) and comparing this with (47), we see that we only need to verify that

( ∑_{τ_{j,k}∼τ_{j,k′}}(∥F_{j,k}∥_{4}∥F_{j,k′}∥_{4})^{2})^{1/2} ≲ (∥(∑_{k} |F_{j,k}|^{2})^{1/2}∥_{4})^{2} (50)

where

F_{j,k} = ( ∑_{m∈τ_{j,k}} |ψ̂_{m} ∗ f|^{2})^{1/2}.

By polarizing we may majorize the left-hand side of (50) by

(∑_{k} ∥F_{j,k}∥_{4}^{4})^{1/2}.

The claim (50) then follows from interchanging the l^{4} and L^{4} norms and using the inclusion l^{4} ⊂ l^{2}.

Thus it suffices to consider the bilinear estimate. (For an earlier partial equivalence between the two estimates, see [3]).

The next step is to use the restriction hypothesis R^{∗}(2 × 2 → q) to obtain something very close to (46). More precisely, we have Proposition 5.3. If q < 2 is such that R^{∗}(2 × 2 → q) holds, then

(∑_{Q} ∥(∑_{m} ψ̂_{m} ∗ f)(∑_{m′} ψ̂_{m′} ∗ g)∥_{L^{q}(Q)}^{2})^{1/2} ≲ N^{1/2}∥(∑_{m} |ψ̂_{m} ∗ f|^{2})^{1/2}∥_{4}∥(∑_{m′} |ψ̂_{m′} ∗ g|^{2})^{1/2}∥_{4} (51)

where the notation is as in Lemma 5.2, and Q ranges over a partition of R^{2} into N-cubes.

Proof We need to control the left-hand side of (51) by a square-function whose summands have frequency support on eccentric rectangles. To do this, we will first control the left-hand side of (51) by a different square-function whose summands have frequency support on disks. To finish up we will use the pointwise estimate in Lemma 7.2.

For this argument it will be notationally convenient to parameterize frequency space by spatial variables (ξ_{1}, ξ_{2}, ξ_{3}) rather than spacetime variables (ξ_{1}, ξ_{2}, τ), and similarly for physical space.

We subdivide frequency space R^{3} into a equally spaced collection of 1/N cubes I. Each set E_{m} is covered by about N of these cubes I, with different E_{m} being covered by essentially disjoint collections of cubes I. We abuse notation and write I ⊂ E_{m} if I forms part of the cover of E_{m}.

We may therefore write for each m

ψ̂_{m} ∗ f(x) = ∑_{I⊂E_{m}} ϕ̂_{I} ∗ f_{m}(x)

where ϕ_{I} is a bump function adapted to I and f_{m} = ψ̂_{m} ∗ f. We can arrange matters so that ϕ_{I}(ξ) = ϕ(ξ − ξ_{I}), where ξ_{I} is the center of I.

We consider the contribution of a single cube Q in (51). To begin with, let us assume that Q contains the origin.

We have

∑_{I⊂E_{m}} ϕ̂_{I} ∗ f_{m}(x) =

∑_{I⊂E_{m}} ∫ e^{−2πix·ξ}ϕ_{I}(ξ)f̌_{m}(ξ) dξ = e^{−2πix·ξ_{I}} ∫ e^{−2πix·(ξ−ξ_{I})}ϕ(ξ − ξ_{I})f̌_{m}(ξ) dξ.

As in [35], we perform a Taylor expansion of the phase

e^{−2πix·(ξ−ξ_{I})} = ∑_{γ} \frac{1}{γ!}(\frac{−2πix}{N})^{γ}(N(ξ − ξ_{I}))^{γ}

where γ is a multiindex. The term γ = 0 should be viewed as the main term. We thus have

∑_{m} ∑_{I⊂E_{m}} ϕ̂_{I} ∗ f_{m}(x) = ∑_{γ} \frac{1}{γ!}(\frac{−2πix}{N})^{γ} ∑_{m} ∑_{I⊂E_{m}} e^{−2πix·ξ_{I}} ∫ ϕ^{γ}(ξ − ξ_{I})f̌_{m}(ξ) dξ

where

ϕ^{γ}(ξ) = (Nξ)^{γ}ϕ(ξ).

A similar expansion holds for g. We can therefore write the contribution

∥(∑_{m} ψ̂_{m} ∗ f)(∑_{m′} ψ̂_{m′} ∗ g)∥_{L^{q}(Q)}

of Q to (51) as

∥ ∑_{γ_{1},γ_{2}} \frac{1}{γ_{1}!γ_{2}!}(\frac{−2πix}{N})^{γ_{1}+γ_{2}} ∏_{t=1}^{2} ∑_{m} ∑_{I⊂E_{m}} a_{m,I,t,γ_{t}}e^{−2πix·ξ_{I}} ∥_{L^{q}(Q)},

where

a_{m,I,t,γ_{t}} = ∫ ϕ^{γ_{t}}(ξ − ξ_{I})f̌_{m}^{t}(ξ) dξ = ϕ̂_{I}^{γ_{t}} ∗ f_{m}^{t}(0).

where we make the convention f^{1} = f, f^{2} = g to simplify the notation.

By the triangle inequality and crudely estimating |\frac{−2πix}{N}|by some constant C, we can majorize the above by

∑_{γ_{1},γ_{2}} \frac{C^{|γ_{1}|+|γ_{2}|}}{γ_{1}!γ_{2}!} ∥∏_{t=1}^{2} ∑_{m} ∑_{I⊂E_{m}} a_{m,I,t,γ_{t}}e^{−2πix·ξ_{I}} ∥_{L^{q}(Q)}.

We now invoke the hypothesis R^{∗}(2 × 2 → q), as discretized in Lemma 5.1 of [35] (with α = 0, n = 3, R = N, andr = 1), to estimate this by

N^{2} ∑_{γ_{1},γ_{2}} \frac{C^{|γ_{1}|+|γ_{2}|}}{γ_{1}!γ_{2}!} ∏_{t=1}^{2}(∑_{m} ∑_{I⊂E_{m}} |a_{m,I,t,γ_{t}}|^{2})^{1/2}.

Strictly speaking one has to generalize Lemma 5.1 of [35] since the points ξ_{I} are only within 1/N of the cone rather than being on the cone itself, but the argument extends trivially.

Expanding the definition of a_{m,I,t,γ_{t}} we thus have

∥(∑_{m} ψ̂_{m} ∗ f)(∑_{m′} ψ̂_{m′} ∗ g)∥_{L^{q}(Q)} ≲ N^{2} ∑_{γ_{1},γ_{2}} \frac{C^{|γ_{1}|+|γ_{2}|}}{γ_{1}!γ_{2}!} ∏_{t=1}^{2}(∑_{m} ∑_{I⊂E_{m}} |ϕ̂_{I}^{γ_{t}} ∗ f_{m}^{t}(0)|^{2})^{1/2}

This formula applies when Q contains the origin. By translation invariance we may generalize it to

∥(∑_{m} ψ̂_{m} ∗f)(∑_{m′} ψ̂_{m′} ∗g)∥_{L^{q}(Q)} ≲ N^{2} ∑_{γ_{1},γ_{2}} \frac{C^{|γ_{1}|+|γ_{2}|}}{γ_{1}!γ_{2}!} ∏_{t=1}^{2}(∑_{m} ∑_{I⊂E_{m}} |ϕ̂_{I}^{γ_{t}} ∗f_{m}^{t}(y)|^{2})^{1/2}

where Q is now arbitrary and y ∈ Q. If we average this in L^{2} over all y in Q we obtain

∥(∑_{m} ψ̂_{m}∗f)(∑_{m′} ψ̂_{m′}∗g)∥_{L^{q}(Q)} ≲ N^{2}N^{−3/2} ∑_{γ_{1},γ_{2}} \frac{C^{|γ_{1}|+|γ_{2}|}}{γ_{1}!γ_{2}!} ∥∏_{t=1}^{2}(∑_{m} ∑_{I⊂E_{m}} |ϕ̂_{I}^{γ_{t}}∗f_{m}^{t}|^{2})^{1/2}∥_{L^{2}(Q)}.

Summing this in l^{2} over all cubes Q and using the triangle inequality, we thus obtain

lhs of (51) ≲ N^{2}N^{−3/2} ∑_{γ_{1},γ_{2}} \frac{C^{|γ_{1}|+|γ_{2}|}}{γ_{1}!γ_{2}!} ∥∏_{t=1}^{2}(∑_{m} ∑_{I⊂E_{m}} |ϕ̂_{I}^{γ_{t}} ∗ f_{m}^{t}|^{2})^{1/2}∥_{L^{2}(R^{n})}.

By Hölder’s inequality, we may dominate this by

N^{2}N^{−3/2} ∑_{γ_{1},γ_{2}} \frac{C^{|γ_{1}|+|γ_{2}|}}{γ_{1}!γ_{2}!} ∏_{t=1}^{2}∥(∑_{m} ∑_{I⊂E_{m}} |ϕ̂_{I}^{γ_{t}} ∗ f_{m}^{t}|^{2})^{1/2}∥_{4}.

By Lemma 7.2 and the fact that ϕ^{γ_{t}} is a bump function with a norm growing at most exponentially in γ_{t}, we have the pointwise bound

∑_{I⊂E_{m}} |ϕ̂_{I}^{γ_{t}} ∗ f_{m}^{t}|^{2} ≲ C^{γ_{t}}M|f_{m}^{t}|^{2}.

Inserting this bound into the above, and observing that the γ_{1}, γ_{2} summations are just a convergent series of constants, we see that

lhs of (51) ≲ N^{1/2}∏_{t=1}^{2}∥(∑_{m} M|f_{m}^{t}|^{2})^{1/2}∥_{4}.

The claim then follows from the vector-valued maximal function inequality of Fefferman and Stein [11].

This estimate does not have the desired gain of N^{−τ}, but it does improve the L^{2} control to L^{q} control, at least on N-cubes^{4}. To finish the proof we will need to interpolate this estimate with another estimate which has the desired gain, but only has L^{∞} control on N-cubes. We will use a variant of the estimate in [3], namely

Proposition 5.4. If (45) held for some τ ≥ 0, then, then one has

\begin{matrix}(∑_{Q} ∥(∑_{m} ψ̂_{m} ∗ f) & (∑_{m′} ψ̂_{m′} ∗ g)∥_{L^{∞}(Q)}^{2})^{1/2} \\ ≲ N^{\frac{1}{4} −τ+ε}∥(∑_{m} |ψ̂_{m} ∗ f|^{2})^{1/2}∥_{4}∥(∑_{m′} |ψ̂_{m′} ∗ g|^{2})^{1/2}∥_{4} \end{matrix} (52)

for all test functions f, g, where Q is as in the previous Proposition.

Proof By Hölder’s inequality and symmetry it suffices to prove the linear estimate

(∑_{Q} ∥ ∑_{m} ψ̂_{m} ∗ f∥_{L^{∞}(Q)}^{4})^{1/4} ≲ \sqrt{N}^{\frac{1}{4} −τ} ∥(∑_{m} |ψ̂_{m} ∗ f|^{2})^{1/2}∥_{4}. (53)

The m represent a partition of the set S_{N^{2}} into N sectors of angular width 1/N. We introduce a coarser partition, dividing the 1/N neighbourhood S_{N} of the cone into N^{1/2} sectors Ẽ_{l}, of angular width N^{−1/2}; to each m we may associate a parent l such that E_{m} is essentially contained in Ẽ_{l}. We abuse notation and write E_{m} ⊂ Ẽ_{l} as shorthand for saying that Ẽ_{l} is the parent of E_{m}.

The left-hand side of (53) then becomes

(∑_{Q} ∥ ∑_{l} f_{l}∥_{L^{∞}(Q)}^{4})^{1/4} (54)

4For previous usage of these types of norms in the study of the wave equation, see [33].

where

f_{l} = ∑_{m:E_{m}⊂Ẽ_{l}} ψ̂_{m} ∗ f.

From the triangle inequality, Hölder’s inequality, and the fact that l ranges over a set with cardinality N^{1/2}, we have

∥ ∑_{l} f_{l}∥_{L^{∞}(Q)} ≲ ∑_{l} ∥f_{l}∥_{L^{∞}(Q)} ≲ N^{3/8}(∑_{l} ∥f_{l}∥_{L^{∞}(Q)}^{4})^{1/4}.

On the other hand, f_{l} has frequency support in Ẽ_{l}, which is a 1 × N^{−1/2} × N^{−1} rectangle. Thus by the localized form of Lemma 7.3 in the Appendix, we essentially have

∥f_{l}∥_{L^{∞}(Q)} ≲ N^{−3/8}∥f_{l}∥_{L^{4}(Q)}.

Strictly speaking we also have to have a rapidly decreasing contribution from cubes other than Q on the right-hand side, but we will ignore this irrelevant complication. Combining these estimates we see that we may majorize (54) by

(∑_{Q} ∥ ∑_{l} f_{l}∥_{L^{∞}(Q)}^{4})^{1/4} ≲ (∑_{l} ∥f_{l}∥_{4}^{4})^{1/4}.

Thus to conclude the proof, we need to show that

(∑_{l} ∥f_{l}∥_{4}^{4})^{1/4} ≲ \sqrt{N}^{\frac{1}{4} −τ} ∥(∑_{m} |ψ̂_{m} ∗ f|^{2})^{1/2}∥_{4}.

By raising both sides to the fourth power and expanding, this is

∑_{l} ∥f_{l}∥_{4}^{4} ≲ \sqrt{N}^{1−4τ} ∑_{m} ∑_{m′} ∥(ψ̂_{m} ∗ f)(ψ̂_{m′} ∗ f)∥_{2}^{2}.

The right-hand side is clearly larger than

∑_{l} \sqrt{N}^{1−4τ} ∑_{m:E_{m}⊂Ẽ_{l}} ∑_{m′:E_{m′} ⊂Ẽ_{l}} ∥(ψ̂_{m} ∗ f)(ψ̂_{m′} ∗ f)∥_{2}^{2}.

Thus we need only show that

∥f_{l}∥_{4}^{4} ≲ \sqrt{N}^{1−4τ} ∑_{m:E_{m}⊂Ẽ_{l}} ∑_{m′:E_{m′} ⊂Ẽ_{l}} ∥(ψ̂_{m} ∗ f)(ψ̂_{m′} ∗ f)∥_{2}^{2}

uniformly in l. Factorizing the right-hand side and taking fourth roots, this is

∥f_{l}∥_{4} ≲ \sqrt{N}^{\frac{1}{4} −τ} ∥( ∑_{m:E_{m}⊂Ẽ_{l}} |ψ̂_{m} ∗ f|^{2})^{1/2}∥_{4}. (55)

By changing the definition of ψ̂_{m} slightly, we may replace ψ̂_{m} ∗ f in (55) with ψ̂_{m} ∗ f_{l}. By a rotation we may assume that Ẽ_{l} is within O(1/\sqrt{N}) of the e_{1}, e_{3} plane. We now repeat the argument in Lemma 5.2, applying the conformal linear transformation

(ξ_{1}, ξ_{2}, τ) ↦ (\frac{τ + ξ_{1}}{2} − N \frac{τ − ξ_{1}}{2} , \sqrt{N}ξ_{2}, \frac{τ + ξ_{1}}{2} + N \frac{τ − ξ_{1}}{2} ).

This transformation sends the E_{m} to 1 × 1/\sqrt{N} × 1/N rectangles distributed in an essentially regular fashion along a large portion of the C/N neighbourhood of the upper unit light cone. Also the norms on the left and right-hand side of (55) scale in the same way. Thus (55) is just a consequence of the hypothesis (45), with N replaced by \sqrt{N}.

Proposition 5.4 represents a gain of \frac{1}{4} + τ over the N^{1/2} estimate (which would follow from the results of [20]). Despite the quite crude estimates used in the above argument, the sharp hypothesis of τ = 1/4 would give a sharp inequality, as can be seen by taking f to be a bump function.

Combining these two Propositions using Hölder’s inequality and the identity

\frac{1}{2} = (1 − \frac{κ}{2})\frac{1}{q} + \frac{κ}{2} \frac{1}{∞}

we thus obtain

Corollary 5.5. If (45) held for some τ ≥ 0, then (46) (and hence (45), by Lemma 5.2) holds with τ replaced by \frac{κ}{16} + \frac{κτ}{4} + ε.

The map

τ ↦ \frac{κ}{16} + \frac{κτ}{4}

is a contraction with fixed point τ = κ/(16−4κ). Thus by applying the above corollary a sufficiently large (but finite) number of times, starting with Mockenhaupt’s estimate [20], which gives τ = 0 for instance, we obtain the result.

6. Applications of the square function estimate

In this section we apply the square function estimate (45) just proven to various problems in harmonic analysis. The implications of this estimate are fairly well known, so this section will mainly consist of citing references and stating the exponents.

As observed in [3], the estimate (45) can be inserted directly as a replacement for estimate (1.1) in [20] to obtain an improved estimate on the Bochner-Riesz multiplier for the cone, defined by the multiplier

m^{α}(ξ, τ) = ϕ(τ)(1 − \frac{|ξ|}{τ})_{+}^{α},

where ϕ is a bump function on [1, 2] and α > 0. Indeed, we have

Proposition 6.1. [20, 3] If (45) holds for some τ, then the Bochner-Riesz operator of order α is bounded on L^{4} when α > 1/8 − τ/2.

Thus we have boundedness on L^{4} when α > 1/8 − 1/476. Note that the optimal exponent τ = 1/4 in (45) implies the optimal L^{4} (and L^{p}) result for the Bochner-Riesz operator.

Similarly, this estimate can be used to improve on the local smoothing estimate LS(4 → 4, 1/8 + ε) proven in [21]. Indeed, we have

Proposition 6.2. If (45) holds for some τ, then we have LS(4 → 4, 1/8−τ/2+ε).

We sketch the argument very briefly as follows. In the notation of [21], (45) becomes (ignoring negligible errors)

∥F f∥_{4} ≲ 2^{j( \frac{1}{8} − \frac{τ}{2} )}∥(∑_{ν} |F_{ν}f_{ν}|^{2})^{1/2}∥_{4}.

From the Kakeya estimate in [21] and duality we have

∥(∑_{ν} |F_{ν}f_{ν}|^{2})^{1/2}∥_{4} ≲ 2^{εj}∥(∑_{ν} |f_{ν}|^{2})^{1/2}∥_{4}.

Finally, from the square function estimates in [9] we have

∥(∑_{ν} |f_{ν}|^{2})^{1/2}∥_{4} ≲ ∥f∥_{4}.

Combining all the estimates gives the result.

In particular, we have smoothing for α > 1/8 − 1/476. As before, the optimal exponent τ = 1/4 in (45) would imply the optimal local smoothing conjecture.

Finally, we mention an observation of Oberlin, Smith, and Sogge [23, 24] regarding convolution with the helix in R^{3}:

T f(x) = ∫ f(x_{1} − cos t, x_{2} − sin t, x_{3} − t)ϕ(t) dt

where ϕ is a bump function. It is easy to see that T maps L^{4} to the Sobolev space L_{1/6}^{4}, and in [23] it was conjectured that this could be improved to L_{1/4}^{4}.

Proposition 6.3. [24] If (45) holds for some τ, then T maps L^{4} to L_{α}^{4} for all α < 1/6 + τ/3.

Note that the Fourier transform of the helix is concentrated near the light cone, so T is really a cone multiplier in disguise.

We thus have smoothing for the helix for α < 1/6 + 1/714. A sharp exponent for (45) would imply an essentially sharp result for the helix.

It is extremely likely that all these results extend to multipliers which are singular on surfaces similar to the cone. Based on the arguments of this paper, one sufficient condition seems to be that portions of the surface whose normals differ by 2^{−j} can be parabolically rescaled by an amount 1 × 2^{j} × 2^{2j} to be of disjoint conic type. This may not be the most general condition.

7. Appendix: some elementary harmonic analysis

In this section we state some elementary results which were used repeatedly in the paper.

We begin with a well-known quasi-orthogonality property of functions with disjoint frequency support. Define a rectangle to be the product of n (possibly halfinfinite or infinite) intervals in R^{n}.

Lemma 7.1. Let R_{k} be a collection of rectangles in frequency space such that the dilates 2R_{k} are almost disjoint, and suppose that f_{k} are a collection of functions whose Fourier transforms are supported on R_{k}. Then for all 1 ≤ p ≤ ∞ we have

(∑_{k} ∥f_{k}∥_{p}^{p^{∗}})^{1/p^{∗}} ≲ ∥ ∑_{k} f_{k}∥_{p} ≲ (∑_{k} ∥f_{k}∥_{p}^{p_{∗}})^{1/p_{∗}},

where p_{∗} = min(p, p′), p^{∗} = max(p, p′).

Proof Let P_{k} be a smooth Fourier multiplier adapted to 2R_{k} which equals 1 on R_{k}. We claim that

∥ ∑_{k} P_{k}F_{k}∥_{p} ≲ (∑_{k} ∥F_{k}∥_{p}^{p_{∗}})^{1/p_{∗}}

(∑_{k} ∥ ∑_{k} P_{k}F ∥_{p}^{p^{∗}})^{1/p^{∗}} ≲ ∥F ∥_{p}

for arbitrary functions F_{k}, F; the lemma then follows by setting F_{k} = P_{k}F_{k} = f_{k}, F = ∑_{k} F_{k}.

The latter estimate is dual to the former (with p replaced by p′), so it suffices to prove the former estimate. By interpolation it suffices to prove this estimate for p = 1, p = 2, and p = ∞. When p = 2 the estimate is immediate from Plancherel’s theorem. When p = 1 or p = ∞ the lemma follows from the triangle inequality and the estimates

∥P_{k}F_{k}∥_{1} ≲ ∥F_{k}∥_{1}, ∥P_{k}F_{k}∥_{∞} ≲ ∥F_{k}∥_{∞},

which follow from Young’s inequality and standard estimates on the kernel of P_{k}.

We also recall a pointwise estimate of Rubio de Francia, which is closely related to the standard square function estimate for equally-spaced cubes in frequency space [9], [7].

Lemma 7.2. [25] Let {I} be a collection of equally spaced cubes I, and let ϕ_{I}(ξ) = ϕ(ξ − ξ_{I}) be bump functions adapted to I, where ξ_{I} denotes the center of I. Then for any function f we have the pointwise estimate

(∑_{I} |ϕ̂_{I} ∗ f|^{2})^{1/2} ≤ C(ϕ)(M|f|^{2})^{1/2}

where M is the Hardy-Littlewood maximal function, and C(ϕ) depends only on the dimension n and finitely many of the derivatives of ϕ (after rescaling I to be of unit length).

We also give a standard variant of Sobolev’s inequality.

Lemma 7.3. Let p < q. If f is a function whose Fourier transform is supported on a rectangle Q, then ∥f∥_{q} ≲ |Q|^{1/p−1/q}∥f∥_{p}.

Proof We observe that the claim is invariant under affine transformations, so we may take Q to be the unit cube. In this case we may write f = f ∗ ψ where ψ is a Schwarz function whose Fourier transform equals one on the unit cube. The claim then follows from Young’s inequality.

By using the rapid decay of ψ in the above proof, we see that we may essentially localize this estimate to rectangles q dual to Q. More precisely, we have

∥f∥_{L^{q}(q)} ≲ ∑_{q′} c_{q,q′}|Q|^{1/p−1/q}∥f∥_{L^{p}(q′)}

where q′ ranges over a disjoint set of rectangles dual to Q, and c_{q,q′} is rapidly decreasing for q′ away from q.

Note that Hölder’s inequality provides a “dual” to the above lemma: If f is a function supported in a set E, then ∥f∥_{p} ≲ |E|^{1/p−1/q}∥f∥_{q}.

8. Acknowledgements

The first author was partially supported by NSF grant DMS-9706764. The second author was partially supported by the Spanish DGICYT (grant number PB97-0030) and the European Comission via the TMR network (Harmonic Analysis). We thank Sergiu Klainerman, Hart Smith, Andreas Seeger, and Chris Sogge for informing us of some of the applications mentioned above.

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Department of Mathematics, UCLA, Los Angeles, CA 90024

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

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid (Spain).

E-mail address: ana.vargas@uam.es