On the Bilinear Hilbert Transform

Michael T. Lacey^{1}

Abstract.

In joint work with C. Thiele, the author has shown that A. Calderón’s bilinear Hilbert transform extends to a bounded operator on certain products of L^{p} spaces. This article illustrates the method of proof by giving a complete proof of an inequality which is slightly weaker than the original conjecture of Calderón concerning this transform.

1991 Mathematics Subject Classification: 42A50

Keywords and Phrases: hilbert transform, time–frequency analysis

1 Introduction

This note discusses a recently developed theory for the bilinear Hilbert transform, defined by

Hfg(x) := lim_{ϵ→0} ∫_{|y|>ϵ} f(x + y)g(x − y) \frac{dy}{y} .

A conjecture of A. Calderón concerned possible extensions of this transform to a bounded operator on products of L^{p} spaces. In this regard, note that the term dy/y is dimensionless, so that any inequalities satisfied by Hfg are those of Hölder’s inequality. In collaboration with C. Thiele, the author has established

Theorem 1.1. H extends to a bounded operator on L^{p} × L^{q} into L^{r} if

1/p + 1/q = 1/r,  1 < p, q ≤ ∞,  2/3 < r < ∞.

In particular, H maps L^{2} × L^{2} into L^{1}, which is notable as the linear Hilbert transform does not preserve L^{1}. This was the form of Calderón’s origonal conjecture dating from 1964.

The interest in the Theorem lies in the method of proof, as it can be seen as an outgrowth and rëexamination of a group of sophisticated and subtle techniques invented first by L. Carleson [1] and later by C. Fefferman [3]—those in the celebrated proof of the pointwise convergence of Fourier series. The central point is to exploit orthogonality in a situation which is highly sensitive to the temporal and frequency aspects of the functions under consideration.

1The author has been supported by an NSF Grant, DMS—9706884.

Documenta Mathematica · Extra Volume ICM 1998 · II · 647–656

To explain a significant part of the method of proof, we limit our discussion to a single instance of the Theorem above, one that is free of some of the technicalities of the general case treated in [4, 5, 6, 7]. The next section provides background for the rest of the paper. Then we present the geometric–combinatorial model of the bilinear Hilbert transform and prove that it maps L^{2} × L^{2} into weak L^{1}.

I am grateful to L. Grafakos for taking the time to carefully read this manuscript and providing me with a long list of improvements.

2 Preliminaries

The Fourier transform is taken to be

Ff(ξ) = f̂(ξ) := ∫ e^{−2πixξ}f(x) dx.

We refer to x as the time variable and ξ as the frequency variable. The inner product will be denoted by ⟨f, g⟩ = ∫ fḡ dx.

The linear Hilbert transform is given by the principal value of convolution with 1/y.

Hf(x) := lim_{ϵ→0} \frac{1}{π} ∫_{|y|>ϵ} f(x − y) \frac{dy}{y} .

Context will distinguish linear and bilinear forms of the transform. We recall that

FHf(ξ) = Ff(ξ) lim_{ϵ→0} \frac{1}{π} ∫_{ϵ<|y|<1/ϵ} e^{−2πiyξ} \frac{dy}{y} = iFf(ξ) sign(ξ).

But F is unitary, so that H is bounded on L^{2}. This calculation also shows that the singularity of 1/y has the effect of distinguishing the origin in frequency—sign(ξ) has a jump discontinuity at 0.

Deeper properties of H are addressed by decomposing the non–local kernel 1/y into a sum of localized kernels. Fix a symmetric Schwartz function ψ which resolves the identity in that

∑_{j=−∞}^{∞} ψ(2^{j}y) ≡ 1,  y ≠ 0.  (2.2)

Then, let ψ_{j}(y) = y^{−1}ψ(2^{j}y), which is the same as ψ_{j}(y) = 2^{j}ψ_{1}(2^{j}y). Each ψ_{j} gives rise to an operation

H_{j}f(x) = ∫ f(x − y)ψ_{j}(y) dy.  (2.3)

This is a local operation in that ψ_{j} is no longer a singular kernel. In fact, |ψ_{j}(y)| ≤ C2^{j}, and it decays rapidly for |y| > 2^{−j}. Each H_{j} trivially maps L^{p} into itself for 1 ≤ p ≤ ∞. We say that H_{j} as scale 2^{j}.

Carleson’s Theorem can now be recalled in a form specific to our purposes. To prove the pointwise convergence of Fourier series, the maximum partial Fourier sums must be controlled. On the real line, this is equivalent to providing a control on the maximal operator

Cf(x) := sup_{λ}|H[e^{2πiλ·}f](x)|.

Carleson proved that C maps L^{2} into itself. See [1].

The aspect of C that distinguishes it from other operators of harmonic analysis is its invariance under conjugation of f by exponentials. Conjugation being dual to translation, we see that C has no distinguished point in frequency, which is a feature shared with the bilinear Hilbert transform. Denote, for the moment, f^{λ}(x) = e^{2πiλx}f(x). Then one readily sees that Hf^{λ}g^{λ}(x) = e^{4πiλx}Hfg(x). That is, Hfg commutes with conjugation, and so again it has no distinguished points in frequency. This suggests that the analysis of the two operators should be intimately related.

It also suggests that such an analysis cannot distinguish points in the frequency variable. We follow such a path. Each scale of the 1/y kernel requires a separate decomposition of f, g that is sensitive to both temporal and frequency aspects of the functions. It is convienent to introduce this with elements of the geometry of the phase plane.

Let φ be a Schwartz function on L^{2} norm one, with Fourier transform supported in [−\frac{1}{2}, \frac{1}{2}], so that

|φ̂(ξ)|^{2} + |φ̂(ξ + \frac{1}{2})|^{2} ≡ constant − \frac{1}{2} ≤ ξ < \frac{1}{2}.

Let s = I_{s} × ω_{s} be a rectangle of area 1 and set

φ_{s}(x) := e^{2πic(ω_{s})x}|I_{s}|^{−1/2}φ(\frac{s − c(I_{s})}{|I_{s}|}),  (2.4)

where c(J) denotes the center of the interval J. With this definition, I_{s} plays the role of the temporal variable and ω_{s} of the frequency. The initial set of rectangles we are interested in are

S_{1} := {(n, n + 1) × (m/2, m/2 + 1) | m, n ∈ ℤ},

which have scale 1. To make our considerations independent of scale, for each integer j, let S_{j} be the rectangles which are images of those in S_{1} under the area preserving dilation (x, ξ) → (2^{−j}x, 2^{j}ξ). And set I_{j}f = \frac{1}{2} ∑_{s∈S_{j}}⟨f, φ_{s}⟩φ_{s}. Each I_{j} is just a change of scale applied to I_{1}. And I_{1} is in fact the identity on L^{2}. See [2, Section 3.4.4] for a full discussion of this fact.

Now, for the transform, take a particular scale of the transform, as given in (2.3). We have

H_{j}fg = I_{j}[ H_{j}(I_{j}f, I_{j}g) ] = ∑_{s1,s2,s3∈S_{j}}⟨f, φ_{s1}⟩⟨g, φ_{s2}⟩⟨H_{j}φ_{s1}φ_{s2}, φ_{s3}⟩. (2.5)

The point underlying this expression is that the triple sum over S_{j} diagonalizes, and the “diagonal terms,” collected over 2^{j}, have useful combinatorial structures.

The significant aspect of the diagonalization takes place on the Fourier side, which is made explicit by choosing the kernel ψ in (2.2) so that Fψ is supported on (−10, −4] ∪ [4, 10). Recall that φ is supported on [−\frac{1}{2}, \frac{1}{2}] and therefore φ_{s} is supported on ω_{s}.

Then the final inner product in (2.5) is determined in part by the Fourier support of H_{j}φ_{s1}φ_{s2}. We claim that this function is zero unless ±(ω_{s1} − ω_{s2}) ∩ [4 · 2^{j}, 10 · 2^{j}) ≠ 0, and then the Fourier transform is supported in ω_{s1} + ω_{s2}. That is, the three frequency intervals ω_{sk} depend in fact on only one parameter. A formal calculation verifies the claim. Expand φ_{s1} in a dual variable τ_{1} and φ_{s2} in τ_{2}, then

H_{j}φ_{s1}φ_{s2}(x) = ∫∫ e^{2πi(τ_{1}+τ_{2})x}φ̂_{s1}(τ_{1})φ̂_{s2}(τ_{2})ψ̂(2^{−j}(τ_{1} − τ_{2})) dτ_{1}dτ_{2}.

In this expression, recall that Fφ_{s} is supported on ω_{s}. Note that τ_{1} + τ_{2} is the frequency variable for the left hand side, and then our claims follow.

The second diagonalization is of a more trivial nature. Recall that φ is a Schwartz function, so that φ_{s} is highly localized around I_{s}. In addition, ψ has rapid decay away from the origin. From these considerations, it follows that for all s1, s2, s3,

|⟨H_{j}φ_{s1}φ_{s2}, φ_{s3}⟩| ≤ C_{N}|I_{s1}|^{−1/2}[1 + |I_{s1}|^{−1} max_{j≠k} dist(I_{sj}, I_{sk})]^{−N}, N ≥ 1.

A final diagonalization procedure, which we do not present here, reduces the sum over j in (2.5) to a sum of “model sums” as defined in the next section. This is worth doing because the model sums can be analysized using only natural geometric–combinatorial considerations: There are no ad hoc features of the argument for the model sums.

3 Model Sums

In this section we state and prove a theorem which is general enough to prove the most important single inequality for the bilinear Hilbert transform. But this will require some definitions—we give them and illustrate with special cases which contain all of the difficulty of the general case.

A collection of intervals G is a grid if for all I, I′ ∈ G, we have I ∩ I′ equal to ∅, I or I′, and I ⊂_{≠} I′ implies 2|I| ≤ |I′|. The special cases of interest are collections of dyadic and triadic intervals.

A collection S of rectangles s = R_{s} × ρ_{s} are called tiles if for all s, s′ ∈ S we have |s| ≤ 4|s′|, and in addition, {R_{s} | s ∈ S} and {ρ_{s} | s ∈ S} are grids. For an example consider triadic rectangles s = R_{s} × ρ_{s} of area one. Each ρ_{s} is a union of three triadic intervals of equal length, ρ_{s1}, ρ_{s2}, ρ_{s3}. Then all of the rectangles R_{s} × ρ_{s}, R_{s} × ρ_{sj}, 1 ≤ j ≤ k form a collection of tiles.

This is most relevant, because of the connection to the decomposition in (2.5), and the diagonalization that was discussed there. For that reason, we make a further definition along these lines: A collection of tiles S are called tri–tiles if to each s = R_{s} × ρ_{s} ∈ S there are three tiles sj = R_{s} × ρ_{sj}, 1 ≤ j ≤ 3, with these properties. For all s ∈ S and all j, (a) ρ_{sj} ⊂ ρ_{s}, (b) ξ_{1} < ξ_{2} < ξ_{3}, for all ξ_{j} ∈ ρ_{sj}, and (c) {s, s1, s2, s3 | s ∈ S} is a collection of tiles.

The tri–tiles will describe the location of functions in the time–frequency plane. A collection of functions {ϕ_{sj} | s ∈ S, 1 ≤ j ≤ 3} are adapted to a collection of tri–tiles S if for all s ∈ S and all j, ∥ϕ_{sj}∥_{2} ≤ 1, there is an affine map α_{j} on ℝ so that Fϕ_{sj} is suported on α_{j}(ρ_{sj}), and we have

|ϕ_{sj}(x)| ≤ \frac{C_{0}}{\sqrt{|R_{s}|}}(1 + \frac{|x − c(I_{s})|}{|R_{s}|})^{−10},  (3.6)
⟨ϕ_{sj}, ϕ_{s′j}⟩ = 0 if I_{s} ≠ I_{s′}, ω_{sj} = ω_{s′j}.  (3.7)

One should compare these conditions to the definition of φ_{s} in (2.4).

The discrete combinatorial model of the bilinear Hilbert transform is

H^{S}f_{1}f_{2}(x) = ∑_{s∈S}|R_{s}|^{−1/2}ϕ_{s3}(x) ∏_{j=1}^{2}⟨f_{j}, ϕ_{sj}⟩.

Here, S is a set of tri–tiles and the ϕ_{sj} are adapted to S. Compare this sum to (2.5).

These operators obey the same inequalities as in Theorem 1.1. But for this exposition, we restrict our attention to that case which follows from purely L^{2} arguments, that is

Lemma 3.8. The operator H^{S} maps L^{2} × L^{2} into weak L^{1}. The norm of the operator depends only on the constant C_{0} in (3.6).

3.1 The Key Lemma

We state and prove the key Lemma in the proof of Lemma 3.8. But first we shall have to delineate the structures with which the Lemma must be stated. The tri– tiles admit a partial order. Thus we write s < s′ if ρ_{s} ⊃ ρ_{s′} and I_{s′} ⊃ I_{s}. We note that s and s′ intersect as rectangles if and only if they are comparable under ‘<.’

We say that a collection of tri–tiles T ⊂ S is a tree with top t if for every s ∈ T, s < t. (The top need not be in the tree and tops are not unique.) It is easy to see that this partial order does not admit a cycle, so our use of the phrase tree conforms to common usage. We note that any collection S of tri–tiles is a union of trees. Simply let S^{∗} denote those elements of S that are maximal under ‘<,’ and for each t ∈ S^{∗}, let T_{t} be the maximal tree in S with top t. S is the union of the T_{t}, t ∈ S^{∗}.

We refine the notion of a tree by saying that T is a j–tree if T is a tree with top t and ρ_{sj} ∩ ρ_{t} = ∅ for all s ≠ t ∈ T. Under this condition, the functions {ϕ_{sj} | s ∈ T} are orthogonal. If on the other hand we were to assume that ρ_{sk} ∩ ρ_{t} ≠ ∅ for all s ∈ T, it follows from the grid structure that T is a j–tree for j ≠ k. Now, for s ≠ t ∈ T, at most one of the three intervals ρ_{sk} can intersect ρ_{t}, so an arbitrary tree is a union of at most three subtrees which are j–trees for two choices of j.

Trees have important analytic properties. The time intervals in a tree refine that of the top. The frequency intervals increase in length, but only at the rate dictated by uncertainty. And all the frequency intervals contain that of the top, hence the trees have localized the frequency variables in the transform. For a k–tree set

Δ(T, k) := [|R_{t}|^{−1} ∑_{s∈T}|⟨f_{k}, ϕ_{sk}⟩|^{2}]^{1/2},

Here, we specifically include the case of k = 3, as well as k = 1, 2, for ultimately we will form the inner product of Hf_{1}f_{2} with a well–chosen third function f_{3}. For an arbitrary collection of tiles S, we set

k–size(S) = sup{Δ(T, k) : T ⊂ S is a k–tree.}.

We note that a singleton T = {t} is a k–tree for all k. So k–size(S) dominates the terms |R_{s}|^{−1/2}|⟨f_{k}, ϕ_{sk}⟩| for all s ∈ S.

It should be noted, for we will relie upon this fact latter, that for any collection of tri–tiles S, we have k–size(S) ≤ K∥f_{k}∥_{∞}.

Now, fix k = 1, 2 or 3 and let T ⊂ S be a k–tree for j ≠ k and let t denote the top of the tree. Then, by applying Cauchy–Schwarz,

|⟨H^{T}f_{1}f_{2}, f_{3}⟩| ≤ ∑_{s∈T} \frac{|⟨f_{j}, ϕ_{sj}⟩|}{\sqrt{|R_{s}|}} ∏_{j≠k}|⟨f_{k}, ϕ_{sk}⟩| ≤ |R_{t}| ∏_{j=1}^{3} k–size(S).  (3.9)

This is the central estimate on a tree.

Lemma 3.10. Let k = 1, 2 or 3. Let f_{k} be a Schwartz function and S any collection of tri–tiles. Then S is a union of S_{1} and S_{2} which have these two properties. Let S_{1}^{∗} denote the elements in S_{1} that are maximal under <. We have

∑_{t∈S_{1}^{∗}}|R_{t}| ≤ C_{1}k–size(S)^{−2}∥f_{k}∥_{2}^{2},  (3.11)
k–size(S_{2}) ≤ \frac{1}{2}[k–size(S)].  (3.12)

The constant C_{1} depends only on C_{0} in (3.6).

Note that the second collection S_{2} is better in that it has smaller size. But we have controlled the number of trees that are in S_{1} the first collection—this is the critical condition, the one that orthogonality gives us.

For the proof of the Lemma it suffices to consider the case of k–size(S) = 1. We construct S_{1} in two similar steps, with the construction being motivated by the particulars of the issues of orthogonality with which we conclude this argument.

It is required to distinguish between two types of k–trees. A k–tree T with top t will be called a left tree if for all s ≠ t ∈ T, ρ_{sk} lies to the left of ρ_{t}. Note that for s, s′ in a left tree T, the relation s < s′ implies that ρ_{sk} lies to the left of ρ_{s′k}. A right tree has a corresponding definition.

We construct S_{1ℓ} ⊂ S as a union of trees T̃_{l} with tops t(l), for integers l ≥ 1. Each T̃_{l} will be associated to a left tree T_{l}. The construction is inductive. We take T_{1} ⊂ S^{1} to be a left tree which satisfies several conditions. First, Δ(T_{1}, k) ≥ 1/4. Second, T_{1} is maximal with respect to inclusion. Third, the top t(1) is to be maximal with respect to ‘<.’ Finally, ρ_{t(1)} is to be left most—that is min{ξ | ξ ∈ ρ_{t(1)}} is minimal among the maximal tops. After selecting T_{1}, we take T̃_{1} to be the maximal tree with top t(1) in S. We remove T̃_{1} from S and repeat this procedure until there is no left tree T_{l} ⊂ S meeting these criteria. The union of the T̃_{l} is then S_{1ℓ}.

Under this construction, it is obvious that Δ(T, k) < 1/4 for all left trees T ⊂ S − S_{1ℓ}. Moreover—and this is the essential combinatorial observation—the T_{l} satisfy this disjointness condition.

If s ∈ T_{l}, s′ ∈ T_{l′} and ρ_{sk} ⊂_{≠} ρ_{s′k} then R_{t(l)} ∩ R_{s′} = ∅.  (3.13)

Indeed, the grid structure implies that ρ_{t(l)} ⊂ ρ_{s} ⊂ ρ_{sk′}, and so ρ_{t(l)} lies to the left of ρ_{t(l′)}. Thus, the tree T_{l} was constructed first. But then s′ ∉ T̃_{l}, so that we must have s′ ≮ t(l), which is to say R_{t(l)} ∩ R_{s′} = ∅.

We verify that S_{1ℓ} satisfies (3.11) momentarily.

We finally construct S_{1r} as a union of right trees in S − S_{1ℓ}, with the obvious changes in the argument above. We conclude that this collection satisfies (3.11) as well. Then S_{1} is the union S_{1ℓ} ∪ S_{1r} and S_{2} := S − S_{1}. For any left or right tree T ⊂ S_{2}, we have Δ(T, k) < 1/4 so that (3.12) follows.

We have still to establish (3.11), for the collections S_{1ℓ} and S_{1r} defined above. This is the point that orthogonality enters the proof, and in fact we need only consider S_{1ℓ}. For the purposes of this argument, we suppose that S = ⋃_{l} T_{l}, as constructed above. The properties of the T_{l} that we need are

Δ(T_{l}, k) ≥ 1/4 for all l,

\frac{|⟨f_{k}, ϕ_{sk}⟩|}{\sqrt{|R_{s}|}} ≤ 1 for all s ∈ S,

and the trees satisfy the disjointness condition (3.13). We demonstrate the inequality

∑_{l=1}^{∞}|R_{t(l)}| ≤ K∥f_{k}∥_{2}^{2}.  (3.14)

For the proof of (3.14), it suffices to assume that S is finite, so that the number below is finite.

B := ∥∑_{s∈S} ⟨f_{k}, ϕ_{sk}⟩ϕ_{sk}∥_{2}.

We show that B ≤ K∥f_{k}∥_{2}, which proves the Lemma as the next inequality shows.

∑_{l=1}^{∞}|R_{t(l)}| ≤ ∑_{s∈S}|⟨f_{k}, ϕ_{sk}⟩|^{2}
= ⟨f_{k}, ∑_{s∈S}⟨f_{k}, ϕ_{sk}⟩ϕ_{sk}⟩
≤ ∥f_{k}∥_{2}B.

Note that we are exploiting the self–dual nature of the problem.

Now, we expand B^{2}, to get a diagonal term D and off–diagonal term O. The diagonal term is D = ∑_{s∈S}|⟨f_{k}, ϕ_{sk}⟩|^{2}, which is no more than B∥f_{k}∥_{2} as we have just seen. The off–diagonal term is O := 2 ∑_{s∈S}|⟨f_{k}, ϕ_{sk}⟩|O(s), where

O(s) := ∑_{s′∈S(s)}|⟨ϕ_{sk}, ϕ_{s′k}⟩||⟨f_{k}, ϕ_{s′k}⟩|,

and we use the notation S(s) := {s′ ∈ S | ρ_{sk} ⊂_{≠} ρ_{s′k}}. This is justified by (3.7) and the fact that the Fourier transform of ϕ_{sk} is supported on an affine image of ρ_{sk}. Hence the inner product of ϕ_{sk} and ϕ_{s′k} is zero unless ρ_{sk} and ρ_{s′k} intersect. But then we can assume that one interval is contained in another due to the grid structure.

Note that by Cauchy–Schwarz, O ≤ 2B^{1/2}[∑_{s∈S} O(s)^{2}]^{1/2} and we claim that for each tree T_{l}

∑_{s∈T_{l}} O(s)^{2} ≤ K|R_{t(l)}|.  (3.15)

This will complete the proof of the Lemma, for we will then have B^{2} ≤ KB∥f∥_{2}.

To see the claim, fix a tree top t(l) and consider s ∈ T_{l}. We observe that (3.6) implies that

|⟨ϕ_{sk}, ϕ_{s′k}⟩| ≤ K \begin{matrix}√ & \overline{|R_{s′}|} \\ √ & \overline{|R_{s}|} \end{matrix}(1 + dist(R_{s}, R_{s′})|R_{s}|^{−1})^{−5}.

A detailed proof of the estimate is left to the reader, but note that the right hand side is only slightly bigger than ∥ϕ_{s′k}∥_{L^{1}(R_{s′})}∥ϕ_{sk}∥_{L^{∞}(R_{s′})}. But also note that (3.13) implies that every s′ ∈ S(s) must satisfy R_{t(l)} ∩ R_{s′} = ∅. Hence,

O(s) ≤ K|R_{s}|^{−1/2} ∑_{s′∈S(s)}(1 + dist(R_{s}, R_{s′})|R_{s}|^{−1})^{−5}|R_{s}|

≤ K|R_{s}|^{1/2} ∫_{R_{t(l)}^{c}}(1 + dist(R_{s}, x)|R_{s}|^{−1})^{−5} \frac{dx}{|R_{s}|}
≤ K|R_{s}|^{1/2}(1 + dist(R_{s}, R_{t(l)}^{c})|R_{s}|^{−1})^{−4}.

But the tree structure imposes restrictions on the intervals R_{s}: They are in relation to R_{t(l)} as the dyadic intervals [j2^{−n}, (j + 1)2^{−n}], 0 ≤ j < 2^{n} are to [0, 1]. Thus, one sees that (3.15) holds. The proof is done.

3.2 Application of the Key Lemma

The Key Lemma, together with some considerations of a more familiar nature, give Lemma 3.8. It suffices to prove the inequality

|{x | H^{S}f_{1}f_{2}(x) ≥ 2}| ≤ K,  (3.16)

for all f_{1}, f_{2} of L^{2} norm one and any collection of tri–tiles S. K depends only on C_{0} in (3.6). This is due to an invariance of the model operators under dilation. As this is a commonplace reduction, we omit the easy argument to pass from the inequality above to Lemma 3.8.

As f_{1} and f_{2} are Schwartz functions, S has finite 1 and 2 size, say no more than 2^{−n_{0}}. If n_{0} ≥ 0, there is nothing for us to do at this point. So assume that n_{0} < 0. We iteratively apply Lemma 3.10 in the following fashion. Apply Lemma 3.10 for those k = 1, 2 for which k–size(S) ≥ 2^{−n_{0}−1}. We see that S = S_{n_{0}} ∪ S^{n_{0}+1} where

∑_{t∈S_{n_{0}}^{∗}}|R_{t}| ≤ C2^{2n_{0}} and k–size(S^{n_{0}+1}) ≤ 2^{−n_{0}−1}, k = 1, 2.

The point that distinguishes the case n_{0} < 0 is that the sum of the tops of the trees is less than a constant.

Continuing this procedure, we may write S = ⋃_{n=−∞}^{0} S_{n}, where k–size(S_{n}) ≤ 2^{−n} for all −∞ < n ≤ 0 and k = 1, 2. And for n strictly less than zero,

∑_{t∈S_{n}^{∗}}|R_{t}| ≤ C2^{2n}, n < 0.

There is no such control for S_{0}, and we shall return to this collection momentarily. Now for a fixed tree T with top t in S_{n}, away from the interval 2R_{t}, we have

|H^{T}f_{1}f_{2}(x)| ≤ sup_{s∈T} ∏_{1}^{2} \frac{|⟨f_{j}, ϕ_{sj}⟩|}{\sqrt{|R_{s}|}} ∑_{s∈T} \sqrt{|R_{s}|}|ϕ_{s3}(x)|

≤ K2^{−2n}(1 + \frac{dist(x, R_{t})}{|R_{t}|})^{−10}, if x ∉ 2R_{t},

as (3.6) and the tree structure easily imply. Thus, if we set E_{n} = ⋃_{t∈S_{n}^{∗}} 2^{−n}R_{t}, then we have |E_{n}| ≤ K2^{n}. Thus, the union of these sets has bounded measure, so we need only estimate H^{S_{n}} off of the set E_{n}, but then

∥H^{S_{n}}f_{1}f_{2}∥_{L^{1}(E_{n}^{c})} ≤ K2^{3n}.

Consequently, the collection S^{0} = ⋃_{n=−∞}^{−1} S_{n} satisfies (3.16).

Therefore, we can assume that the 1 and 2 size of S is at most 1. This is the case in which the Key Lemma is decisive. Set G := {x | H^{S}f_{1}f_{2}(x) > 1/2}, which is a set of finite measure as we take f_{1} and f_{2} to be Schwartz functions. We can assume that |G| > K′ for otherwise there is nothing to prove. Then take f_{3} to be a Schwartz functions approximating |G|^{−1/2}𝕀_{G} so well that

|G|^{1/2} ≤ 2|⟨H^{S}f_{1}f_{2}, f_{3}⟩|,

but in addition |f_{3}(x)| ≤ 2|G|^{−1/2} for all x. It follows that the 3–size(S) can be assumed to be at most one, provided K′ is large enough.

By applying Lemma 3.10 inductively, we can write S as ⋃_{n=0}^{∞} S_{n} where each S_{n} has k–size at most 2^{−n}, for 1 ≤ k ≤ 3, and

∑_{t∈S_{n}^{∗}}|R_{t}| ≤ C2^{2n}.

But then by (3.9) it follows that

|⟨H^{S_{n}}f_{1}f_{2}, f_{3}⟩| ≤ K2^{−3n} ∑_{t∈S_{n}^{∗}}|R_{t}| ≤ K2^{−n}.

And this is summable over n > 0, from which we conclude that |G|^{1/2} ≤ K. The proof is done.

References

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[2] I. Daubechies. Ten Lectures on Wavelets. CBMS–NSF regional conference series in applied mathematics. SIAM (1992) Philadelphia.

[3] C. Fefferman. “Pointwise convergence of Fourier series.” Ann. Math. 98 (1973) 551–571.

[4] M.T. Lacey, C.M. Thiele. “Bounds for the bilinear Hilbert transform on L^{p}.” Proc. Nat. Acad. Sci. 94 (1997) 33—35.

[5] M.T. Lacey, C.M. Thiele. “On Calderòn’s Conjecture.” To appear in Ann. Math.

[6] M.T. Lacey, C.M. Thiele. “On Calderòn’s Conjecture for the bilinear Hilbert transform.” Proc. Nat. Acad. Sci. 95 (1998) 4828–4830.

[7] M.T. Lacey, C.M. Thiele. “L^{p} Bounds for the bilinear Hilbert transform, p > 2.” Ann. Math. 146 (1997) 693—724.

Michael T. Lacey School of Mathematics Georgia Institute of Technology Atlanta GA 30332 USA

                     email: lacey@math.gatech.edu
                     web: www.math.gatech.edu/~lacey