A BILINEAR HILBERT TRANSFORM ESTIMATE

TERENCE TAO

Abstract. We prove the simplest boundedness estimate on the bilinear Hilbert transform, namely that it maps L^{∞} × L^{∞} → L^{1} on compacta.

1. Introduction

Let H(f_{1}, f_{2}) denote the bilinear Hilbert transform

H(f_{1}, f_{2})(x) = p.v. ∫ f_{1}(x + t)f_{2}(x − t) \frac{dt}{t}.  (1)

We shall show the estimate

∥H(f_{1}, f_{2})∥_{L^{1}([0,1])} ≲ ∥f_{1}∥_{L^{∞}([0,1])}∥f_{2}∥_{L^{∞}([0,1])}  (2)

for all test functions f_{1}, f_{2} supported on [0, 1]. This is certainly not the strongest estimate that the bilinear Hilbert transform satisfies, but it is one of the simplest to show. Our arguments shall be based on those in [2].

The bilinear Hilbert transform has two remarkable exact localization properties, one in time and the other in frequency. The time localization property states that if f_{1} is supported at x_{1} and f_{2} is supported at x_{2}, then H(f_{1}, f_{2}) is supported at (x_{1} + x_{2})/2. The frequency localization property states that if f̂_{1} is supported at ξ_{1} and f̂_{2} is supported at ξ_{2}, then (an appropriate interpretation of) { \atop H(f_{1}, f_{2})} is supported at ξ_{1} + ξ_{2}.

The frequency localization properties are most evident in the Fourier multiplier description of H(f_{1}, f_{2}), namely

{ \atop H(f_{1}, f_{2})(ξ)} = C ∫ δ(ξ − ξ_{1} − ξ_{2})sgn(ξ_{2} − ξ_{1})f̂_{1}(ξ_{1})f̂_{2}(ξ_{2}) dξ_{1}dξ_{2}.  (3)

There is a further localization property in phase space arising from the smoothness properties of the kernel (or multiplier). Informally, suppose f_{i} is supported near x_{i} and has frequency about ξ_{i} for i = 1, 2. Then the principle of stationary phase suggests that there will be some cancellation in (1) (or (3)) unless

|x_{1} − x_{2}||ξ_{1} − ξ_{2}| ≲ 1.  (4)

To make this more precise we need a dyadic decomposition of either the first or second factor in (4).

To decompose the first factor we would split the operator into pieces for which |x_{1}−x_{2}| ∼ 2^{−k}. This would correspond to considering pieces of H(f_{1}, f_{2}) such as

∫ f_{1}(x + t)f_{2}(x − t)2^{k}ψ(2^{k}t) dt

where ψ is some odd bump function supported on {t : |t| ∼ 1}. However, although this decomposition is intuitive and gives excellent spatial localization properties, its frequency localization properties are not ideal. As it turns out, it is preferable to decompose H(f_{1}, f_{2}) in frequency space rather than physical space, mainly because Littlewood-Paley theory is technically easier when performed in the frequency domain. In other words, we shall decompose H(f_{1}, f_{2}) into pieces for which |ξ_{1} − ξ_{2}| ∼ 2^{k}; this will then localize space imperfectly to the region |x_{1} − x_{2}|. One can then see hints of Littlewood-Paley theory appear, for if one fixes ξ_{1} (for instance), then the contributions of the different scales k use different lacunary frequency ranges^{1} of the g frequency variable ξ_{2}.

Of course, there is no reason why the frequency ξ_{1} should be fixed. In practice there would be contributions to H(f, g) from many frequencies ξ_{1} and many times x_{1}, and around each such frequency and location we would have a “tree” of scales k for which one had Littlewood-Paley type behaviour in the g variable. The trick is then to pick these trees carefully so that they are “orthogonal” in some sense, so that one can apply Littlewood-Paley theory on the entire collection of trees at once rather than on just one tree at a time. This will allow us to estimate the bilinear Hilbert transform effectively.

We now turn to the details. The first step is to use duality to convert (2) from a bilinear operator estimate to a trilinear form estimate. Not only does this make the estimate more symmetric, it will introduce a third function f_{3} which will be needed later on. Basically, we shall need the phase space portraits of all three functions f_{1}, f_{2}, f_{3} in order to decide how to cut up the phase plane into appropriate trees.

Let Λ denote the trilinear form

Λ(f_{1}, f_{2}, f_{3}) = ∫ ∫ f_{1}(x + t)f_{2}(x − t)f_{3}(x) dx\frac{dt}{t}.

By duality, (2) is equivalent to showing the more symmetric estimate

|Λ(f_{1}, f_{2}, f_{3})| ≲ 1  (5)

for all test functions f_{1}, f_{2}, f_{3} on [0, 1] such that ∥f_{i}∥_{∞} ≲ 1.

By the Fourier inversion formula, we can rewrite Λ as

Λ(f_{1}, f_{2}, f_{3}) = C ∫_{IR_{3}} δ(ξ_{1} + ξ_{2} + ξ_{3})f̂_{1}(ξ_{1})f̂_{2}(ξ_{2})f̂_{3}(ξ_{3})sgn(ξ_{1} − ξ_{2}) dξ.

This multiplier is supported on the plane {(ξ_{1}, ξ_{2}, ξ_{3}) : ξ_{1} + ξ_{2} + ξ_{3} = 0} and is singular on the line

l = {(t, t, −2t) : t ∈ IR}.

By a Whitney decomposition we may cover the set IR^{3}\l by dyadic cubes w = w_{1} × w_{2} × w_{3} such that

100|w_{i}| ≤ dist(w, l) ≤ 1000|w_{i}|,  (6)

for i = 1, 2, 3. We can then approximate Λ by

∑_{w} ∫ δ(ξ_{1} + ξ_{2} + ξ_{3})f̂_{1}(ξ_{1})f̂_{2}(ξ_{2})f̂_{3}(ξ_{3})m_{w}(ξ) dξ  (7)

1Of course, these frequency ranges are centered around ξ_{1} rather than around 0 as is customary in standard Littlewood-Paley theory. Indeed, the 0 frequency plays no distinguished role in the theory of the bilinear Hilbert transform.

where w ranges over all such cubes and m is a bump function adapted to \frac{1}{2}w. Λ cannot be decomposed in this way, but it can be written as a finite linear combination of expressions which are of this form for some (possibly translated and/or dilated) dyadic mesh. It thus suffices to show that (7) is bounded^{2}.

We may of course restrict the cubes w to those satisfying

w ∩ {(ξ_{1}, ξ_{2}, ξ_{3}) : ξ_{1} + ξ_{2} + ξ_{3} = 0} ≠ 0.  (8)

For each dyadic interval w_{i}, let Δ_{w_{i}} be a Fourier multiplier whose symbol is a bump function supported on w_{i} which equals 1 on \frac{1}{2}w_{i}. By the Fourier inversion formula, we may rewrite (7) as

∑_{w} ∫ ∫ ∏_{i=1}^{3}(Δ_{w_{i}}f_{i})(x_{i} + t)m̂_{w}(x) dxdt.

Define a tile to be any dyadic rectangle P = I_{P} × w_{P} with area 1, and define the tile semi-norms ∥f∥_{P} by

∥f∥_{P} = |I_{P} |^{−1} ∫ |Δ_{w_{P}} f(x)|(1 + \frac{dist(x, I_{P})}{|I_{P} |})^{−100} dx.  (9)

One can view ∥f∥_{P} as the average value of f on the time-frequency tile P.

Define a tri-tile to be a triplet P⃗ = (P_{1}, P_{2}, P_{3}) of tiles with I_{P_{1}} = I_{P_{2}} = I_{P_{3}} = IP⃗ and wP⃗ = w_{P_{1}} × w_{P_{2}} × w_{P_{3}} satisfying (6) and (8). Whenever P⃗ appears in an expression, we understand P_{1}, P_{2}, P_{3} to be the components of P⃗.

From the rapid decay of m̂_{w} we have the estimate

| ∫ ∫ ∏_{i=1}^{3}(Δ_{w_{P_{i}}}f_{i})(x_{i} + t)m̂_{w}(x) dxdt| ≲ ∑_{P⃗:wP⃗ =w} |IP⃗ |∥f_{1}∥_{P_{1}}∥f_{2}∥_{P_{2}}∥f_{3}∥_{P_{3}}

for all w. We can thus reduce ourselves to showing that

∑_{P⃗} |IP⃗ |∥f_{1}∥_{P_{1}}∥f_{2}∥_{P_{2}}∥f_{3}∥_{P_{3}} ≲ 1  (10)

where P⃗ ranges over all tri-tiles whose frequency cube wP⃗ satisfy (7) and (8).

The idea is to use the phase space portrait of f_{1}, f_{2}, f_{3} to decompose the collection of all tri-tiles P⃗ into more manageable pieces on which we can control the various pieces of the summand in (10). This appears to be the only successful way to control the genuinely multi-linear expression (10); more traditional strategies of keeping the operator fixed (or decomposing it in a predetermined manner, e.g. by dyadic decomposition) and decomposing the functions f_{1}, f_{2}, f_{3} do not appear to be flexible enough to achieve a satisfactory decomposition.

Since f_{1}, f_{2}, f_{3} are supported on [0, 1], we shall restrict our attention to those tiles P⃗ whose spatial interval IP⃗ is contained in [−C, C]. The contribution of the remaining tiles is treated similarly, but the estimates improve rapidly as one moves away from [0, 1], and summing up is quite easy.

2Note that this strategy will easily extend to more general multipliers than sgn(ξ_{1} − ξ_{2}).

In order to prove (10) we have to do two things. Firstly, we have to obtain good estimates on the individual tile semi-norms ∥f_{i}∥_{P}. Secondly, we have to work out how the three components P_{1}, P_{2}, P_{3} of a tri-tile P⃗ are related to each other.

To achieve the former task, we need the following lemma, which is unfortunately not perfectly correct, but is morally true. Later on we shall indicate substitutes for this lemma which are rigorously true and how to use them in the ensuing argument. This estimate can be viewed as a generalized L^{2} Littlewood-Paley estimate.

Lemma 1.1. (Informal) Let i = 1, 2, 3, and let P be a collection of tiles which are finitely overlapping in the time-frequency plane, and such that all spatial intervals I_{P} are contained in a single interval I_{0}. Then we have the Bessel inequality

∑_{P} |I_{P} |∥f_{i}∥_{P}^{2} ≲ |I_{0}|.  (11)

Proof (Informal) It suffices to show that

∑_{P} |I_{P} |^{1/2}c_{P} ∥f_{i}∥_{P} ≲ |I_{0}|^{1/2}

for all l^{2} sequences c_{P}. By the converse to Hölder’s inequality we may write

|I_{P} |^{1/2}∥f_{i}∥_{P} = ⟨f_{i}, Δ_{w_{P}} ψ_{P} ⟩

where ψ_{P} is a function satisfying the pointwise estimate

|ψ_{P} (x)| ≲ |I_{P} |^{−1/2}(1 + \frac{dist(x, I_{P})}{|I_{P} |})^{−100}.  (12)

It thus suffices to show that

|⟨f_{i}, ∑_{P} c_{P} Δ_{w_{P}} ψ_{P} ⟩| ≲ |I_{0}|^{1/2}.

The second term in the inner product is morally supported on I_{0}. Since f_{i} is in L^{∞}, it morally suffices by Cauchy-Schwarz to show that

∥ ∑_{P} c_{P} Δ_{w_{P}} ψ_{P} ∥_{2} ≲ 1.

By (12), each Δ_{w_{P}} ψ_{P} is in L^{2} uniformly in P. It would thus suffice to show that the Δ_{w_{P}} ψ_{P} are orthogonal whenever the tiles P are disjoint. If the P had disjoint frequency intervals then this is clear from the orthogonality of the Δ_{w_{P}}. If the P had disjoint spatial intervals then we do not quite have orthogonality because the Δ_{w_{P}} ψ_{P} have something of a tail outside of I_{P}; however if we pretend that these functions are perfectly supported in I_{P} then we do get true orthogonality whenever two tiles P are disjoint (either spatially or in frequency), and the claim follows.

One may hope to make this lemma rigorous using the Cotlar-Knapp-Stein lemma, but unfortunately this will not work as we have a three-parameter set of tiles rather than a one-parameter set, and the lemma as stated is actually false. However, substitutes exist; see below.

Now we need to understand how the triplets (P_{1}, P_{2}, P_{3}) are arranged with respect to each other. We need some definitions.

Definition 1.2. Let P and P′ be tiles. We write P′ < P if I_{P′} ⊊ I_{P} and w_{P} ⊆ 3w_{P′}, and P′ ≤ P if P′ < P or P′ = P. We write P′ ≲ P if I_{P′} ⊆ I_{P} and w_{P} ⊆ 1000w_{P′}. We write P′ ≲′ P if P′ ≲ P and P′ ≰ P.

The point of these definitions is that tiles P′ which satisfy P′ ≤ P are clustered very close to P, whereas tiles P′ satisfying P′ ≲′ P are separated from P in a lacunary fashion.

The crucial geometric fact about our collection of tri-tiles is

Lemma 1.3. Let P⃗ = (P_{1}, P_{2}, P_{3}) and P⃗′ = (P′_{1}, P′_{2}, P′_{3}) be tri-tiles such that P′_{1} ≤ P_{1}. Then P′_{2} ≲ P_{2} and P′_{3} ≲ P_{3}. If we also assume that |P′_{1}| ≪ |P_{1}|, then we may improve the previous statements to P′_{2} ≲′ P_{2} and P′_{3} ≲′ P_{3}. Similar statements hold with the roles of 1, 2, 3 permuted.

Proof This is clear from (6), (8), the above definitions, and some linear algebra.

In other words, once we anchor one of the tiles P_{1}, the other tiles P_{2}, P_{3} behave in an essentially lacunary manner. In particular, P_{1} determines P_{2} and P_{3} up to a finite factor. Similarly for permutations of 1,2,3.

These two lemmas are all we will need to prove (10). To illustrate the technique, we first prove a toy version of (10).

Definition 1.4. For any collection P⃗ of tri-tiles and i = 1, 2, 3, we define the i-size size_{i}(P⃗) of P⃗ to be

size_{i}(P⃗) = sup_{P_{i}^{0}}(|I_{P_{i}^{0}}|^{−1/2} ∑_{P⃗ ∈P⃗:P_{i}≲′P_{i}^{0}} |IP⃗ |∥f_{i}∥_{P_{i}}^{2})^{1/2}) + sup_{P⃗ ∈P⃗} ∥f_{i}∥_{P_{i}}  (13)

where P_{i}^{0} range over all tiles.

From (11) we see that size_{i}(P⃗) ≲ 1 for all collections P⃗ of tri-tiles and all i = 1, 2, 3. Define a tree T of type i and top P⃗_{T} to be a collection of tri-tiles {P⃗ } such that P_{i} ≤ P_{T,i}

for all P⃗ in the tree. We write I_{T} for IP⃗_{T}.

Proposition 1.5. For any tree T, we have

∑_{P⃗ ∈T} |IP⃗ |∥f_{1}∥_{P_{1}}∥f_{2}∥_{P_{2}}∥f_{3}∥_{P_{3}} ≲ |I_{T} |size_{1}(T)size_{2}(T)size_{3}(T).  (14)

Proof We may assume that T is a 1-tree. By (13) we have ∥f_{i}∥_{P_{i}} ≲ size_{i}(T). This allows us to deal with the contribution of tri-tiles T for which |IP⃗ | ∼ |I_{T} |, since by Lemma 1.3 there are only O(1) such tiles.

It remains to deal with the tiles for which |IP⃗ | ≪ |I_{T} |. For these tiles, we majorize (14) by

size_{1}(T) ∑_{P⃗ ∈T:|IP⃗ |≪|I_{T} |} |IP⃗ |∥f_{2}∥_{P_{2}}∥f_{3}∥_{P_{3}}.

The claim then follows from Cauchy-Schwarz, (11), and the observation from (1.3) that the P_{2} and the P_{3} are both essentially disjoint.

The estimate (14) allows us to control the contribution to (10) coming from a single tree. To control all of (10), we must therefore decompose the collection P⃗ of all tri-tiles into trees T whose size and spatial extent |I_{T} | are under control.

More precisely, we shall prove the following combinatorial consequence of (11), allowing us to use the phase space portrait of f_{i} to cut up P⃗ into trees of controlled i-size and spatial extent |I_{T} |:

Proposition 1.6. Let n ∈ Z, i = 1, 2, 3, and P⃗ be a collection of tri-tiles such that size_{i}(P⃗) ≤ 2^{−n}. Then there exists a collection T of trees in P⃗ such that

∑_{T ∈T} |I_{T} | ≲ 2^{2n}  (15)

and

size_{i}(P⃗\ ⋃_{T ∈T} T) ≤ 2^{−n−1}.  (16)

Assuming this proposition for the moment, let us complete the proof of (10).

Let P⃗ be the set of all tri-tiles. From (11) we have size_{i}(P⃗) ≲ 1. By applying the above proposition repeatedly, we can decompose

P⃗ = ⋃_{n=−C}^{∞} P⃗_{n,i}

for each i = 1, 2, 3, where size_{i}(P⃗_{n,i}) ≤ 2^{−n} and each P⃗_{n,i} is the union of a collection T_{n,i} of trees satisfying (15). We can then split (10) as

∑_{n_{1},n_{2},n_{3}≥−C} ∑_{P⃗ ∈P⃗_{n_{1},1}∩P⃗_{n_{2},2}capP⃗_{n_{3},3}} |IP⃗ |∥f_{1}∥_{P_{1}}∥f_{2}∥_{P_{2}}∥f_{3}∥_{P_{3}}.

By symmetry we may restrict ourselves to the contribution when n_{1} ≤ n_{2} ≤ n_{3}. In this case we cut up the inner sum using the trees in T_{n_{1},1}. By (14) the contribution of each such tree T to the inner sum is O(|I_{T} |2^{−n_{1}}2^{−n_{2}}2^{−n_{3}}), since

size_{i}(T ∩ P⃗_{n_{1},1} ∩ P⃗_{n_{2},2} ∩ P⃗_{n_{3},3}) ≤ size_{i}(P⃗_{n_{i},i}) ≤ 2^{−n_{i}}.

By (15) we can thus estimate the n_{1} ≥ n_{2} ≥ n_{3} portion of (10) by

C ∑_{−C≤n_{1}≤n_{2}≤n_{3}} 2^{2n_{1}}2^{−n_{1}}2^{−n_{2}}2^{−n_{3}}.

This sum converges, so we are done.

It remains to prove Proposition 1.6, which is the only portion of the argument which really uses the combinatorics of tile intersections. Note that that this proposition only involves the function f_{i}, and is independent of the other two functions. The idea is to keep removing trees from P⃗ until (16) is satisfied.

Let P⃗^{∗} consist of those tri-tiles P⃗ in P⃗ such that

∥f_{i}∥_{P_{i}} ≥ 2^{−n−2};

for these tiles we thus have

∥f_{i}∥_{P_{i}} ∼ 2^{−n}  (17)

by the size hypothesis. Let P⃗^{∗∗} be those tri-tiles in P⃗^{∗} for which P_{i} is maximal with respect to the ordering <.

By construction, the tiles {P_{i} : P⃗ ∈ P⃗^{∗∗}} are disjoint. From this and (11) (with I_{0} = [−C, C]) we see that

∑_{P⃗ ∈P⃗^{∗∗}} |IP⃗ | ≲ 2^{2n}.  (18)

For each P⃗ ∈ P⃗^{∗∗} we associate the i-tree

T = {P⃗′ ∈ P⃗^{∗} : P′_{i} ≤ P_{i}}.

From (18) we see that one can remove these trees T from P⃗ and place them into T while respecting (15). After removing these trees, we have eliminated all elements of P⃗^{∗}, so that we have

∥f_{i}∥_{P_{i}} < 2^{−n−2}  (19)

for all remaining multi-tiles P⃗.

If P is a tile, let ξ_{P} denote the center of w_{P}. If P and P′ are tiles, we write P′ ≲^{+} P if P′ ≲′ P and ξ_{P′} > ξ_{P}, and P′ ≲^{−} if P′ ≲′ P and ξ_{P′} < ξ_{P}. If T is a tree, write ξ_{T,i} for ξ_{P_{T,i}}.

We now perform the following algorithm. We consider the set of all trees T in P⃗ such that

P_{i} ≲^{+} P_{T,i} for all P⃗ ∈ T  (20)

and

∑_{P⃗ ∈T} |IP⃗ |∥f_{j}∥_{P_{i}}^{2} ≥ 2^{−2n−5}|I_{T} |.  (21)

If there are no trees obeying (20) and (21), we terminate the algorithm. Otherwise, we choose T among all such trees so that ξ_{T,i} is maximal, and that T is maximal with respect to set inclusion. Let T′ denote the i-tree

T′ = {P⃗ ∈ P⃗ : P_{i} ≤ P_{T,i}}.

We remove both T and T′ from P⃗, and add them to T. (These two trees are allowed to overlap). Then one repeats the algorithm until we run out of trees obeying (20) and (21).

Since P⃗ is finite, this algorithm terminates in a finite number of steps, producing trees T_{1}, T′_{1}, T_{2}, T′_{2}, …, T_{M}, T′_{M}. By construction the T_{s} are disjoint; we claim that the tiles P_{i} in ⋃_{s=1}^{M} T_{s} are only finitely overlapping.

To see this, suppose that we had tri-tiles P⃗ ∈ T_{s}, P⃗′ ∈ T_{s′} such that P′_{i} ≤ P_{i}. We shall show that |IP⃗ | ∼ |IP⃗′|, which will imply the finite overlap condition.

Suppose for contradiction that |w_{P′_{i}}| ≫ |w_{P_{i}}|. Since P_{i} ≲ P_{T_{s},i} and P′_{i} ≲^{+} P_{T_{s′},i}, we thus see that ξ_{T_{s′},i} < ξ_{T_{s},i}. By our selection algorithm this implies that s < s′.

Also, since |w_{P′_{i}}| ≫ |w_{P_{i}}|, I_{P′_{i}} ⊆ I_{T_{s}}, and P_{i} ≲ P_{T_{s},i} we see that P′_{i} ≤ P_{T_{s},i}. Since s < s′, this means that P⃗′ ∈ T′_{s}. But T′_{s} and T_{s′} are disjoint by construction, which is a contradiction.

Thus the tiles in T_{s} are disjoint. From (11) and (21) we thus have

∑_{s=1}^{M} |I_{T_{s}}| ≲ 2^{2n}.

Since T′_{s} has the same top as T_{s}, we may thus add all the T_{s} and T′_{s} to T while respecting (15).

Now consider the set P⃗ of remaining tri-tiles. We note that

∑_{P⃗ ∈T:P_{i}≲^{+}P_{T,i}} |IP⃗ |∥f_{j}∥_{P_{i}}^{2} < 2^{−2n−5}|I_{T} |  (22)

for all trees T in P⃗, since otherwise the portion of T which obeyed (20) would be eligible for selection by the above algorithm.

We now repeat the previous algorithm, but replace ≲^{+} by ≲^{−} and select the trees T so that ξ_{T,i} is minimized rather than maximized. This yields a further collection of trees to add to T while still respecting (15), and the remaining collection of tiles P⃗ has the property that

∑_{P⃗ ∈T:P_{i}≲^{−}P_{T,i}} |IP⃗ |∥f_{j}∥_{P_{i}}^{2} < 2^{−2m−5}|I_{T} |  (23)

for all trees T in P⃗. Combining (19), (22), and (23) we see that

size_{i}(T) ≤ 2^{−m−1}

for all trees T in P⃗, and we are done.

2. Remarks

To make the above argument rigorous we need to fix up (11). It turns out that a satisfactory and rigorous substitute for (11) is as follows.

Definition 2.1. Let 1 ≤ i ≤ n. Two trees T, T′ are said to be strongly i-disjoint if • P_{i} ≠ P′_{i} for all P⃗ ∈ T, P⃗′ ∈ T′.

• Whenever P⃗ ∈ T, P⃗′ ∈ T′ are such that w_{P_{i}} ⊊ w_{P′_{i}}, then one has IP⃗′ ∩ I_{T} = ∅, and similarly with T and T′ reversed.

An inspection of the construction in Lemma 1.6 shows that the trees T_{m} used there are not only finitely overlapping, but (after a finite refinement) can be made strongly i-disjoint.

Lemma 2.2. Let i = 1, 2, 3, n ∈ Z, and let T be a collection of trees in P⃗ which are mutually strongly i-disjoint and such that

size_{i}(T) ∼ 2^{−n} for all T ∈ T.  (24)

Let I_{0} be an interval such that I_{T} ⊆ I_{0} for all T ∈ T. Then we have

∑_{T ∈T} |I_{T} | ≲ 2^{2n} min(1, |I_{0}|).  (25)

This lemma is easily implied by (11), and it turns out that it is the only consequence of (11) that one needs in the previous argument. The proof is the same as that in (11), with the strong disjointness being used to control the errors arising from ψ_{P} not being totally supported in I_{P}, and (24) used to control the analogues of the co-efficients c_{P} which arise from the converse to Hölder’s inequality.

References

[1] M. Lacey, C. Thiele L^{p} estimates on the bilinear Hilbert transform for 2 < p < ∞. Ann. Math. 146, pp. 693-724, [1997]

[2] C. Muscalu, T. Tao, C. Thiele, Multi-linear operators given by singular multipliers, to appear.

School of Mathematics, UNSW, Sydney NSW 2052 AUSTRALIA

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