New York Journal of Mathematics New York J. Math.

New York Journal of Mathematics

New York J. Math. 14(2008)145–192.

Uniform estimates for some paraproducts

Xiaochun Li

Abstract. We establishL^p×L^qtoL^restimates for some general para- products that arise in the study of the bilinear Hilbert transform along curves.

Contents

1. Introduction 145

2. Main results 147

3. A telescoping argument 149

4. Time frequency analysis 153

4.1. Definitions 154

4.2. Reduction 158

4.3. Principal lemmas 159

4.4. The size estimate for a tree 165

4.5. Proof of Lemma 4.11 174

5. Proof of Theorem 2.2 175

5.1. Proof of Lemma 5.1 176

5.2. A truncated trilinear form 181

5.3. Preliminary lemmata 183

5.4. Proof of Lemma 5.10 190

References 191

1. Introduction

It is an important theme of current research in analysis to decompose more complicated operators, such as the Cauchy integral on Lipschitz curves [1], as a sum of simpler operators. This theme has taken special prominence in

Received January 17, 2008.

Mathematics Subject Classification. Primary 42B20, 42B25. Secondary 46B70, 47B38.

Key words and phrases. paraproduct, uniform estimate.

Research was partially supported by NSF grant DMS-0456976.

ISSN 1076-9803/08

145

multilinear Harmonic Analysis, beginning with the work of Lacey and Thiele [12], which expressed the bilinear Hilbert transforms as a sum of modulated paraproducts. This theme has found much broader application as well.

The bilinear Hilbert transforms have a bilinear symbol given by restriction to a half-plane, with slope that depends upon the transform in question.

In considering more complicated symbols, one is led to paraproducts which have a complicated underlying description. One then seeks certain estimates of these paraproducts that areuniformin the parametrizations. This line of investigation was started in [23], the results of which give a new, multilinear proof of the boundedness of the Calderon commutator, fulfilling a program of study of Calderon [1]. It was further extended in work of the author and Grafakos [8,9,14], in the study of the disc as a bilinear multiplier. Muscalu, Tao and Thiele [16,15,17] gave alternate proofs (and more general proofs) of these results in the multilinear operator setting.

In this paper, we continue this line of study, considering certain uniform estimates that are motivated by an analysis of a bilinear Hilbert transform along polynomial curves. Namely, consider the operators

(1.1) (f, g)−→p.v.

_∞

−∞f(x−y)g(x−p(y)) dy y ,

for some polynomial p(y). The study of these operators leads to subtle questions in multilinear analysis, stationary phase methods, and paraprod- ucts. An initial investigation into operators of this type is given in [6], where the polynomial is taken to be a square, and the singular kernel is mollified to e^i|y|−β/|y| for someβ > 0. Without this modification, a signif- icant difficulty might be encountered. There is a natural analogue of the bilinear Hilbert transform along parabolas in the ergodic theory setting, that is, the nonconventional ergodic average _N¹ _N−1

n=0 f(Tⁿx)g(Tⁿ²x). In [7], Furstenberg proved that the characteristic factor of the trilinear ergodic averages _N¹ _N−1

n=0 f(T^an)g(T^bn)h(T^cn) for alla, b, c∈Z is characteristic for the previous nonconventional ergodic average. We are indebted to M. Lacey for bringing these Furstenberg theorems to our attention. Thus a possible method for the bilinear Hilbert transform along a parabola is to understand the trilinear Hilbert transform first. Unfortunately, it turns out the trilinear Hilbert transform is very difficult to handle. It is very interesting to find a proof for the bilinear Hilbert transform along curves without using any information of the trilinear Hilbert transform. It might be possible to obtain such a way by combining time-frequency analysis and the known results for the trilinear oscillatory integrals. This investigation will appear in another paper.

The paraproducts that arise have a richer parametrization than what has been considered before. The question of uniform estimates is the main focus of this article. In the next section, a class of paraproducts are introduced.

They are parametrized by:

• the width of the frequency window associated to the paraproducts, denoted by L₁ andL₂ below,

• the overlap of the frequency window associated to the paraproducts, denoted by M₁ and M₂ below,

• a modulation of the frequency window, denoted by the (lower case) parameters n₁, n₂,2^m below.

Prior results have concentrated on the uniformity of estimates with respect toM₁, M₂fromL^p×L^qtoL^rforr≥1 andL₁ =L₂[16]. The principal point of this article is to get the estimates for 1/2 < r <1 and arbitrary L₁, L₂. Another new point of this article is the (weak) uniformity that we establish inL₁, L₂ and the modulation parameters 2^m (see Theorem2.2below). This novelty is forced upon us by the stationary phase methods that one must use in the analysis of (1.1). One of anticipated applications of our theorems is the bilinear multiplier problems associated to the symbol defined by a characteristic function of a suitable domain with a smooth boundary.

Acknowledgements. The author would like to thank his wife, Helen, and his son, Justin, for being together through the hard times in the past two years. And he is also very thankful to Michael Lacey for his constant support and encouragement.

2. Main results

Letj∈Z,L₁, L₂ be positive integers andM₁, M₂ be integers.

ω_1,j = [2^L1j+M1/2,2·2^L1j+M1] and

ω_2,j= [−2^L2j+M2,2^L2j+M2].

Let Φ₁ be a Schwartz function whose Fourier transform is a standard bump function supported on [1/2,2], and Φ₂ be a Schwartz function such that Φ₂ is a standard bump function supported on [−1,1] and Φ₂(0) = 1. For ∈ {1,2} and n₁, n₂ ∈Z, define Φ_,j,n by

Φ_,j,n(ξ) =

e^2πin(·)Φ(·) ξ 2^Lj+M

.

It is clear that Φ_,j,n is supported on ω_,j. For locally integrable functions f’s, we definef_,j’s by

f_,j,n(x) =f∗Φ_,j,n(x).

We define a paraproduct to be

(2.1) Π_{L1,L2,M1,M2,n1,n2}(f₁, f₂)(x) =

j∈Z

2

=1

f_,j,n(x).

Another paraproduct we should introduce is the following. For ∈ {1,2}, let ω_,j denote the set {ξ : 2^Lj+M/2 ≤ |ξ| ≤ 2·2^Lj+M}. Let m be a nonnegative integer and define Φ_,j,m by

Φ_,j,m(ξ) =

e^2πi2m(·)Φ₁(·) ξ 2^Lj+M

. Letf_,j,m be the function defined by

f_,j,m(x) =f∗Φ_,j,m(x).

We define a paraproduct to be

(2.2) Π_{L1,L2,M1,M2,m}(f₁, f₂)(x) =

j∈Z

2

=1

f_,j,m(x).

One reason we study these paraproducts is that one will encounter such paraproducts in the study of the bilinear Hilbert transforms along polyno- mial curves. We have the following uniform estimates for these paraprod- ucts.

Theorem 2.1. For anyp₁>1,p₂>1with 1/p₁+ 1/p₂= 1/r, there exists a constant C independent of M₁, M₂, n₁, n₂ such that

(2.3) Π_{L1,L2,M1,M2,n1,n2}(f₁, f₂)

r ≤C

1 +|n₁|₁₀

1 +|n₂|₁₀

f_1p1f_2p2, for all f₁∈L^p1 and f₂∈L^p2.

Theorem 2.2. Let Π_{L1,L2,M1,M2,m}(f₁, f₂) be as in (2.2). Suppose that for allj,

(2.4) 2^L2j+M2 ≥2^L1j+M1+m.

For anyε >0,p₁>1,p₂ >1with1/p₁+1/p₂ = 1/r, there exists a constant C independent of m, M₁, M₂, L₁, L₂ such that

(2.5) Π_{L1,L2,M1,M2,m}(f₁, f₂)

r≤C2^εmf_1p1f_2p2, for all f₁∈L^p1 and f₂∈L^p2.

The case when L₁ = L₂ and r > 1 was proved in [16]. The constant C in Theorem 2.1 may depend on L₁, L₂. It is easy to see by the following argument that C is O(max{2^L1,2^L2}). It is possible to get a much bet- ter upper bound such as O

log(1 + max{L₂/L₁, L₁/L₂})

by tracking the constants carefully in the proof we will provide. But we do not pursue the sharp constant in this article. The independence ofM₁, M₂ is the most im- portant issue. In Sections 3,4, we give a proof for Theorem2.1. The proof of Theorem 2.2 will be given in Section 5. By using Theorem 2.1, we get the L^r bound for Π_{L1,L2,M1,M2,m} with a operator norm O(2^10m). Unfortu- nately sometimes this is not enough for our application. The desired norm isO(2^εm) for a very small positive numberε. It might be possible to remove the condition (2.4) or get the uniform estimate for Π_{L1,L2,M1,M2,m}in which

the operator norm is independent ofm. The uniform estimate fromL²×L² toL¹is trivial and (2.4) is redundant for this case. In Section5, we see that the uniform estimates for Π_{L1,L2,M1,M2,m} can be achieved forp₁, p₂ >2 and 1 < r < 2 (see Proposition 5.7) and (2.4) is superfluous for Theorem 2.2 when p₁, p₂>2 and 1< r <2 (see Corollary5.3).

3. A telescoping argument

We now start to prove Theorem 2.1. To prove Theorem 2.1, we first introduce a definition of admissible trilinear form. And we should show that by a telescoping argument used in [8, 23], we can reduce the problem to estimates for an admissible trilinear form. And thusL^r estimates forr >1 can be obtained by Littlewood–Paley theorem. The r < 1 case is more complicated. We have to use the time frequency analysis to deal with this case in Section 4.

Definition 3.1. An admissible trilinear form is a trilinear form (3.1) Λ_{L1,L2,M1,M2,n1,n2}(f₁, f₂, f₃) =

j∈Z

3

=1

f_,j,n(x)dx,

where n₃ = 0, f_,j,n = f∗Φ_,j,n and Φ_,j,n is a function whose Fourier transform is supported onω_,j such that:

(1) Eachω_,j is an interval inRsuch that the distance from the origin to the interval is not more than 3|ω_,j|. And{ω_,j}_j forms a sequence of lacunary intervals, that is, |ω_,j|/|ω_,j+1| ≤ 1/2 for all j ∈ Z. More- over, |ω_3,j| ≥Cmax{|ω_1,j|,|ω_2,j|} for some constant C independent of M₁, M₂, n₁, n₂.

(2) There are at least two indices ∈ {1,2,3} such thatΦ_,j,n satisfies

(3.2) Φ_,j,n(0) = 0

(3.3)

D^α

Φ_,j,n

|ω_,j|ξ≤ C_N(1 +|n|)^α (1 +|ξ|)^N ,

for allξ ∈Rand all nonnegative integersα, N. If an index in{1,2,3}

satisfies (3.2) and (3.3), we call the index a good index in the trilinear form Λ_{L1,L2,M1,M2,n1,n2}. For the index which is not a good index, we call it a bad index in the trilinear form Λ_{L1,L2,M1,M2,n1,n2}.

(3) If ∈ {2,3} is a bad index, then Φ_,j,n satisfies (3.3). Moreover, among the other two good indices=, at least one of them satisfies

|ω,j| ≤ Cmin{|ω_1,j|,|ω_2,j|,|ω_3,j|} for some constant C independent of f₁,f₂,f₃,M₁,M₂,n₁,n₂.

(4) If 1 is a bad index, thenΦ_1,j,n1 satisfies (3.4) Φ_1,j,n1(x) =

m(j) k=0

Φ_1,j+k,n1(x), where m(j) is some nonnegative integer.

Lemma 3.2. Let f₃ be a locally integrable function. Then

Π_{L1,L2,M1,M2,n1,n2}(f₁, f₂)(x)f₃(x)dx

is a sum of finitely many admissible trilinear forms such that the number of admissible trilinear forms in the sum is no more than a constant C inde- pendent of M₁, M₂, n₁, n₂.

Proof. For∈ {1,2}, writeω_,j as [a_,j, b_,j]. Ifb_2,j< b_1,j/16, then|ω_2,j|<

|ω_1,j|/6 and the distance fromω_1,j+ω_2,jto the origin is not less than|ω_1,j|/4.

In this case, simply letω_3,j be a small neighborhood of−(ω_1,j+ω_2,j) and the Fourier transform ofΦ_3,jis a suitable bump function adapted toω_3,j, then we have the desired lemma. Thus we now only consider the caseb_2,j≥b_1,j/16.

Let ω_3,j be [−18b_2,j,18b_2,j]. And Φ_3,j be a Schwartz function such that its Fourier transform is a bump function adapted toω_3,j andΦ_3,j(ξ) = 1 for all ξ∈[−17b_2,j,17b_2,j]. Then

Π(f₁, f₂)(x)f₃(x)dx=

j∈Z

3

=1

f_,j,n(x)dx,

where f_3,j,n3(x) =f₃∗Φ_3,j(x) and n₃ = 0. Let Φ₂ be a Schwartz function such that Φ₂ is a bump function on [−1,1] and Φ₂(ξ) = 1 for all ξ ∈ [−3/4,3/4]. And define Φ_2,j by Φ_2,j(ξ) = Φ₂(ξ/b_2,j). Let f_2,j = f ∗Φ_2,j. We also denote f_3,j,n3 by f_3,j. We can replacef_2,j,n2 by f_2,j because

j∈Z

f_,j,n1(x)

f_2,j,n2−f_2,j

(x)f_3,j(x)dx

is an admissible trilinear form. Hence the only thing we need to show is that Λ(f₁, f₂, f₃) =

j∈Z

f_,j,n1(x)f_2,j(x)f_3,j(x)dx

is admissible. For any real number x, let [x] denote the largest integer not exceeding x. Let m(j) be the integer defined by

m(j) =

(L₂j+M₂)−(L₁j+M₁) + 6 L₂

.

By b_2,j ≥ b_1,j/16, we see that m(j) ≥ 0. By a telescoping argument, Λ(f₁, f₂, f₃) equals to

j∈Z

f_1,j,n1(x)

m(j) k=0

f_2,j−k(x)f_3,j−k(x)−f_2,j−k−1(x)f_3,j−k−1(x)

dx,

since

f_1,j,n1(x)f_{2,j−m(j)−1}(x)f_{3,j−m(j)−1}(x)dx= 0 due to the following sim- ple fact on the support of Fourier transform of each function in the integrand, i.e.,

suppf_1,j,n1+ suppf_{2,j−m(j)−1}

∩

−

suppf_{3,j−m(j)−1}

=∅. By a change of variables j→j+k, we have that Λ(f₁, f₂, f₃) is equal to

j∈Z m(j)

k=0

f_1,j+k,n1(x)

f_2,j(x)f_3,j(x)−f_2,j−1(x)f_3,j−1(x)

dx,

wherem(j) is the integer defined by m(j) =

(L₂j+M₂)−(L₁j+M₁) + 6 L₁

.

We write this integral as a sum of three parts Λ₁,Λ₂,Λ₃, where Λ₁ =

j∈Z

_m(j) k=0

f_1,j+k,n1(x)

f_2,j(x)

f_3,j(x)−f_3,j−1(x) dx,

Λ₂ =

j∈Z

_m(j) k=0

f_1,j+k,n1(x)

·

f_2,j(x)−f_2,j−1(x)

f_3,j−1(x)−f_3,j−8(x) dx, Λ₃ =

j∈Z

_m(j) k=0

f_1,j+k,n1(x)

f_2,j(x)−f_2,j−1(x)

f_3,j−8(x)dx.

It is clear that Λ₂ is an admissible trilinear form. Write Λ₁ as Λ₁₁+ Λ₁₂, where

Λ₁₁=

j∈Z

_m(j) k=0

f_1,j+k,n1(x)

f_2,j(x)−f_2,j−1(x)

f_3,j(x)−f_3,j−1(x) dx,

Λ₁₂=

j∈Z

_m(j) k=0

f_1,j+k,n1(x)

f_2,j−1(x)

f_3,j(x)−f_3,j−1(x) dx.

Clearly, Λ₁₁ is an admissible trilinear form. Notice that supp

_m(j)−10−[L2/L1] k=0

f_1,j+k,n1

⊆[0,2⁻²2^L2j+M2] = [0,2⁻²2^−L2b_2,j], and

supp f_3,j−f_3,j−1

⊆[−18b_2,j,18b_2,j]\[−16·2^−L2b_2,j,16·2^−L2b_2,j].

Thus Λ₁₂ is equal to

j∈Z

_m(j) k=m(j)−10−[L2/L1]

f_1,j+k,n1(x)

·f_2,j−1,n2(x)

f_3,j,n3(x)−f_3,j−1,n3(x) dx, which is obviously a finite sum of admissible trilinear forms. As for Λ₃, observe that

supp

_m(j)−100−[L2/L1] k=0

f_1,j+k,n1

⊆[0,2⁻⁸⁰2^L2j+M2] = [0,2⁻⁸⁰2^−L2b_2,j], and

supp f_2,j−f_2,j−1

⊆[−b_2,j, b_2,j]\[−2^−L2−1b_2,j,2^−L2−1b_2,j].

Thus Λ₃ is equal to

j∈Z

_m(j) k=m(j)−100−[L2/L1]

f_1,j+k,n1(x)

f_2,j−f_2,j−1(x)

f_3,j−8(x)dx, which is a finite sum of admissible trilinear forms.

Lemma 3.3. Let Λ_{L1,L2,M1,M2,n1,n2} be an admissible trilinear form. Then for any real numbers p₁, p₂, p₃ >1 with1/p₁+ 1/p₂+ 1/p₃ = 1, there exists C independent of M₁, M₂, n₁, n₂ such that

(3.5)

Λ_{L1,L2,M1,M2,n1,n2}(f₁, f₂, f₃)≤C(1 +|n₁|)¹⁰(1 +|n₂|)¹⁰f₁p1f₂p2f₃p3, for all f₁∈L^p1, f₂ ∈L^p2 and f₃ ∈L^p3.

Proof. If there is no bad index in the trilinear form, take₀to be any integer in{1,2,3}. Otherwise, let₀be a bad index. Applying the Cauchy–Schwarz inequality, Λ_{L1,L2,M1,M2,n1,n2} is dominated by

sup

j∈Z f_0,j,n

0 =0

j

f_,j,n²_1/2 dx.

Using H¨older’s inequality, we dominate the trilinear form by sup

j∈Z f_0,j,n

0 p1

=0

j

f_,j,n²_1/2 p

.

The Littlewood–Paley theorem yields that for =₀

j

f_,j,n²_1/2 p

≤C(1 +|n|)¹⁰f_p. If₀ ∈ {2,3}, then by (3.3), we have

supj∈Z f_0,j,n

0≤(1 +|n₀|)¹⁰M(f₀),

which clearly yields the lemma. We now only need to consider the case ₀= 1. It suffices to prove that

(3.6)

sup

j

m(j) k=0

f₁∗Φ_1,j+k,n1 p1

≤C(1 +|n₁|)¹⁰f_1p1.

Notice that ω_1,j’s are essentailly disjoint intervals and Fourier transform of _m(j)

k=0 Φ_1,j+k,n1 is supported on a bounded interval depending on j. The left-hand side of (3.6) is less than

C

M

j

f₁∗Φ_1,j,n1 p1

. It is easy to verify that

jf₁∗Φ_1,j,n1 is a bounded operator onL² associ- ated to a standard Calder´on–Zygmund kernel by paying at most a cost of (1 +|n₁|)¹⁰ in the corresponding estimates. Thus by a standard Calder´on–

Zygmund argument, we have for any real numberp >1, there is a constant C independent of M₁, M₂, n₁, n₂ such that

j

f ∗Φ_1,j,n1

p≤C(1 +|n₁|¹⁰)f_p

holds for allf ∈L^p, which yields (3.6). Therefore we complete the proof of

the lemma.

Combining Lemma3.2 and Lemma 3.3, we obtain (2.3) forp₁, p₂, r >1.

To finish the proof of Theorem2.1, we need to provide a proof ofL^restimate with 1/2< r≤1 for (2.3), which will be given in Section 4.

4. Time frequency analysis

In this section we prove (2.3) with 1/2 < r ≤ 1 for the paraproducts by time frequency analysis, which was used for establishing L^p (uniform) estimates for the bilinear Hilbert transforms in [9,12,13,14,15,16,17,23].

LetF be a measurable set in R. X(F) denotes the set of all measurable functions supported on F such that the L^∞ norms of the functions are no more than 1. A function in X(F) can be considered essentially as the characteristic function 1_F.

To obtain Theorem2.1, by Lemma3.3, an interpolation argument in [15], and the scaling invariance, it is sufficient to prove that for any p₁, p₂ > 1 such that 1/p₁+ 1/p₂ ≥1 and any measurable set F₃ ⊆ R with |F₃| = 1, there exists a subset F₃ ⊂F₃ such that |F₃| ≥1/2 and

(4.1)

Π_{L1,L2,M1,M2,n1,n2}(f₁, f₂)(x)f₃(x)dx

≤C(1 +|n₁|)¹⁰(1 +|n₂|)¹⁰|F₁|^1/p1|F₂|^1/p2 holds for all f₁ ∈ X(F₁), f₂ ∈ X(F₂), f₃ ∈ X(F₃), where C is a constant independent off₁, f₂, f₃,M₁, M₂, n₁, n₂.

If 2^L2j+M2 < 2^L1j+M1/8, let ω_3,j = [−19·2^L1j+M1/8,−2^L1j+M1/8] and Φ_3,j be a Schwartz function whose Fourier transform is a bump function adapted toω_3,jsuch thatΦ_3,j(ξ) = 1 for allξ∈[−9·2^L1j+M1/4,−2^L1j+M1/4].

If 2^L2j+M2 ≥2^L1j+M1/8, letω_3,j= [−18·2^L2j+M2,18·2^L2j+M2] and Φ_3,j be a Schwartz function whose Fourier transform is a bump function adapted to ω_3,j such that Φ_3,j(ξ) = 1 for all ξ ∈ [−17·2^L2j+M2,17·2^L2j+M2]. Let n₃ = 0, Φ_3,j,n3 = Φ_3,j, f_3,j,n3(x) = f₃ ∗Φ_3,j,n3(x). Define a trilinear form Λ_{L1,L2,M1,M2,n1,n2} by

(4.2) Λ_{L1,L2,M1,M2,n1,n2}(f₁, f₂, f₃) =

j∈Z

3

=1

f_,j,n(x)dx.

Clearly Λ_{L1,L2,M1,M2,n1,n2} =

Π_{L1,L2,M1,M2,n1,n2}(f₁, f₂)(x)f₃(x)dx. Thus to prove (4.1), it suffices to prove the following lemma.

Lemma 4.1. Letp₁, p₂ >1such that 1/p₁+1/p₂ ≥1andΛ_{L1,L2,M1,M2,n1,n2} be the trilinear form defined by (4.2). Let F₁, F₂, F₃ be measurable sets in R with |F₃|= 1. Then there exists a subset F₃ ⊆ F₃ such that |F₃| >1/2 and there exists a constantC independent of F₁, F₂, F₃,f₁,f₂, f₃, M₁,M₂, n₁, n₂ such that

(4.3) Λ_{L1,L2,M1,M2,n1,n2}(f₁, f₂, f₃)

≤C(1 +|n₁|)¹⁰(1 +|n₂|)¹⁰|F₁|^1/p1|F₂|^1/p2 holds for all f₁ ∈X(F₁), f₂ ∈X(F₂), f₃ ∈X(F₃).

Lemma4.1and Lemma3.3implies the estimates (2.3) by an interpolation argument in [15]. Therefore we obtain Theorem2.1once we finish a proof of Lemma4.1. The following subsections are devoted to proof of Lemma 4.1.

4.1. Definitions. To prove Lemma4.1, we introduce some definitions first.

Let ψ be a nonnegative Schwartz function such that ψ is supported in [−1/100,1/100] and satisfiesψ(0) = 1. Let ψ_k(x) = 2^kψ(2^kx) for anyk∈Z. Forj∈Zand∈ {1,2,3}, definek_jto be an integer such that|ω_,j| ∼2^kj.

Denote min_∈{1,2,3}k_j by k_j. And define

I_kj,n = [2^−kjn,2^−kj(n+ 1)].

Define

1^∗_j,n(x) =1_Ikj,n∗ψ_kj(x).

It is easy to see that

Λ_{L1,L2,M1,M2,n1,n2}(f₁, f₂, f₃) =

j∈Zn∈Z

1^∗_j,n(x) 3

=1

f_,j,n(x)dx.

For an integer γ with 0 ≤ γ < 2¹⁰⁰, let Z(γ) be the set of all integers congruent to γ modulo 2¹⁰⁰. For S⊂Z(γ)×Zwe define

(4.4) Λ_S(f₁, f₂, f₃) =

R(j,n)∈S

1^∗_j,n(x) 3

=1

f_,j,n(x)dx.

Λ_S depends on L₁, L₂, M₁, M₁, n₁, n₂. We suppress this dependence for no- tational convenience. Note that there are finite congruence classes modulo 2¹⁰⁰. We will therefore concentrate on proving Lemma 4.1 for the trilinear form Λ_S.

In time-frequency space, each functionf_,j,n for∈ {1,2,3} corresponds to a box I_kj,n×ω_,j. The most difficult situation is when only one of boxes is the Heisenberg box, i.e., |I_k,j,n||ω_,j| ∼ 1. In this situation, we can use the John–Nirenberg type argument to get the equivalence of L^p estimates of Littlewood–Paley type square functions for only one of functions. For other two functions, there is no such an equivalence and an extra cost for it has to been paid if one estimates the BMO norm. It turns out that the L^p equivalence for at least one of three functions is the most crucial key to solve the problem. Our proof will heavily rely on this equivalence for one of functions.

Letpbe a positive number close to 1. To obtain the Lemma4.1, it suffices to prove (4.3) for p₁ ≥ p, p₂ ≥ p and 1/p₁+ 1/p₂ ≥ 1. For simplicity, we only deal with the casen₁ =n₂ =n₃= 0. The general case can be handled in the same way by paying at most a cost of (1 +|n₁|)¹⁰(1 +|n₂|)¹⁰ in the constants.

We now start to prove that for n₁ = n₂ = 0, any 1 < p < 2 and any measurable set F₃ with |F₃| = 1 in R, there exists a subset F₃ of F₃ with

|F₃| ≥1/2 such that

(4.5) Λ_S(f₁, f₂, f₃)≤C|F₁|^1/p1|F₂|^1/p2

holds for all p₁ ≥ p, p₂ ≥ p with 1/p₁ + 1/p₂ ≥ 1, f₁ ∈ X(F₁), f₂ ∈ X(F₂), f₃ ∈ X(F₃), where the constant C is independent of S, F₁,F₂, F₃, f₁,f₂, f₃,M₁,M₂. Let us introduce some definitions first.

Definition 4.2. Letp >1. Define the exceptional set Ω by

(4.6) Ω =

3

=1

x∈R:M_p M1_F

(x)> C₀|F|^1/p

whereM f is the Hardy–Littlewood maximal function of f and M_pf equals to

M(|f|^p)_1/p .

By this definition, for the measurable set F₃ with |F₃| = 1, we take F₃ =F₃\Ω. IfC₀ is chosen sufficiently large we see that|F₃| ≥ |F₃|/2.

Definition 4.3. Given S ⊂ Z(γ) × Z and s = (j, n) ∈ S. Let k_s = min_∈{1,2,3}{k_j}. The dyadic interval [2^−ksn,2^−ks(n+ 1)] is called the time interval of s. We denote it byI_s.

Definition 4.4. LetS be a subset ofZ(γ)×Z. We say that S is a convex set inZ(γ)×Zif for anys∈Z(γ)×ZwithI_s1 ⊆I_s⊆I_s2 for somes₁, s₂ ∈S, we have s∈S.

Definition 4.5. LetT⊂S. If there is t∈Tsuch thatI_s⊂I_tholds for all s∈ T, then T is called a tree with top t. T is called a maximal tree with toptinS if there does not exist a larger tree inSwith the same top strictly containing T.

Definition 4.6. LetTbe a tree in S. Define scl(T) the set of scale indices of Tby

scl(T) ={j∈Z:∃n∈Z,s.t.(j, n)∈T}. Forj ∈scl(T), thej-th shadow of Tis defined by

Sh_j(T) = I_s:s= (j, n)∈T . Define an approximation of1_Shj(T) by

1^∗_Sh

j(T)(x) =1_Shj(T)∗ψ_kj(x).

Definition 4.7. Let (j, n) =s∈S and ∈ {1,2,3}. And let 1^∗∗_j,n(x) =

I_kj,n

2^kj

(1 + 2^2kj|x−y|²)²⁰⁰dy.

Define a seminormf_j,n by (4.7) f

j,n =f

s = 1

|I_s|^1/p1^∗∗_j,nf_,j,n

p+ 1

|I_s|^1/p2^−kj1^∗∗_j,nDf_,j,n

p

whereDf_,j,n is the derivative off_,j,n. Define ζ(j, M, K) by

(4.8) ζ(j, M, K) =

L₁j+M₁−M₂−6 L₂

+

L₁ L₂

M+K,

whereL= 2¹⁰⁰,K is an integer between−10Land 10LandM is an integer between 0 and 6L. For∈ {2,3}, we define aζ seminormf

j,n,ζ by f

j,n,ζ = (4.9)

f_j,n+ sup

M,K

1

|I_s|^1/p

1^∗∗_j,nf_,ζ(j,M,K),0

p+|I_s|1^∗∗_j,nDf,ζ(j,M,K),0

p

. For= 1, let theζ seminormf₁

j,n,ζ=f₁

j,n.

Definition 4.8. Let T ⊂S be a tree and t = (j_T, n_T) ∈ T be the top of T. Denote by I_T the time interval of the top of treeT.

(a) In the case |ω_2,j| ≤ |ω_1,j|/6 for all j ∈ scl(T), define Δ^∗(T) for ∈ {1,3} by

(4.10) Δ^∗(T)(x) =

(j,n)∈T

1^∗∗_j,nf_,j,n(x)²_1/2 .

For = 2, define

(4.11) Δ^∗₂(T)(x) =1^∗∗_jT,nTf_2,jT,n(x).

And in this case, for ∈ {1,2,3}, define the-size ofT by

(4.12) size(T) = 1

|I_T|^1/pΔ^∗(T)

p+f

jT,nT.

(b) In the case|ω_2,j|>|ω_1,j|/6 for allj∈scl(T), for = 2,3, letf_,j,T= f_,j,0 ifj ∈scl(T) andf_,j,T≡0 if j /∈scl(T). Define:

Δ^∗(T) =

(j,n)∈T

1^∗∗_j,n

f_,j,T−f_,j−L,T

(x)²_1/2 (4.13)

+

(j,n)∈T

1^∗∗_j,n

f_,j,n−f_,j,0

(x)²_1/2 .

And define Δ^∗₁(T) by (4.14) Δ^∗₁(T)(x) =

(j,n)∈T

1^∗∗_j,nf_1,j,n1(x)²_1/2 .

In this case, for ∈ {1,2,3}, define the-size ofT by (4.15) size(T) = 1

|I_T|^1/pΔ^∗(T)

p+f

jT,nT,ζ.