New York Journal of Mathematics

New York J. Math. 14 (2008) 145–192.

Uniform estimates for some paraproducts

Xiaochun Li

Abstract. We establish L^{p} ×L^{q} to L^{r} estimates for some general paraproducts 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

145^{ISSN 1076-9803/08}

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.

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 are uniform in 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)) \frac{dy}{y} ,

for some polynomial p(y). The study of these operators leads to subtle questions in multilinear analysis, stationary phase methods, and paraproducts. 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 some β > 0. Without this modification, a significant 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 \frac{1}{N} ∑_{n=0}^{N−1} f(T^{n}x)g(T^{n^{2}}x). In [7], Furstenberg proved that the characteristic factor of the trilinear ergodic averages \frac{1}{N} ∑_{n=0}^{N−1} f(T^{an})g(T^{bn})h(T^{cn}) for all a, 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 L_{1} and L_{2} below,

• the overlap of the frequency window associated to the paraproducts, denoted by M_{1} and M_{2} below,

• a modulation of the frequency window, denoted by the (lower case)

parameters n_{1}, n_{2}, 2^{m} below. Prior results have concentrated on the uniformity of estimates with respect to M_{1}, M_{2} from L^{p}×L^{q} to L^{r} for r ≥ 1 and L_{1} = L_{2} [16]. The principal point of this article is to get the estimates for 1/2 < r < 1 and arbitrary L_{1}, L_{2}. Another new point of this article is the (weak) uniformity that we establish in L_{1}, L_{2} and the modulation parameters 2^{m} (see Theorem 2.2 below). 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

Let j ∈ ℤ, L_{1}, L_{2} be positive integers and M_{1}, M_{2} be integers.

ω_{1,j} = [2^{L_{1}j+M_{1}}/2, 2 · 2^{L_{1}j+M_{1}}]

and

ω_{2,j} = [−2^{L_{2}j+M_{2}}, 2^{L_{2}j+M_{2}}].

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

Φ̂_{ℓ,j,n_{ℓ}}(ξ) = (e^{2πin_{ℓ}(·)}Φ̂_{ℓ}(·))(\frac{ξ}{2^{L_{ℓ}j+M_{ℓ}}} ).

It is clear that Φ̂_{ℓ,j,n_{ℓ}} is supported on ω_{ℓ,j}. For locally integrable functions f_{ℓ}’s, we define f_{ℓ,j}’s by

f_{ℓ,j,n_{ℓ}}(x) = f_{ℓ} ∗ Φ_{ℓ,j,n_{ℓ}}(x).

We define a paraproduct to be

(2.1)  Π_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}}(f_{1}, f_{2})(x) = ∑_{j∈ℤ} ∏_{ℓ=1}^{2} f_{ℓ,j,n_{ℓ}}(x).

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

Φ̂_{ℓ,j,m}(ξ) = (e^{2πi2^{m}(·)}Φ̂_{1}(·))(\frac{ξ}{2^{L_{ℓ}j+M_{ℓ}}} ).

Let f_{ℓ,j,m} be the function defined by

f_{ℓ,j,m}(x) = f_{ℓ} ∗ Φ_{ℓ,j,m}(x).

We define a paraproduct to be

(2.2)  Π_{L_{1},L_{2},M_{1},M_{2},m}(f_{1}, f_{2})(x) = ∑_{j∈ℤ} ∏_{ℓ=1}^{2} f_{ℓ,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 polynomial curves. We have the following uniform estimates for these paraproducts.

Theorem 2.1. For any p_{1} > 1, p_{2} > 1 with 1/p_{1} + 1/p_{2} = 1/r, there exists a constant C independent of M_{1}, M_{2}, n_{1}, n_{2} such that

(2.3) ∥Π_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}}(f_{1}, f_{2})∥_{r} ≤ C(1+|n_{1}|)^{10}(1+|n_{2}|)^{10}∥f_{1}∥_{p_{1}}∥f_{2}∥_{p_{2}},

for all f_{1} ∈ L^{p_{1}} and f_{2} ∈ L^{p_{2}}.

Theorem 2.2. Let Π_{L_{1},L_{2},M_{1},M_{2},m}(f_{1}, f_{2}) be as in (2.2). Suppose that for all j,

(2.4)  2^{L_{2}j+M_{2}} ≥ 2^{L_{1}j+M_{1}+m}.

For any ε > 0, p_{1} > 1, p_{2} > 1 with 1/p_{1}+1/p_{2} = 1/r, there exists a constant C independent of m, M_{1}, M_{2}, L_{1}, L_{2} such that

(2.5)  ∥Π_{L_{1},L_{2},M_{1},M_{2},m}(f_{1}, f_{2})∥_{r} ≤ C2^{εm}∥f_{1}∥_{p_{1}}∥f_{2}∥_{p_{2}},

for all f_{1} ∈ L^{p_{1}} and f_{2} ∈ L^{p_{2}}.

The case when L_{1} = L_{2} and r > 1 was proved in [16]. The constant C in Theorem 2.1 may depend on L_{1}, L_{2}. It is easy to see by the following argument that C is O(max{2^{L_{1}}, 2^{L_{2}}}). It is possible to get a much better upper bound such as O( log(1 + max{L_{2}/L_{1}, L_{1}/L_{2}})) by tracking the constants carefully in the proof we will provide. But we do not pursue the sharp constant in this article. The independence of M_{1}, M_{2} is the most important issue. In Sections 3, 4, we give a proof for Theorem 2.1. 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_{L_{1},L_{2},M_{1},M_{2},m} with a operator norm O(2^{10m}). Unfortunately sometimes this is not enough for our application. The desired norm is O(2^{εm}) for a very small positive number ε. It might be possible to remove the condition (2.4) or get the uniform estimate for Π_{L_{1},L_{2},M_{1},M_{2},m} in which the operator norm is independent of m. The uniform estimate from L^{2} × L^{2} to L^{1} is trivial and (2.4) is redundant for this case. In Section 5, we see that the uniform estimates for Π_{L_{1},L_{2},M_{1},M_{2},m} can be achieved for p_{1}, p_{2} > 2 and 1 < r < 2 (see Proposition 5.7) and (2.4) is superfluous for Theorem 2.2 when p_{1}, p_{2} > 2 and 1 < r < 2 (see Corollary 5.3).

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 thus L^{r} estimates for r > 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_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}}(f_{1}, f_{2}, f_{3}) = ∫ ∑_{j∈ℤ} ∏_{ℓ=1}^{3} f̃_{ℓ,j,n_{ℓ}}(x)dx,

where n_{3} = 0, f̃_{ℓ,j,n_{ℓ}} = f_{ℓ} ∗ Φ̃_{ℓ,j,n_{ℓ}} and Φ̃_{ℓ,j,n_{ℓ}} is a function whose Fourier transform is supported on ω̃_{ℓ,j} such that:

(1) Each ω̃_{ℓ,j} is an interval in ℝ such that the distance from the origin to the interval is not more than 3|ω̃_{ℓ,j}|. And {ω̃_{ℓ,j}}_{j} forms a sequence of lacunary intervals, that is, |ω̃_{ℓ,j}|/|ω̃_{ℓ,j+1}| ≤ 1/2 for all j ∈ ℤ. Moreover, |ω̃_{3,j}| ≥ C max{|ω̃_{1,j}|, |ω̃_{2,j}|} for some constant C independent of M_{1}, M_{2}, n_{1}, n_{2}.

(2) There are at least two indices ℓ ∈ {1, 2, 3} such that Φ̃_{ℓ,j,n_{ℓ}} satisfies

(3.2)   ; Φ̃_{ℓ,j,n_{ℓ}}(0) = 0

(3.3)  |D^{α}( ; Φ̃_{ℓ,j,n_{ℓ}}(|ω̃_{ℓ,j}|ξ))| ≤ \frac{C_{N}(1 + |n_{ℓ}|)^{α}}{(1 + |ξ|)^{N}} ,

for all ξ ∈ ℝ and all nonnegative integers α, 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_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}}. For the index which is not a good index, we call it a bad index in the trilinear form Λ_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}}.

(3) If ℓ ∈ {2, 3} is a bad index, then Φ̃_{ℓ,j,n_{ℓ}} satisfies (3.3). Moreover, among the other two good indices ℓ′ ≠ ℓ, at least one of them satisfies |ω_{ℓ′,j}| ≤ C min{|ω̃_{1,j}|, |ω̃_{2,j}|, |ω̃_{3,j}|} for some constant C independent of f_{1}, f_{2}, f_{3}, M_{1}, M_{2}, n_{1}, n_{2}.

(4) If 1 is a bad index, then Φ̃_{1,j,n_{1}} satisfies

(3.4)  Φ̃_{1,j,n_{1}}(x) = ∑_{k=0}^{m′(j)} Φ_{1,j+k,n_{1}}(x),

where m′(j) is some nonnegative integer.

Lemma 3.2. Let f_{3} be a locally integrable function. Then

∫ Π_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}}(f_{1}, f_{2})(x)f_{3}(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 independent of M_{1}, M_{2}, n_{1}, n_{2}.

Proof. For ℓ ∈ {1, 2}, write ω_{ℓ,j} as [a_{ℓ,j}, b_{ℓ,j}]. If b_{2,j} < b_{1,j}/16, then |ω_{2,j}| < |ω_{1,j}|/6 and the distance from ω_{1,j}+ω_{2,j} to the origin is not less than |ω_{1,j}|/4. In this case, simply let ω̃_{3,j} be a small neighborhood of −(ω_{1,j} +ω_{2,j}) and the Fourier transform of Φ̃_{3,j} is a suitable bump function adapted to ω̃_{3,j}, then we have the desired lemma. Thus we now only consider the case b_{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 to ω_{3,j} and Φ̂_{3,j}(ξ) = 1 for all ξ ∈ [−17b_{2,j}, 17b_{2,j}]. Then

∫ Π(f_{1}, f_{2})(x)f_{3}(x)dx = ∫ ∑_{j∈ℤ} ∏_{ℓ=1}^{3} f_{ℓ,j,n_{ℓ}}(x)dx,

where f_{3,j,n_{3}}(x) = f_{3} ∗ Φ_{3,j}(x) and n_{3} = 0. Let Φ̃_{2} be a Schwartz function

such that ˆ Φ̃_{2} is a bump function on [−1, 1] and ˆ Φ̃_{2}(ξ) = 1 for all ξ ∈

[−3/4, 3/4]. And define Φ_{2,j} by Φ̂_{2,j}(ξ) = ˆ Φ̃_{2}(ξ/b_{2,j}). Let f_{2,j} = f ∗ Φ_{2,j}. We also denote f_{3,j,n_{3}} by f_{3,j}. We can replace f_{2,j,n_{2}} by f_{2,j} because

∫ ∑_{j∈ℤ}f_{ℓ,j,n_{1}}(x)(f_{2,j,n_{2}} − 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′(f_{1}, f_{2}, f_{3}) = ∫ ∑_{j∈ℤ}f_{ℓ,j,n_{1}}(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) = [\frac{(L_{2}j + M_{2}) − (L_{1}j + M_{1}) + 6}{L_{2}}] .

By b_{2,j} ≥ b_{1,j}/16, we see that m(j) ≥ 0. By a telescoping argument, Λ′(f_{1}, f_{2}, f_{3}) equals to

∫ ∑_{j∈ℤ}f_{1,j,n_{1}}(x) ∑_{k=0}^{m(j)} (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,n_{1}}(x)f_{2,j−m(j)−1}(x)f_{3,j−m(j)−1}(x)dx = 0 due to the following simple fact on the support of Fourier transform of each function in the integrand, i.e.,

(suppf̂_{1,j,n_{1}} + suppf̂_{2,j−m(j)−1}) ∩ ( − (suppf̂_{3,j−m(j)−1})) = ∅.

By a change of variables j → j + k, we have that Λ′(f_{1}, f_{2}, f_{3}) is equal to

∫ ∑_{j∈ℤ} ∑_{k=0}^{m′(j)} f_{1,j+k,n_{1}}(x)(f_{2,j}(x)f_{3,j}(x) − f_{2,j−1}(x)f_{3,j−1}(x))dx,

where m′(j) is the integer defined by

m′(j) = [\frac{(L_{2}j + M_{2}) − (L_{1}j + M_{1}) + 6}{L_{1}}] .

We write this integral as a sum of three parts Λ_{1}, Λ_{2}, Λ_{3}, where

Λ_{1} = ∫ ∑_{j∈ℤ}( ∑_{k=0}^{m′(j)} f_{1,j+k,n_{1}}(x))f_{2,j}(x)(f_{3,j}(x) − f_{3,j−1}(x))dx,

Λ_{2} = ∫ ∑_{j∈ℤ}( ∑_{k=0}^{m′(j)} f_{1,j+k,n_{1}}(x))
· (f_{2,j}(x) − f_{2,j−1}(x))(f_{3,j−1}(x) − f_{3,j−8}(x))dx,
Λ_{3} = ∫ ∑_{j∈ℤ}( ∑_{k=0}^{m′(j)} f_{1,j+k,n_{1}}(x))(f_{2,j}(x) − f_{2,j−1}(x))f_{3,j−8}(x)dx.

It is clear that Λ_{2} is an admissible trilinear form. Write Λ_{1} as Λ_{11} + Λ_{12}, where

Λ_{11} = ∫ ∑_{j∈ℤ}( ∑_{k=0}^{m′(j)} f_{1,j+k,n_{1}}(x))(f_{2,j}(x)−f_{2,j−1}(x))(f_{3,j}(x)−f_{3,j−1}(x))dx,

Λ_{12} = ∫ ∑_{j∈ℤ}( ∑_{k=0}^{m′(j)} f_{1,j+k,n_{1}}(x))f_{2,j−1}(x)(f_{3,j}(x) − f_{3,j−1}(x))dx.

Clearly, Λ_{11} is an admissible trilinear form. Notice that

supp( ∑_{k=0}^{m′(j)−10−[L_{2}/L_{1}]} f̂_{1,j+k,n_{1}}) ⊆ [0, 2^{−2}2^{L_{2}j+M_{2}}] = [0, 2^{−2}2^{−L_{2}}b_{2,j}],

and

supp(f̂_{3,j} − f̂_{3,j−1}) ⊆ [−18b_{2,j}, 18b_{2,j}]\[−16 · 2^{−L_{2}}b_{2,j}, 16 · 2^{−L_{2}}b_{2,j}].

Thus Λ_{12} is equal to

∫ ∑_{j∈ℤ}( ∑_{k=m′(j)−10−[L_{2}/L_{1}]}^{m′(j)} f_{1,j+k,n_{1}}(x))

· f_{2,j−1,n_{2}}(x)(f_{3,j,n_{3}}(x) − f_{3,j−1,n_{3}}(x))dx,

which is obviously a finite sum of admissible trilinear forms. As for Λ_{3}, observe that

supp( ∑_{k=0}^{m′(j)−100−[L_{2}/L_{1}]} f̂_{1,j+k,n_{1}}) ⊆ [0, 2^{−80}2^{L_{2}j+M_{2}}] = [0, 2^{−80}2^{−L_{2}}b_{2,j}],

and

supp(f̂_{2,j} − f̂_{2,j−1}) ⊆ [−b_{2,j}, b_{2,j}]\[−2^{−L_{2}−1}b_{2,j}, 2^{−L_{2}−1}b_{2,j}].

Thus Λ_{3} is equal to

∫ ∑_{j∈ℤ}( ∑_{k=m′(j)−100−[L_{2}/L_{1}]}^{m′(j)} f_{1,j+k,n_{1}}(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_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}} be an admissible trilinear form. Then for any real numbers p_{1}, p_{2}, p_{3} > 1 with 1/p_{1} + 1/p_{2} + 1/p_{3} = 1, there exists C independent of M_{1}, M_{2}, n_{1}, n_{2} such that (3.5) |Λ_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}}(f_{1}, f_{2}, f_{3})| ≤ C(1 + |n_{1}|)^{10}(1 + |n_{2}|)^{10}∥f_{1}∥_{p_{1}}∥f_{2}∥_{p_{2}}∥f_{3}∥_{p_{3}},

for all f_{1} ∈ L^{p_{1}}, f_{2} ∈ L^{p_{2}} and f_{3} ∈ L^{p_{3}}.

Proof. If there is no bad index in the trilinear form, take ℓ_{0} to be any integer in {1, 2, 3}. Otherwise, let ℓ_{0} be a bad index. Applying the Cauchy–Schwarz inequality, Λ_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}} is dominated by

∫ sup_{j∈ℤ} |f̃_{ℓ_{0},j,n_{ℓ_{0}}} | ∏_{ℓ≠ℓ_{0}} ( ∑_{j} |f̃_{ℓ,j,n_{ℓ}} |^{2})^{1/2}dx.

Using Hölder’s inequality, we dominate the trilinear form by

∥ sup_{j∈ℤ} |f̃_{ℓ_{0},j,n_{ℓ_{0}}} |∥_{p_{1}} ∏_{ℓ≠ℓ_{0}} ∥( ∑_{j} |f̃_{ℓ,j,n_{ℓ}} |^{2})^{1/2}∥_{p_{ℓ}}.

The Littlewood–Paley theorem yields that for ℓ ≠ ℓ_{0}

∥( ∑_{j} |f̃_{ℓ,j,n_{ℓ}} |^{2})^{1/2}∥_{p_{ℓ}} ≤ C(1 + |n_{ℓ}|)^{10}∥f_{ℓ}∥_{p_{ℓ}}.

If ℓ_{0} ∈ {2, 3}, then by (3.3), we have

sup_{j∈ℤ} |f̃_{ℓ_{0},j,n_{ℓ_{0}}} | ≤ (1 + |n_{ℓ_{0}}|)^{10}M(f_{ℓ_{0}}),

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

(3.6)  ∥ sup_{j} | ∑_{k=0}^{m′(j)} f_{1} ∗ Φ_{1,j+k,n_{1}} |∥_{p_{1}} ≤ C(1 + |n_{1}|)^{10}∥f_{1}∥_{p_{1}}.

Notice that ω_{1,j}’s are essentailly disjoint intervals and Fourier transform of ∑_{k=0}^{m′(j)} Φ_{1,j+k,n_{1}} is supported on a bounded interval depending on j. The left-hand side of (3.6) is less than

C∥M( ∑_{j} f_{1} ∗ Φ_{1,j,n_{1}})∥_{p_{1}}.

It is easy to verify that ∑_{j} f_{1} ∗ Φ_{1,j,n_{1}} is a bounded operator on L^{2} associated to a standard Calderón–Zygmund kernel by paying at most a cost of (1 + |n_{1}|)^{10} in the corresponding estimates. Thus by a standard Calderón– Zygmund argument, we have for any real number p > 1, there is a constant C independent of M_{1}, M_{2}, n_{1}, n_{2} such that

∥ ∑_{j} f ∗ Φ_{1,j,n_{1}} ∥_{p} ≤ C(1 + |n_{1}|^{10})∥f∥_{p}

holds for all f ∈ L^{p}, which yields (3.6). Therefore we complete the proof of the lemma. □

Combining Lemma 3.2 and Lemma 3.3, we obtain (2.3) for p_{1}, p_{2}, r > 1. To finish the proof of Theorem 2.1, we need to provide a proof of L^{r} estimate 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].

Let F 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 Theorem 2.1, by Lemma 3.3, an interpolation argument in [15], and the scaling invariance, it is sufficient to prove that for any p_{1}, p_{2} > 1 such that 1/p_{1} + 1/p_{2} ≥ 1 and any measurable set F_{3} ⊆ ℝ with |F_{3}| = 1, there exists a subset F′_{3} ⊂ F_{3} such that |F′_{3}| ≥ 1/2 and

(4.1) | ∫ Π_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}}(f_{1}, f_{2})(x)f_{3}(x)dx|

≤ C(1 + |n_{1}|)^{10}(1 + |n_{2}|)^{10}|F_{1}|^{1/p_{1}}|F_{2}|^{1/p_{2}}

holds for all f_{1} ∈ X(F_{1}), f_{2} ∈ X(F_{2}), f_{3} ∈ X(F′_{3}), where C is a constant independent of f_{1}, f_{2}, f_{3}, M_{1}, M_{2}, n_{1}, n_{2}.

If 2^{L_{2}j+M_{2}} < 2^{L_{1}j+M_{1}}/8, let ω_{3,j} = [−19 · 2^{L_{1}j+M_{1}}/8, −2^{L_{1}j+M_{1}}/8] and Φ_{3,j} be a Schwartz function whose Fourier transform is a bump function adapted to ω_{3,j} such that Φ̂_{3,j}(ξ) = 1 for all ξ ∈ [−9·2^{L_{1}j+M_{1}}/4, −2^{L_{1}j+M_{1}}/4]. If 2^{L_{2}j+M_{2}} ≥ 2^{L_{1}j+M_{1}}/8, let ω_{3,j} = [−18 · 2^{L_{2}j+M_{2}}, 18 · 2^{L_{2}j+M_{2}}] and Φ_{3,j} be a Schwartz function whose Fourier transform is a bump function adapted to ω_{3,j} such that Φ̂_{3,j}(ξ) = 1 for all ξ ∈ [−17 · 2^{L_{2}j+M_{2}}, 17 · 2^{L_{2}j+M_{2}}]. Let n_{3} = 0, Φ_{3,j,n_{3}} = Φ_{3,j}, f_{3,j,n_{3}}(x) = f_{3} ∗ Φ_{3,j,n_{3}}(x). Define a trilinear form Λ_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}} by

(4.2)  Λ_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}}(f_{1}, f_{2}, f_{3}) = ∫ ∑_{j∈ℤ} ∏_{ℓ=1}^{3} f_{ℓ,j,n_{ℓ}}(x)dx.

Clearly Λ_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}} = ∫ Π_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}}(f_{1}, f_{2})(x)f_{3}(x)dx. Thus to prove (4.1), it suffices to prove the following lemma.

Lemma 4.1. Let p_{1}, p_{2} > 1 such that 1/p_{1}+1/p_{2} ≥ 1 and Λ_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}} be the trilinear form defined by ( 4.2). Let F_{1}, F_{2}, F_{3} be measurable sets in ℝ with |F_{3}| = 1. Then there exists a subset F′_{3} ⊆ F_{3} such that |F′_{3}| > 1/2 and there exists a constant C independent of F_{1}, F_{2}, F_{3}, f_{1}, f_{2}, f_{3}, M_{1}, M_{2}, n_{1}, n_{2} such that

(4.3) |Λ_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}}(f_{1}, f_{2}, f_{3})|

≤ C(1 + |n_{1}|)^{10}(1 + |n_{2}|)^{10}|F_{1}|^{1/p_{1}}|F_{2}|^{1/p_{2}}

holds for all f_{1} ∈ X(F_{1}), f_{2} ∈ X(F_{2}), f_{3} ∈ X(F′_{3}).

Lemma 4.1 and Lemma 3.3 implies the estimates (2.3) by an interpolation argument in [15]. Therefore we obtain Theorem 2.1 once we finish a proof of Lemma 4.1. The following subsections are devoted to proof of Lemma 4.1.

4.1. Definitions. To prove Lemma 4.1, we introduce some definitions first. Let ψ be a nonnegative Schwartz function such that ψ̂ is supported in [−1/100, 1/100] and satisfies ψ̂(0) = 1. Let ψ_{k}(x) = 2^{k}ψ(2^{k}x) for any k ∈ ℤ. For j ∈ ℤ and ℓ ∈ {1, 2, 3}, define k_{jℓ} to be an integer such that |ω_{ℓ,j}| ∼ 2^{k_{jℓ}}.

Denote min_{ℓ∈{1,2,3}} k_{jℓ} by k_{j}. And define

I_{k_{j},n} = [2^{−k_{j}}n, 2^{−k_{j}}(n + 1)].

Define

1_{j,n}^{∗}(x) = 1_{I_{k_{j} ,n}} ∗ ψ_{k_{j}}(x).

It is easy to see that

Λ_{L_{1},L_{2},M_{1},M_{2},n_{1},n_{2}}(f_{1}, f_{2}, f_{3}) = ∫ ∑_{j∈ℤ} ∑_{n∈ℤ}1_{j,n}^{∗}(x) ∏_{ℓ=1}^{3} f_{ℓ,j,n_{ℓ}}(x)dx.

For an integer γ with 0 ≤ γ < 2^{100}, let ℤ(γ) be the set of all integers congruent to γ modulo 2^{100}. For S ⊂ ℤ(γ) × ℤ we define

(4.4)  Λ_{S}(f_{1}, f_{2}, f_{3}) = ∫_{ℝ} ∑_{(j,n)∈S} 1_{j,n}^{∗}(x) ∏_{ℓ=1}^{3} f_{ℓ,j,n_{ℓ}}(x)dx.

Λ_{S} depends on L_{1}, L_{2}, M_{1}, M_{1}, n_{1}, n_{2}. We suppress this dependence for notational convenience. Note that there are finite congruence classes modulo 2^{100}. We will therefore concentrate on proving Lemma 4.1 for the trilinear form Λ_{S}.

In time-frequency space, each function f_{ℓ,j,n} for ℓ ∈ {1, 2, 3} corresponds to a box I_{k_{j},n} × ω_{ℓ,j}. The most difficult situation is when only one of boxes is the Heisenberg box, i.e., |I_{k,j,n}||ω_{ℓ,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.

Let p be a positive number close to 1. To obtain the Lemma 4.1, it suffices to prove (4.3) for p_{1} ≥ p, p_{2} ≥ p and 1/p_{1} + 1/p_{2} ≥ 1. For simplicity, we only deal with the case n_{1} = n_{2} = n_{3} = 0. The general case can be handled in the same way by paying at most a cost of (1 + |n_{1}|)^{10}(1 + |n_{2}|)^{10} in the constants.

We now start to prove that for n_{1} = n_{2} = 0, any 1 < p < 2 and any measurable set F_{3} with |F_{3}| = 1 in ℝ, there exists a subset F′_{3} of F_{3} with |F′_{3}| ≥ 1/2 such that

(4.5)  |Λ_{S}(f_{1}, f_{2}, f_{3})| ≤ C|F_{1}|^{1/p_{1}}|F_{2}|^{1/p_{2}}

holds for all p_{1} ≥ p, p_{2} ≥ p with 1/p_{1} + 1/p_{2} ≥ 1, f_{1} ∈ X(F_{1}), f_{2} ∈ X(F_{2}), f_{3} ∈ X(F′_{3}), where the constant C is independent of S, F_{1}, F_{2}, F_{3}, f_{1}, f_{2}, f_{3}, M_{1}, M_{2}. Let us introduce some definitions first.

Definition 4.2. Let p > 1. Define the exceptional set Ω by

(4.6)  Ω = ⋃_{ℓ=1}^{3} {x ∈ ℝ : M_{p}(M1_{F_{ℓ}})(x) > C_{0}|F_{ℓ}|^{1/p}}

where Mf is the Hardy–Littlewood maximal function of f and M_{p}f equals to (M(|f|^{p}))^{1/p}.

By this definition, for the measurable set F_{3} with |F_{3}| = 1, we take F′_{3} = F_{3}\Ω. If C_{0} is chosen sufficiently large we see that |F′_{3}| ≥ |F_{3}|/2.

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

Definition 4.4. Let S be a subset of ℤ(γ) × ℤ. We say that S is a convex set in ℤ(γ)×ℤ if for any s ∈ ℤ(γ)×ℤ with I_{s_{1}} ⊆ I_{s} ⊆ I_{s_{2}} for some s_{1}, s_{2} ∈ S, we have s ∈ S.

Definition 4.5. Let T ⊂ S. If there is t ∈ T such that I_{s} ⊂ I_{t} holds for all s ∈ T, then T is called a tree with top t. T is called a maximal tree with top t in S if there does not exist a larger tree in S with the same top strictly containing T.

Definition 4.6. Let T be a tree in S. Define scl(T) the set of scale indices of T by

scl(T) = {j ∈ ℤ : ∃n ∈ ℤ, s. t. (j, n) ∈ T}.

For j ∈ scl(T), the j-th shadow of T is defined by

Sh_{j}(T) = ⋃ {I_{s} : s = (j, n) ∈ T}.

Define an approximation of 1_{Sh_{j}(T)} by

1_{Sh_{j}(T)}^{∗}(x) = 1_{Sh_{j}(T)} ∗ ψ_{k_{j}}(x).

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

1_{j,n}^{∗∗}(x) = ∫_{I_{k_{j} ,n}} \frac{2^{k_{j}}}{(1 + 2^{2k_{j}}|x − y|^{2})^{200}} dy.

Define a seminorm ∥f_{ℓ}∥_{j,n} by

(4.7) ∥f_{ℓ}∥_{j,n} = ∥f_{ℓ}∥_{s} = \frac{1}{|I_{s}|^{1/p}} ∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{p} + \frac{1}{|I_{s}|^{1/p}} ∥2^{−k_{jℓ}}1_{j,n}^{∗∗}Df_{ℓ,j,n_{ℓ}} ∥_{p}

where Df_{ℓ,j,n_{ℓ}} is the derivative of f_{ℓ,j,n_{ℓ}}.

Define ζ(j, M, K) by

(4.8)  ζ(j, M, K) = [\frac{L_{1}j + M_{1} − M_{2} − 6}{L_{2}}] + [\frac{L_{1}}{L_{2}} ]M + K,

where L = 2^{100}, K is an integer between −10L and 10L and M is an integer between 0 and 6L. For ℓ ∈ {2, 3}, we define a ζ seminorm ∥f_{ℓ}∥_{j,n,ζ} by

∥f_{ℓ}∥_{j,n,ζ} = (4.9)

∥f_{ℓ}∥_{j,n} + sup_{M,K} \frac{1}{|I_{s}|^{1/p}} (∥1_{j,n}^{∗∗}f_{ℓ,ζ(j,M,K),0}∥_{p} + ∥|I_{s}|1_{j,n}^{∗∗}Df_{ℓ,ζ(j,M,K),0}∥_{p}).

For ℓ = 1, let the ζ seminorm ∥f_{1}∥_{j,n,ζ} = ∥f_{1}∥_{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 tree T.

(a) In the case |ω_{2,j}| ≤ |ω_{1,j}|/6 for all j ∈ scl(T), define Δ_{ℓ}^{∗}(T) for ℓ ∈ {1, 3} by

(4.10)  Δ_{ℓ}^{∗}(T)(x) = ( ∑_{(j,n)∈T} |1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}}(x)|^{2})^{1/2}.

For ℓ = 2, define

(4.11)  Δ_{2}^{∗}(T)(x) = |1_{j_{T},n_{T}}^{∗∗}f_{2,j_{T},n_{ℓ}}(x)|.

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

(4.12)  size_{ℓ}(T) = \frac{1}{|I_{T}|^{1/p}} ∥Δ_{ℓ}^{∗}(T)∥_{p} + ∥f_{ℓ}∥_{j_{T},n_{T}}.

(b) In the case |ω_{2,j}| > |ω_{1,j}|/6 for all j ∈ scl(T), for ℓ = 2, 3, let f_{ℓ,j,T} = f_{ℓ,j,0} if j ∈ scl(T) and f_{ℓ,j,T} ≡ 0 if j ∉ scl(T). Define:

(4.13)  Δ_{ℓ}^{∗}(T) = ( ∑_{(j,n)∈T} |1_{j,n}^{∗∗}(f_{ℓ,j,T} − f_{ℓ,j−L,T})(x)|^{2})^{1/2}

+ ( ∑_{(j,n)∈T} |1_{j,n}^{∗∗}(f_{ℓ,j,n_{ℓ}} − f_{ℓ,j,0})(x)|^{2})^{1/2}.

And define Δ_{1}^{∗}(T) by

(4.14)  Δ_{1}^{∗}(T)(x) = ( ∑_{(j,n)∈T} |1_{j,n}^{∗∗}f_{1,j,n_{1}}(x)|^{2})^{1/2}.

In this case, for ℓ ∈ {1, 2, 3}, define the ℓ-size of T by

(4.15)  size_{ℓ}(T) = \frac{1}{|I_{T}|^{1/p}} ∥Δ_{ℓ}^{∗}(T)∥_{p} + ∥f_{ℓ}∥_{j_{T},n_{T},ζ}.

Let P be a subset of S. Define the ℓ-size^{∗} of P by

(4.16)  size_{ℓ}^{∗}(P) = sup_{T:T⊂P} size_{ℓ}(T),

where T ranges over all trees in P.

In the definition of 1_{j,n}^{∗∗}, we can replace the exponent 200 by a larger number 2^{100} to define a new function. We denote this function by ˜1_{j,n}^{∗}. If 1_{j,n}^{∗∗} is replaced by ˜1_{j,n}^{∗} in the definition of Δ_{ℓ}^{∗}(T), we denote the corresponding function by Δ_{ℓ}(T).

Definition 4.9. Let S be a subset of ℤ(γ) × ℤ. Suppose that S is a union of trees T ∈ F. Define count(S) by

(4.17)  count(S) = ∑_{T∈F} |I_{T}|.

4.2. Reduction. Let S be a subset of ℤ(γ) × ℤ. For Ω defined in (4.6), we define

(4.18)  S(Ω) = {s ∈ S : I_{s} ⊈ Ω}.

The following lemma indicates that we only need to seek the upper bound for the trilinear form Λ_{S(Ω)}.

Lemma 4.10. Let n_{1} = n_{2} = 0 and f_{3} ∈ X(F′_{3}). For all functions f_{1} ∈ X(F_{1}), f_{2} ∈ X(F_{2}), the following inequality holds.

(4.19) |Λ_{S}(f_{1}, f_{2}, f_{3}) − Λ_{S(Ω)}(f_{1}, f_{2}, f_{3})|

≤ C min {1, |F_{1}|^{1/p}} min {1, |F_{2}|^{1/p}},

where C is a constant independent of S, F_{1}, F_{2}, F_{3}, f_{1}, f_{2}, f_{3}, M_{1}, M_{2}.

Proof. Notice that if s = (j, n) ∈ S(Ω)^{c}, then I_{s} ⊆ Ω. Let S_{L}(Ω) be defined by

S_{L}(Ω) = {s ∈ S(Ω)^{c} : 2^{L}I_{s} ⊆ Ω, but 2^{L+1}I_{s} ⊈ Ω}.

We see that S(Ω)^{c} = ∪_{L=0}^{∞}S_{L}(Ω). Let J_{L} be the set of all time intervals I_{s}’s for s ∈ S_{L}(Ω). It is easy to see that J_{L} is a collection of disjoint intervals and ∑_{J∈J_{L}} |J| ≤ |Ω| < 1. Hence, it suffices to show that for any J ∈ J_{L} and any (j, n) = s ∈ S_{L}(Ω) such that I_{s} = J, we have

(4.20) | ∫ 1_{j,n}^{∗}(x) ∏_{ℓ} f_{ℓ,j,n_{ℓ}}(x)dx|

≤ C2^{−L} min {1, |F_{1}|^{1/p}} min {1, |F_{2}|^{1/p}}|J|,

where C is a constant independent of f_{1}, f_{2}, f_{3}, M_{1}, M_{2}, since (4.19) follows by summing all L’s and J’s together.

We now prove (4.20). Since F′_{3} = F_{3}\Ω and f_{3} ∈ X(F′_{3}), we get for any (j, n) ∈ S and any positive integer N,

(4.21) |1_{j,n}^{∗}(x)f_{3,j,n_{ℓ}}(x)| ≤ \frac{C_{N}}{(1 + 2^{k_{j}}dist(x, I_{s}))^{3N}(1 + 2^{k_{j3}}dist(x, Ω^{c}))^{3N}} .

Clearly we have for ℓ ∈ {1, 2} and (j, n) ∈ S,

(4.22)  |f_{ℓ,j,n_{ℓ}}(x)| ≤ ∫ \frac{C_{N}|f_{ℓ}(y)|2^{k_{jℓ}} dy}{(1 + 2^{k_{jℓ}}|x − y|)^{N}} .

By the definition of Ω, we have for ℓ ∈ {1, 2} and (j, n) ∈ S,

(4.23)  |f_{ℓ,j,n_{ℓ}}(x)| ≤ C_{N} min {1, |F_{ℓ}|^{1/p}}(1 + 2^{k_{jℓ}}dist(x, Ω^{c}))^{2}.

Thus (4.21), (4.23) and the fact 2^{k_{j3}} ∼ 2^{max{k_{jℓ}}} yield that the left-hand side of (4.20) is no more than

C_{N}2^{−LN} ∏_{ℓ=1}^{2} min {1, |F_{ℓ}|^{1/p}}|J|

for any positive integer N ≥ 2, which is the desired estimate. □

Hence, to prove (4.5), we only need to prove the following lemma for Λ_{S(Ω)}. The details of the proof of Lemma 4.11 will be given in the next few subsections.

Lemma 4.11. Let n_{1} = n_{2} = 0, 1 < p < 2, F_{3} ⊂ ℝ, and S(Ω) be the set defined in (4.18) and F′_{3} = F \Ω. For all p_{1}, p_{2} ≥ p with 1/p_{1}+1/p_{2} ≥ 1, and all functions f_{1} ∈ X(F_{1}), f_{2} ∈ X(F_{2}), f_{3} ∈ X(F′_{3}), the following inequality holds.

(4.24)  |Λ_{S(Ω)}(f_{1}, f_{2}, f_{3})| ≤ C|F_{1}|^{1/p_{1}}|F_{2}|^{1/p_{2}},

where C is a constant independent of S, F_{1}, F_{2}, F_{3}, f_{1}, f_{2}, f_{3}, M_{1}, M_{2}.

4.3. Principal lemmas. We now state some lemmata which will be used in proof of Lemma 4.11.

Lemma 4.12. Let 1 < q < ∞, ℓ ∈ {1, 2, 3} and T be a tree in S. Then

(4.25)  ∥Δ_{ℓ}^{∗}(T)∥_{q} ≤ C inf_{x∈I_{T}} M_{q}(Mf_{ℓ})(x)|I_{T}|^{1/q},
(4.26)  size_{ℓ}(T) ≤ C inf_{x∈I_{T}} M_{p}(Mf_{ℓ})(x),

where C is a constant independent of f_{ℓ}, T, S, M_{1}, M_{2}.

Proof. (4.25) is a consequence of the following L^{q} estimates of Δ_{ℓ}(T).

(4.27)  ∥Δ_{ℓ}^{∗}(T)∥_{q} ≤ C∥f_{ℓ}∥_{q}.

In fact, one can decompose f_{ℓ} into f_{ℓ}1_{2I_{T}} and f_{ℓ}1_{(2I_{T})^{c}}. For the first function, apply (4.27) to get the desired estimates. For the second function, the desired estimates follow by the fast decay due to Δ_{ℓ}^{∗}(T) is essentially supported on I_{T}.

Note that we consider only the case n_{ℓ} = 0. For n_{ℓ} ≠ 0, the following argument still works if one changes the constant C to C(1 + |n_{ℓ}|)^{5}. We only give the details for the case |ω_{2,j}| ≤ |ω_{1,j}|/2 and ℓ ∈ {1, 3} since other cases can be done in the same way. In this case, we have

Δ_{ℓ}^{∗}(T)(x) = ( ∑_{(j,n)∈T} |1_{j,n}^{∗∗}f_{ℓ,j,0}(x)|^{2})^{1/2}.

Notice that Δ_{ℓ}^{∗}(T)(x) is dominated by

( ∑_{j∈ℤ} |f_{ℓ,j,0}(x)|^{2})^{1/2},

where f_{ℓ,j,0} is defined by f̂_{ℓ,j,0} = f̂_{ℓ}Φ̂_{ℓ,j,0}. Note that Φ̂_{ℓ,j,0} is supported on ω_{ℓ,j} and ω_{ℓ,j}’s are disjoint. Thus the Littlewood–Paley theorem then yields the L^{q} estimates (4.27). To get (4.25), it suffices to show that

∥Δ_{ℓ,out}^{∗}(T)∥_{q} ≤ C inf_{x∈I_{T}} M_{q}(Mf_{ℓ})(x)|I_{T}|^{1/q},

where Δ_{ℓ,out}^{∗}(T) is defined by

Δ_{ℓ,out}^{∗}(T)(x) = ( ∑_{(j,n)∈T} |1_{j,n}^{∗∗}(x)((f1_{(2I_{T})^{c}}) ∗ Φ_{ℓ,j,0})(x)|^{2})^{1/2}.

By the definition of 1_{j,n}^{∗∗} and Φ_{ℓ,j,0}, we have that for any positive integer N,

|1_{j,n}^{∗∗}(x)((f1_{(2I_{T})^{c}}) ∗ Φ_{ℓ,j,0})(x)|

≤ \frac{C_{N}}{(1 + 2^{k_{j}}dist(x, I_{s}))^{100}} ∫_{(2I_{T})^{c}} \frac{|f_{ℓ}(y)|2^{k_{jℓ}}}{(1 + 2^{k_{jℓ}}|x − y|)^{N}} dy.

which is clearly dominated by

\frac{CMf_{ℓ}(x)}{(1 + 2^{k_{j}}dist(x, I_{s}))^{50}(1 + 2^{k_{j}}dist(I_{s}, (2I_{T})^{c}))^{50}} .

Thus for s ∈ T,

∥1_{j,n}^{∗∗}((f1_{(2I_{T})^{c}}) ∗ Φ_{ℓ,j,0})∥_{q}^{q}

≤ \frac{C|I_{s}|}{(1 + 2^{k_{j}}dist(I_{s}, (2I_{T})^{c}))^{25q}} ( inf_{x∈I_{T}} M_{q}(Mf_{ℓ})(x))^{q}.

By the triangle inequality, we obtain that

∥Δ_{ℓ,out}^{∗}(T)∥_{q} ≤ ∑_{s∈T} \frac{C|I_{s}|^{1/q}}{(1 + |I_{s}|^{−1}dist(I_{s}, (2I_{T})^{c}))^{25}} inf_{x∈I_{T}} M_{q}(Mf_{ℓ})(x),

which yields the desired estimate (4.25). Notice that

∥1_{j_{T},n_{T}}^{∗∗}f_{ℓ,j_{T},n_{ℓ}} ∥_{p} + ∥2^{−k_{j_{T}ℓ}}1_{j_{T},n_{T}}^{∗∗}Df_{ℓ,j_{T},n_{ℓ}} ∥_{p} ≤ ∥\frac{CMf_{ℓ}(·)}{(1 + |I_{T}|^{−1}dist(·, I_{T}))^{N}} ∥_{p},

which is clearly dominated by inf_{x∈I_{T}} M_{p}(Mf_{ℓ})(x)|I_{T}|^{1/p}. Therefore we obtain (4.26). □

Lemma 4.13. Suppose that s = (j, n) ∈ S.

If 2^{k_{jℓ}} ∼ 2^{k_{j}}, then

(4.28)  ∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{∞} ≤ C∥f_{ℓ}∥_{j,n}

holds for ℓ ∈ {1, 2, 3}, where C is a constant independent of s, f_{ℓ}, n_{ℓ}. If 2^{k_{j1}} ∼ 2^{k_{j}}, then

(4.29)  ∥1_{j,n}^{∗∗}f_{ℓ,ζ(j,M,K),n_{ℓ}} ∥_{∞} ≤ C∥f_{ℓ}∥_{j,n,ζ}

holds for ℓ ∈ {2, 3}, where ζ(j, M, K) is defined in Definition 4.7 and C is a constant independent of s, f_{ℓ}, n_{ℓ}, ζ, M, K.

Proof. We only prove (4.28) since (4.29) essentially is a consequence of (4.28). Let μ = ∥f_{ℓ}∥_{j,n}. By the definition of the seminorm, we have

(4.30)  ∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{p} + ∥|I_{s}|1_{j,n}^{∗∗}Df_{ℓ,j,n_{ℓ}} ∥_{p} ≤ μ|I_{s}|^{1/p}.

First we prove the BMO estimate for the function, that is

(4.31)  ∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{BMO} ≤ Cμ.

If |I_{s}| ≤ |J|, by (4.30) we have

inf_{c} ∫_{J} |1_{j,n}^{∗∗}(x)f_{ℓ,j,n_{ℓ}}(x) − c|dx ≤ ∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{p}|J|^{1− \frac{1}{p}} ≤ μ|I_{s}|^{\frac{1}{p}} |J|^{1− \frac{1}{p}} ≤ μ|J|.

If |I_{s}| ≥ |J|, by (4.30) we obtain that

inf_{c} ∫_{J} |1_{j,n}^{∗∗}(x)f_{ℓ,j,n_{ℓ}}(x) − c|dx

≤ |J| ∫_{J} |(1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}})′(x)|dx

≤ |J| ∫_{J} |(1_{j,n}^{∗∗})′(x)f_{ℓ,j,n_{ℓ}}(x)|dx + |J| ∫_{J} |1_{j,n}^{∗∗}(x)Df_{ℓ,j,n_{ℓ}}(x)|dx

≤ C|J||I_{s}|^{−1}∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{p}|J|^{1− \frac{1}{p}} + |J|∥1_{j,n}^{∗∗}Df_{ℓ,j,n_{ℓ}} ∥_{p}|J|^{1− \frac{1}{p}}

≤ Cμ|J|^{2− \frac{1}{p}} |I_{s}|^{\frac{1}{p} −1} ≤ Cμ|J|.

Thus we get the BMO estimate (4.31). Interpolating (4.31) and (4.30), we have for any p ≤ q < ∞,

∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{q} ≤ Cμ|I_{s}|^{1/q}.

Notice that an integration by parts and Hölder inequality yield that

∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{∞} ≤ ∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{p′}^{1/2}∥(1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}})′∥_{p}^{1/2},

where 1/p + 1/p′ = 1. Hence the desired estimate (4.28) follows by (4.30) and L^{p′} estimates for the functions. □

Lemma 4.14. Suppose that 2^{k_{jℓ}} ∼ 2^{k_{j}} holds for all (j, n) ∈ S. Then for any tree T in S, we have

(4.32)  ∥Δ_{ℓ}(T)∥_{BMO} ≤ Csize_{ℓ}^{∗}(T),

where C is a constant independent of T, S, L_{1}, L_{2}, M_{1}, M_{2}, f_{ℓ}, n_{ℓ}.

Proof. We only give the a proof for ℓ = 1. Other cases can be handled in the same way. Let μ = size_{ℓ}^{∗}(S). Let J be a dyadic interval and T_{J} = {s ∈ T : I_{s} ⊆ J}. We then dominate inf_{c} ∫_{J} |Δ_{ℓ}(T)(x) − c|dx by a sum of the following three parts.

∫_{J} ( ∑_{s∈T_{J}} |˜1_{j,n}^{∗}(x)f_{ℓ,j,n_{ℓ}}(x)|^{2})^{1/2}dx,

∫_{J} ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|≤|J| \end{matrix}} |˜1_{j,n}^{∗}(x)f_{ℓ,j,n_{ℓ}}(x)|^{2})^{1/2} dx,

and

inf_{c} ∫_{J} |( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} |˜1_{j,n}^{∗}(x)f_{ℓ,j,n_{ℓ}}(x)|^{2})^{1/2} − c| dx.

The first part is clearly dominated by μ|J| because of the Hölder inequality and the fact that μ is the ℓ-size^{∗} of S.

Since p ≤ 2 we estimate the second part by

∥( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|≤|J| \end{matrix}} |˜1_{j,n}^{∗}f_{ℓ,j,n_{ℓ}} |^{2})^{1/2}∥_{L^{p}(J)} |J|^{1− \frac{1}{p}}

≤ ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|≤|J| \end{matrix}} ∥˜1_{j,n}^{∗}f_{ℓ,j,n_{ℓ}} ∥_{L^{p}(J)}^{p})^{1/p} |J|^{1− \frac{1}{p}}

≤ ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|≤|J| \end{matrix}} \frac{C∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{p}^{p}}{(1 + |I_{s}|^{−1}dist(J, I_{s}))^{100}} )^{1/p} |J|^{1− \frac{1}{p}}

≤ μ ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|≤|J| \end{matrix}} \frac{C|I_{s}|}{(1 + |I_{s}|^{−1}dist(J, I_{s}))^{100}} )^{1/p} |J|^{1− \frac{1}{p}} ≤ Cμ|J|.

The third part is estimated by

(inf_{c} ∫_{J} |( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} |˜1_{j,n}^{∗}(x)f_{ℓ,j,n_{ℓ}}(x)|^{2})^{1/2} − c|^{2} dx)^{1/2} |J|^{1/2}

≤ (inf_{c} ∫_{J} | ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} |˜1_{j,n}^{∗}(x)f_{ℓ,j,n_{ℓ}}(x)|^{2} − c| dx)^{1/2} |J|^{1/2}

≤ C (∫_{J} ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} |(|˜1_{j,n}^{∗}(x)f_{ℓ,j,n_{ℓ}}(x)|^{2})′|dx)^{1/2} |J|,

which is dominated by a sum of the following two terms,

R_{1} = C (∫_{J} ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} |I_{s}|^{−1}|˜1_{j,n}^{∗}(x)f_{ℓ,j,n_{ℓ}}(x)|^{2}dx)^{1/2} |J|,

and

R_{2} = C (∫_{J} ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} |˜1_{j,n}^{∗}(x)f_{ℓ,j,n_{ℓ}}(x)||˜1_{j,n}^{∗}(x)Df_{ℓ,j,n_{ℓ}}(x)|dx)^{1/2} |J|,

By Lemma 4.13, we see that for any q ≥ p,

∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{q} ≤ Cμ|I_{T}|^{1/q}.

Thus, by Hölder’s inequality, the first term R_{1} is estimated by

C ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} \frac{|I_{s}|^{−1}}{(1 + |I_{s}|^{−1}dist(J, I_{s}))^{100}} ∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{4}^{2}|J|^{1/2})^{1/2} |J|

≤ Cμ ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} \frac{|I_{s}|^{−1/2}|J|^{1/2}}{(1 + |I_{s}|^{−1}dist(J, I_{s}))^{100}} )^{1/2} |J| ≤ Cμ|J|,

and the second term R_{2} is estimated by

C ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} ∥˜1_{j,n}^{∗}f_{ℓ,j,n_{ℓ}} ∥_{L^{p′}(J)}∥1_{j,n}^{∗∗}Df_{ℓ,j,n_{ℓ}} ∥_{p})^{1/2} |J|

≤ C (μ ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} \frac{|I_{s}|^{\frac{1}{p} −1}∥1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}} ∥_{p′+1}|J|^{\frac{1}{p′(p′+1)}}}{(1 + |I_{s}|^{−1}dist(J, I_{s}))^{100}} )^{1/2} |J|

≤ Cμ ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} \frac{|I_{s}|^{− \frac{1}{p′(p′+1)}} |J|^{\frac{1}{p′(p′+1)}}}{(1 + |I_{s}|^{−1}dist(J, I_{s}))^{100}} )^{1/2} |J| ≤ Cμ|J|.

This completes the proof of (4.32). □

The principal lemma is the following organization lemma.

Lemma 4.15. Let ℓ ∈ {1, 2, 3} and S be a subset of ℤ(γ) × ℤ. S can be partitioned to two parts S_{1} and S_{2} such that S_{1} is a union of maximal trees with

(4.33)  count(S_{1}) ≤ C(size_{ℓ}^{∗}(S))^{−p}|F_{ℓ}|,

and

(4.34)  size_{ℓ}^{∗}(S_{2}) ≤ \frac{1}{2}size_{ℓ}^{∗}(S),

where C is a constant independent of S, M_{1}, M_{2}, f_{ℓ}, F_{ℓ}.

Proof. Let F_{0} be the set of all trees T ⊂ S such that size_{ℓ}(T) > size_{ℓ}^{∗}(S)/2. Recall that I_{T} is the time interval for the top of T. Let I denote the collection of all possible I_{T}’s for trees T ∈ F_{0}. Initially, set S_{1} := ∅, I_{stock} := I, and S_{stock} := S. Take a longest interval J in I_{stock}. By the defintion of I, there must be a tree T ∈ F_{0} whose top is J. Let T̃ be the maximal tree in S_{stock} with the top J. Obviously size_{ℓ}(T̃) ≥ size_{ℓ}^{∗}(S)/2. We remove this maximal tree from S_{stock}. Update S_{stock} := S_{stock}\T̃, S_{1} := S_{1} ∪ T̃, and

I_{stock} := I_{stock}\{I ∈ I_{stock} : I ⊆ J}.

Repeat this procedure until I_{stock} = ∅. Clearly when this process terminates, S_{1} is a union of a trees T̃’s and IT̃’s are disjoint due to the maximality of trees. By (4.26) and the size condition on T̃, we have

inf_{x∈IT̃} M_{p}(Mf_{ℓ})(x) ≥ size_{ℓ}^{∗}(S)/2,

which implies that

⋃ ; T̃ IT̃ ⊆ {x ∈ ℝ : M_{p}(Mf_{ℓ})(x) ≥ size_{ℓ}^{∗}(S)/2}.

Thus the disjointness property of IT̃’s and (weak) L^{q} estimates for 1 ≤ q ≤ ∞ of Hardy–Littlewood maximal functions yield (4.33). Let S_{2} = S\S_{1}. Clearly S_{2} satisfies (4.34). Therefore we complete the proof of Lemma 4.15. □

4.4. The size estimate for a tree. Let S be a convex subset of ℤ(γ)×ℤ. By the definition of S(Ω) in (4.18), it is clear that S(Ω) is convex. Partition S(Ω) into two subsets S^{(1)}(Ω) and S^{(2)}(Ω), where

(4.35)  S^{(1)}(Ω) = {(j, n) ∈ S(Ω) : |ω_{2,j}| ≤ |ω_{1,j}|/6}
(4.36)  S^{(2)}(Ω) = {(j, n) ∈ S(Ω) : |ω_{2,j}| > |ω_{1,j}|/6}.

For any (j, n) ∈ S^{(1)}(Ω), k_{j2} = k_{j} by the definition of k_{j}. And for any (j, n) ∈ S^{(2)}(Ω), 2^{k_{j1}} ∼ 2^{k_{j}}.

Lemma 4.16. For κ ∈ {1, 2}, S^{(κ)}(Ω) is convex.

Proof. We only prove the lemma for κ = 2. One can prove the lemma for κ = 1 similarly. Let s_{1} = (j_{1}, n_{1}), s_{2} = (j_{2}, n_{2}) in S^{(2)}(Ω). And s = (j, n) ∈ ℤ(γ) × ℤ such that I_{s_{2}} ⊆ I_{s} ⊆ I_{s_{1}}. By the convexity of S(Ω) we get s ∈ S(Ω). In order to get s ∈ S^{(2)}(Ω), we need to show that |ω_{2,j}| > |ω_{1,j}|/6. The simple case is the case 2^{k_{j}} = |ω_{1,j}|. In this case,

|ω_{1,j_{2}}|/10 ≤ |ω_{1,j}| ≤ 10|ω_{1,j_{1}}|,

which implies j_{2} ≤ j ≤ j_{1}. Since |ω_{2,j_{1}}| > |ω_{1,j_{1}}|/6 and |ω_{2,j_{2}}| > |ω_{1,j_{2}}|/6, the linearity of the function f(j) = (L_{1}j + M_{1}) − (L_{2}j + M_{2}) yields that |ω_{2,j}| > |ω_{1,j}|/6.

We now turn to another case 2^{k_{j}} = |ω_{2,j}|. Since I_{s} is nested between I_{s_{1}} and I_{s_{2}}, we get |ω_{1,j_{2}}|/10 ≤ |ω_{2,j}| ≤ 10|ω_{1,j_{1}}|. The first half part of this inequality and the definition of k_{j} imply j_{2} ≤ j. And the second half part of the inequality and the fact (j_{1}, n_{1}) ∈ S^{(2)}(Ω) yield j ≤ j_{1}. Thus we get |ω_{2,j}| > |ω_{1,j}|/6 by the linearity of the function f(j). Hence s must be in S^{(2)}(Ω) in either case. This proves the lemma. □

Lemma 4.17. Let κ ∈ {1, 2}, T be a convex tree in S^{(κ)}(Ω) with the top t = (j_{T}, n_{T}) and ∂Sh_{j}(T) be the boundary of the j-th shadow of T. Let Card(∂Sh_{j}(T)) denote the cardinality of the boundary of the j-th shadow. Then

(4.37)  ∑_{j≥j_{T}} 2^{−k_{j}}Card(∂Sh_{j}(T)) ≤ C|I_{T}|,

where C is a constant independent of T.

Proof. This lemma is similar to one technical lemma (Lemma 4.8) in [17]. We give a similar proof. Note that the j-th shadow consists of finite disjoint intervals and its boundary thus contains all endpoints of the intervals. It is sufficient to consider only all left endpoints since the right endpoints can be handled in the same way. Let ∂_{left}(Sh_{j}(T)) denote the collection of all left endpoints of the intervals in the j-th shadow. Let z ∈ ∂_{left}(Sh_{j}(T)) and I_{j}(z) = (z − 2^{−k_{j}}, z − 2^{−k_{j}}/2). To prove (4.37), it suffices to show that the intervals I_{j}(z)’s are disjoint for all possible j, z. Assume that there are j, j′ ∈ scl(T), z ∈ ∂_{left}(Sh_{j}(T)) and z′ ∈ ∂_{left}(Sh_{j′}(T)) such that (j, z) ≠ (j′, z′) and I_{j}(z) ∩ I_{j′}(z′) ≠ ∅. By the nesting property of dyadic intervals and the fact that z − 2^{−k_{j}} is an endpoint of some dyadic intervals, we see that j ≠ j′. Without loss of generality, suppose that j < j′. The fact that I_{j}(z) and I_{j′}(z′) have nonempty intersection then implies z′ ∈ (z − 2^{−k_{j}}, z). Since z is a left endpoint of some intervals in the j-th shadow, z′ can not be in Sh_{j}(T). However, the convexity of T yields that Sh_{j′}(T) ⊆ Sh_{j}(T). This is a contradiction. Therefore we obtain the lemma. □

Lemma 4.18. Let κ ∈ {1, 2}, let T be a convex tree in S^{(κ)}(Ω) and let Λ̃_{T}(f_{1}, f_{2}, f_{3}) be defined by

(4.38)  Λ̃_{T}(f_{1}, f_{2}, f_{3}) = ∑_{j} ∫ ∏_{ℓ=1}^{3} ∑_{n∈T_{j}} F_{ℓ,j,n}(x)dx,

where T_{j} = {n ∈ ℤ : (j, n) ∈ T} and F_{ℓ,j,n} is defined by

(4.39)  F_{ℓ,j,n}(x) = 1_{j,n}^{∗}(x)f_{ℓ,j,n_{ℓ}}(x).

Then we have

(4.40)  |Λ_{T}(f_{1}, f_{2}, f_{3}) − Λ̃_{T}(f_{1}, f_{2}, f_{3})| ≤ Csize_{1}^{∗}(T)size_{2}^{∗}(T)|I_{T}|,

where C is a constant independent of T, S, f_{1}, f_{2}, f_{3}.

Proof. Observe that the difference |Λ_{T} − Λ̃_{T}| by

∑_{j∈scl(T)} ∫ |1_{Sh_{j}(T)}^{∗}(x) − (1_{Sh_{j}(T)}^{∗})^{3}(x)| ∏_{ℓ=1}^{3} |f_{ℓ,j,n_{ℓ}}(x)|dx,

Uniform estimates for some paraproducts 167

which is dominated by

∑_{j∈scl(T)} ∑_{I:|I|=2^{−kj}}

∫_{I} |(1_{Sh_{j}(T)}^{∗}(x) − (1_{Sh_{j}(T)}^{∗})^{3}(x))(˜1_{Sh_{j}(T)}^{∗}(x))^{− \frac{1}{10}} |Π_{j,T}(f_{1}, f_{2}, f_{3})(x)dx, where

(4.41) ˜1_{Sh_{j}(T)}^{∗}(x) = ∫_{Sh_{j}(T)} \frac{2^{k_{j}}}{(1 + 2^{2k_{j}}|x − y|^{2})^{2^{1000}}} dx

and

Π_{j,T}(f_{1}, f_{2}, f_{3})(x) = ∏_{ℓ=1}^{3} |(˜1_{Sh_{j}(T)}^{∗})^{1/30}f_{ℓ,j,n_{ℓ}}(x)|.

Hölder’s inequality, Lemma 4.13 and (4.25) then yield that

(4.42) ∥Π_{j,T}(f_{1}, f_{2}, f_{3})∥_{L^{1}(I)} ≤ Csize_{1}^{∗}(T)size_{2}^{∗}(T)2^{−k_{j}}.

Thus we estimate the difference |Λ_{T} − Λ̃_{T}| by

Csize_{1}^{∗}(T)size_{2}^{∗}(T)

· ∑_{j∈scl(T)} ∑_{I:|I|=2^{−k_{j}}} |I|∥(1_{Sh_{j}(T)}^{∗} − (1_{Sh_{j}(T)}^{∗})^{3})(˜1_{Sh_{j}(T)}^{∗})^{−1/10}∥_{L^{∞}(I)},

By the definition of 1_{Sh_{j}(T)}^{∗}, it is easy to see that it is a smooth approximation of 1_{Sh_{j}(T)} and for any positive interger N the following inequality holds.

|I|∥(1_{Sh_{j}(T)}^{∗} − (1_{Sh_{j}(T)}^{∗})^{3})(˜1_{Sh_{j}(T)}^{∗})^{−1/10}∥_{L^{∞}(I)}

≤ \frac{C_{N}|I|}{(1 + |I|^{−1}dist(I, ∂Sh_{j}(T)))^{N}} .

Summing up all I’s with |I| = 2^{−k_{j}}, we estimate the difference by

Csize_{1}^{∗}(T)size_{2}^{∗}(T) ∑_{j∈scl(T)} 2^{−k_{j}}Card(∂Sh_{j}(T)).

Hence the lemma follows by Lemma 4.17. □ Lemma 4.19. Let T be a convex tree in S^{(2)}(Ω). For ℓ ∈ {2, 3}, let F_{ℓ,j} be defined by

(4.43)  F_{ℓ,j}(x) = 1_{Sh_{j}(T)}^{∗}(x)f_{ℓ,j,0}(x),

if T_{j} ≠ ∅, and F_{ℓ,j} ≡ 0 if T_{j} = ∅. Then we have

(4.44)  sup_{M} ∥( ∑_{j} |F_{ℓ,j−M} − F_{ℓ,j−M−L}|^{2})^{1/2}∥_{p} ≤ Csize_{ℓ}^{∗}(T)|I_{T}|^{1/p},

where L = 2^{100}, M ranges over all integers between 0 and 6L and C is a constant independent of f_{ℓ}, T.

Proof. For simplicity, we only prove the lemma for M = 0. It is easy to see that |F_{ℓ,j} − F_{ℓ,j−L}(x)| is dominated by

|1_{Sh_{j}(T)}^{∗}(x)(f_{ℓ,j,0}(x)−f_{ℓ,j−L,0}(x))|+|(1_{Sh_{j}(T)}^{∗}(x)−1_{Sh_{j−L}(T)}^{∗}(x))f_{ℓ,j−L}(x)|.

Clearly, by the definition of Δ_{ℓ}^{∗}(T) and size_{ℓ}^{∗}(T), we get

∥( ∑_{j} |1_{Sh_{j}(T)}^{∗}(f_{ℓ,j,0} − f_{ℓ,j−L,0})|^{2})^{1/2}∥_{p} ≤ C∥Δ_{ℓ}^{∗}(T)∥_{p} ≤ Csize^{∗}(T)|I_{T}|^{1/p}.

Thus to obtain (4.44), it suffices to show that

(4.45) ∥( ∑_{j} |(1_{Sh_{j}(T)}^{∗} − 1_{Sh_{j−L}(T)}^{∗})f_{ℓ,j−L,0}|^{2})^{1/2}∥_{p} ≤ Csize^{∗}(T)|I_{T}|^{1/p}.

Heuristically one can consider 1_{Sh_{j}(T)}^{∗} as 1_{Sh_{j}(T)}. Then by the nesting property of the j-th shadows due to the convexity of the tree, we see that Sh_{j−L}(T)\Sh_{j}(T)’s are disjoint and this is the reason why we have such an estimate.

Now we go to the technical details. Since p ≤ 2, we estimate the left-hand side of (4.45) by

( ∑_{j∈scl(T)} ∥(1_{Sh_{j}(T)}^{∗} − 1_{Sh_{j−L}(T)}^{∗})f_{ℓ,j−L,0}∥_{p}^{p})^{1/p}.

This is dominated by

( ∑_{j∈scl(T)} ∑_{I:|I|=2^{−kj}}

∫_{I} |(1_{Sh_{j}(T)}^{∗} − 1_{Sh_{j−L}(T)}^{∗})(x)(˜1_{Sh_{j−L}(T)}^{∗}(x))^{− \frac{1}{10}} Π_{j}^{∗}(f_{ℓ})(x)|^{p}dx)^{1/p},

where ˜1_{Sh_{j}(T)}^{∗} is the function defined in (4.41) and

Π_{j}^{∗}(f_{ℓ}) = (˜1_{Sh_{j−L}(T)}^{∗})^{1/10}f_{ℓ,j−L,0}.

Hölder’s inequality, Lemma 4.13 and (4.25) then yield that

(4.46)  ∥Π_{j}^{∗}(f_{ℓ})∥_{L^{p}(I)} ≤ Csize_{1}^{∗}(T)size_{2}^{∗}(T)|I|^{1/p}.

Uniform estimates for some paraproducts 169

Thus we dominate the left-hand side of (4.45) by Csize_{1}^{∗}(T)size_{2}^{∗}(T)

· ( ∑_{j∈scl(T)} ∑_{I:|I|=2^{−k_{j}}} ∥(1_{Sh_{j}(T)}^{∗} − 1_{Sh_{j−L}(T)}^{∗})(˜1_{Sh_{j−L}(T)}^{∗})^{− \frac{1}{10}} ∥_{L^{∞}(I)}|I|)^{1/p}

Since Sh_{j}(T) ⊂ Sh_{j−L}(T), it is easy to see that

|1_{Sh_{j}(T)}^{∗}(x) − 1_{Sh_{j−L}(T)}^{∗}(x)| ≤ C˜1_{Sh_{j−L}(T)}^{∗}(x).

On the other hand, observe that |1_{Sh_{j}(T)}^{∗} − 1_{Sh_{j−L}(T)}^{∗}| is dominated by

dSh_{j}^{∗}(x) = 1_{Sh_{j−L}(T)\Sh_{j}(T)} ∗ ψ_{k_{j−L}}(x) + \frac{C_{N}}{(1 + 2^{k_{j}}dist(x, ∂(Sh_{j}(T))))^{N}} ,

for any positive integer N. Hence the L^{∞}(I) norm of

(1_{Sh_{j}(T)}^{∗} − 1_{Sh_{j−L}(T)}^{∗})(˜1_{Sh_{j−L}(T)}^{∗})^{− \frac{1}{10}}

is estimated by

\frac{C_{N}}{(1 + |I|^{−1}dist(I, Sh_{j−L}(T)\Sh_{j}(T)))^{N}} + \frac{C_{N}}{(1 + |I|^{−1}dist(I, ∂(Sh_{j}(T))))^{N}} .

For those I’s contained in Sh_{j}(T), we have

\frac{1}{(1 + |I|^{−1}dist(I, Sh_{j−L}(T)\Sh_{j}(T)))^{N}} ≤ \frac{1}{(1 + |I|^{−1}dist(I, ∂(Sh_{j}(T))))^{N}} .

For those I’s contained in (Sh_{j−L}(T))^{c}, we get

\frac{1}{(1 + |I|^{−1}dist(I, Sh_{j−L}(T)\Sh_{j}(T)))^{N}}

≤ \frac{1}{(1 + |I|^{−1}dist(I, ∂(Sh_{j−L}(T))))^{N}} .

Thus we have

∑_{I:|I|=2^{−k_{j}}} \frac{1}{(1 + |I|^{−1}dist(I, Sh_{j−L}(T)\Sh_{j}(T)))^{N}}

≤ |I|^{−1}|Sh_{j−L}(T)\Sh_{j}(T)| + Card(∂Sh_{j}(T)) + Card(∂Sh_{j−L}(T)). By the nesting property of j-th shadows, the fact 2^{k_{j}} ∼ 2^{k_{j−L}}, and Lemma 4.17, we obtain that

∑_{j∈scl(T)} ∑_{I:|I|=2^{−k_{j}}} ∥(1_{Sh_{j}(T)}^{∗} − 1_{Sh_{j−L}(T)}^{∗})(˜1_{Sh_{j−L}(T)}^{∗})^{− \frac{1}{10}} ∥_{L^{∞}(I)}|I| ≤ C|I_{T}|,

which yields the desired estimate (4.45). □

Lemma 4.20. Let κ ∈ {1, 2} and T be a convex tree in S^{(κ)}(Ω). Then we have

(4.47)  |Λ_{T}(f_{1}, f_{2}, f_{3})| ≤ Csize_{1}^{∗}(T)size_{2}^{∗}(T)|I_{T}|,

where C is a constant independent of T, S, f_{1}, f_{2}, f_{3}.

Proof. By Lemma 4.18, it is sufficient to show that

(4.48)  |Λ̃_{T}(f_{1}, f_{2}, f_{3})| ≤ Csize_{1}^{∗}(T)size_{2}^{∗}(T)|I_{T}|,

where C is a constant independent of T, S, f_{1}, f_{2}, f_{3}.

We first prove the simple case κ = 1. In this case, k_{j2} = k_{j} for all (j, n) ∈ T. We thus dominate |Λ̃_{T}| by

∫_{ℝ} sup_{j} | ∑_{n∈T_{j}} F_{2,j,n}(x)| ∏_{ℓ≠2} ( ∑_{(j,n)∈T} |F_{ℓ,j,n}(x)|^{2})^{1/2}dx.

By the definition of Δ_{ℓ} and Hölder’s inequality, we estimate |Λ_{T}| by

∥ sup_{(j,n)∈T} |F_{2,j,n_{2}}^{∗}|∥_{∞}∥Δ_{1}(T)∥_{p}∥Δ_{3}(T)∥_{p′} ,

where 1/p + 1/p′ = 1 and F_{ℓ,j,n}^{∗} = 1_{j,n}^{∗∗}f_{ℓ,j,n_{ℓ}}. Lemma 4.13 yields that

∥F_{2,j,n}^{∗}∥_{∞} ≤ size_{2}^{∗}(T).

Clearly the definition of size yields

∥Δ_{1}(T)∥_{p} ≤ size_{1}^{∗}(T)|I_{T}|^{1/p}.

And (4.25) yields

∥Δ_{3}(T)∥_{p′} ≤ C|I_{T}|^{1/p′}.

Putting all of them together, we obtain (4.47) for the case κ = 1.

We now prove the case κ = 2. In this case, 2^{k_{j}} ∼ 2^{k_{j1}} for all (j, n) ∈ T. For simplicity, we only consider the case n_{ℓ} = 0. The general case can be done in the same way by paying a cost of (1 + |n_{ℓ}|)^{10} in the constant. Then we write the trilinear form Λ̃_{T} as

Λ̃_{T}(f_{1}, f_{2}, f_{3}) = ∑_{j∈ℤ}∫ ∏_{ℓ=1}^{3} F_{ℓ,j}(x)dx,

where F_{ℓ,j} is defined in (4.43). Here we take a convenient notation that F_{ℓ,j} is identically zero if j ∉ scl(T). Let L = 2^{100}. By the telescoping argument used in Lemma 3.2, we can write Λ̃_{T} as a finite sum of two types of trilinear forms. One type of them is defined by

(4.49) Λ_{T,1}(f_{1}, f_{2}, f_{3}) = ∫ ∑_{j∈ℤ}F_{1,j+m′(j)−M}(x)Π_{j,L}(F_{2,j}, F_{3,j})(x)dx,

where m′(j) = [(L_{2}j+M_{2}−L_{1}j−M_{1}+6)/L_{1}], M is an integer between 0 and 6L, and Π_{j,L}(F_{2,j}, F_{3,j}) equals either (F_{2,j} − F_{2,j−L})F_{3,j−8L} or F_{2,j−L}(F_{3,j} − F_{3,j−L}). Another type is defined by (4.50)

∫ ∑_{j∈ℤ}( ∑_{k=0}^{m′(j)} F_{1,j+k}(x))(F_{2,j}(x)−F_{2,j−L}(x))(F_{3,j−M}(x)−F_{3,j−M−L}(x))dx,

which is denoted by Λ_{T,2}(f_{1}, f_{2}, f_{3}).

We now prove the estimate for the first type trilinear form Λ_{T,1}. Let us first consider the case

Λ_{T,1}(f_{1}, f_{2}, f_{3}) = ∫ ∑_{j∈ℤ}F_{1,j+m′(j)−M}(x)(F_{2,j} − F_{2,j−L})(x)F_{3,j−8L}(x)dx.

In this case, by the Cauchy–Schwarz inequality, |Λ_{T,1}| is estimated by

∫ ( ∑_{j} |F_{1,j+m′(j)−M}(x)F_{3,j−8L}(x)|^{2})^{1/2}( ∑_{j} |F_{2,j}(x)−F_{2,j−L}(x)|^{2})^{1/2}dx.

Using Hölder’s inequality, we dominate it by

∥( ∑_{j} |F_{1,j+m′(j)−M}F_{3,j−8L}|^{2})^{1/2}∥_{p′} ∥( ∑_{j} |F_{2,j} − F_{2,j−L}|^{2})^{1/2}∥_{p}.

The first factor in this expression is no more than

∥( ∑_{j} | ∑_{n∈T_{j+m′(j)−M}} 1_{j+m′(j)−M,n}^{∗}f_{1,j+m′(j)−M,n_{1}}f_{3,j−8L,0}|^{2})^{1/2}∥_{p′},

which is dominated by

∥( ∑_{j} ∑_{n∈T_{j+m′(j)−M}} |(˜1_{j+m′(j)−M,n}^{∗})^{2}f_{1,j+m′(j)−M,n_{1}}f_{3,j−8L,0}|^{2})^{1/2}∥_{p′}.

We estimate it by

∥( ∑_{(j,n)∈T} |˜1_{j,n}^{∗}f_{1,j,n_{1}} |^{2})^{1/2}∥_{p′} sup_{(j,n)∈T} ∥˜1_{j,n}^{∗}f_{3,ζ(j,M,K),0}∥_{∞},

where K is some integer between −10L and 10L and ζ(j, M, K)is defined as in (4.8). Clearly, ˜1_{j,n}^{∗}f_{3,ζ(j,M,K),0} is bounded. Also by Lemma 4.14 and an interpolation, we have

(4.51)  ∥( ∑_{j} |˜1_{j,n}^{∗}f_{1,j,n_{1}} |^{2})^{1/2}∥_{p′} ≤ Csize_{1}^{∗}(T)|I_{T}|^{1/p′}.

And Lemma 4.19 yields that

(4.52)  ∥( ∑_{j} |F_{2,j} − F_{2,j−L}|^{2})^{1/2}∥_{p} ≤ size_{2}^{∗}(T)|I_{T}|^{1/p}.

(4.51) and (4.52) give us the desired estimate for Λ_{T,1} in the first case. We now consider the case

Λ_{T,1}(f_{1}, f_{2}, f_{3}) = ∫ ∑_{j∈ℤ}F_{1,j+m′(j)−M}(x)F_{2,j−L}(x)(F_{3,j} − F_{3,j−L})(x)dx.

In this case, using the Cauchy–Schwarz inequality, we have that |Λ_{T,1}| is estimated by

∫ ( ∑_{j} |F_{1,j+m′(j)−M}(x)F_{2,j−L}(x)|^{2})^{1/2}( ∑_{j} |F_{3,j}(x) − F_{3,j−L}(x)|^{2})^{1/2}dx.

By Hölder’s inequality, we dominate it by

∥( ∑_{j} |F_{1,j+m′(j)−M}F_{2,j−L}|^{2})^{1/2}∥_{p′} ∥( ∑_{j} |F_{3,j} − F_{3,j−L}|^{2})^{1/2}∥_{p}.

The first factor in this expression is no more than

∥( ∑_{j} ∑_{n∈T_{j+m′(j)−M}} |(˜1_{j+m′(j)−M,n}^{∗})^{2}f_{1,j+m′(j)−M,n_{1}}f_{2,j−L,0}|^{2})^{1/2}∥_{p′}.

We estimate it by

∥( ∑_{(j,n)∈T} |˜1_{j,n}^{∗}f_{1,j,n_{1}} |^{2})^{1/2}∥_{p′} sup_{(j,n)∈T} ∥˜1_{j,n}^{∗}f_{2,ζ(j,M,K),0}∥_{∞},

where K is some integer between −10L and 10L and ζ(j, M, K)is defined as in (4.8). By (4.29) and the definition of size, we see that

(4.53)  sup_{(j,n)∈T} ∥˜1_{j,n}^{∗}f_{2,ζ(j,M,K),0}∥_{∞} ≤ Csize_{2}^{∗}(T).

Lemma 4.19 and (4.26) yield that

(4.54)  ∥( ∑_{j} |F_{3,j} − F_{3,j−L}|^{2})^{1/2}∥_{p} ≤ |I_{T}|^{1/p}.

Putting (4.51), (4.53) and (4.54) together, we thus get the desired estimate for Λ_{T,1} in the second case.

Finally let us estimate Λ_{T,2}. The integrand in (4.50) is dominated by

sup_{j} | ∑_{k=0}^{m′(j)} F_{1,j+k}(x)|( ∑_{j∈ℤ} |(F_{2,j} − F_{2,j−L})(x)|^{2})^{\frac{1}{2}}

· ( ∑_{j∈ℤ} |(F_{3,j−M} − F_{3,j−M−L})(x)|^{2})^{\frac{1}{2}} .

There exist p_{1}, p_{3} ∈ ℝ such that 1/p_{1} + 1/p + 1/p_{3} = 1 and p_{1} > p′, p_{3} > 1. By Hölder’s inequality we dominate Λ_{T,2} by

∥ sup_{j} | ∑_{k=0}^{m′(j)} F_{1,j+k}(x)|∥_{p_{1}} ∥( ∑_{j∈ℤ} |F_{2,j} − F_{2,j−L}|^{2})^{\frac{1}{2}} ∥_{p}

· ∥( ∑_{j∈ℤ} |F_{3,j−M} − F_{3,j−M−L}|^{2})^{\frac{1}{2}} ∥_{p_{3}}.

Just notice that one can simply define the size with respect to any number p_{3} by using L^{p_{3}}, then (4.26) and Lemma 4.19 still hold. Thus we have

(4.55)  ∥( ∑_{j∈ℤ} |F_{3,j−M} − F_{3,j−M−L}|^{2})^{\frac{1}{2}} ∥_{p_{3}} ≤ C|I_{T}|^{1/p_{3}}

Notice that the supports of Fourier transform of F_{1,j+k}’s are essentially disjoint. We thus have

∥ sup_{j} | ∑_{k=0}^{m′(j)} F_{1,j+k}(x)|∥_{p_{1}} ≤ C∥ ∑_{j} F_{1,j}∥_{p_{1}}.

Clearly,

∥ ∑_{j} F_{1,j}∥_{2} ≤ ∥Δ_{1}(T)∥_{2}.

By Lemma 4.14 and an interpolation, we have that

∥Δ_{1}(T)∥_{2} ≤ Csize_{1}^{∗}(T)|I_{T}|^{1/2}.

Thus we get

∥ ∑_{j} F_{1,j}∥_{2} ≤ Csize_{1}^{∗}(T)|I_{T}|^{1/2}.

A routine argument as we did in Lemma 4.14 yields

(4.56)  ∥ ∑_{j} F_{1,j}∥_{BMO} ≤ Csize_{1}^{∗}(T).

Now by an interpolation, we obtain that

(4.57)  ∥ ∑_{j} F_{1,j}∥_{p_{1}} ≤ Csize_{1}^{∗}(T)|I_{T}|^{1/p_{1}}.

Hence the desired estimate for Λ_{T,2} now follows by (4.57), (4.52) and (4.55). Therefore we obtain Lemma 4.20. □

4.5. Proof of Lemma 4.11. We now prove Lemma 4.11. Without loss of generality, we can assume that S is a convex set. Lemma 4.16 then yields that S^{(1)}(Ω) and S^{(2)}(Ω) are convex. By the definition of convexity, we see that the convexity is preserved for a maximal tree in a convex set and the remaining set obtained by removing a maximal tree from a convex set. Thus, applying the organization Lemma 4.15 for S^{(κ)}(Ω) inductively, we decompose

(4.58)  S^{(κ)}(Ω) = ⋃_{σ} S_{σ}^{(κ)},

where κ ∈ {1, 2}, σ ranges over all possible dyadic numbers, S_{σ}^{(κ)} = ∪_{T∈F_{σ}^{(κ)}}T such that F_{σ}^{κ} is a collection of convex trees with

(4.59)  count(S_{σ}^{(κ)}) ≤ Cσ^{−p},

and for both ℓ = 1 and ℓ = 2,

(4.60)  size_{ℓ}^{∗}(S_{σ}^{(κ)}) ≤ σ|F_{ℓ}|^{1/p}.

By Lemma 4.12 and the definition of S(Ω), we know that σ ≤ 1 in order to make S_{σ}^{(κ)} nonempty and we can also sharpen the upper bound in the size estimate for S_{σ}^{(κ)} by

(4.61)  size_{ℓ}^{∗}(S_{σ}^{(κ)}) ≤ min{1, σ|F_{ℓ}|^{1/p}}.

Hence we estimate Λ_{S(Ω)} by

|Λ_{S(Ω)}(f_{1}, f_{2}, f_{3})| ≤ ∑_{κ=1}^{2} ∑_{σ≤1} ∑_{T∈F_{σ}^{(κ)}} |Λ_{T}(f_{1}, f_{2}, f_{3})|.

Lemma 4.20 yields that

|Λ_{S(Ω)}(f_{1}, f_{2}, f_{3})| ≤ ∑_{κ=1}^{2} ∑_{σ≤1} ∑_{T∈F_{σ}^{(κ)}} size_{1}^{∗}(S_{σ}^{(κ)})size_{2}^{∗}(S_{σ}^{(κ)})|I_{T}|.

Applying (4.61) and (4.59), we thus obtain

(4.62) |Λ_{S(Ω)}(f_{1}, f_{2}, f_{3})| ≤ C ∑_{σ≤1} min{1, σ|F_{1}|^{1/p}} min{1, σ|F_{2}|^{1/p}}σ^{−p},

which clearly implies (4.24). Therefore we have proven Lemma 4.11.

5. Proof of Theorem 2.2

We now prove Theorem 2.2. The uniform estimate from L^{2} × L^{2} to L^{1} follows immediately by a change of variables and Littlewood–Paley theory and (2.4) is superfluous. Take this simple idea and we can get the uniform estimate for p_{1}, p_{2} > 2 and 1 < r < 2 in Proposition 5.7 for the case 2^{L_{2}j+M_{2}} < 2^{L_{1}j+M_{1}}/8 or 2^{L_{1}j+M_{1}} < 2^{L_{2}j+M_{2}}/8. For the general case, we pay a cost of m in the operator norm in this range of p_{1}, p_{2}, p to get Lemma 5.4.

For the r < 1 case, we use some idea from Section 4 and one can see that technically it is much simpler than what we did in Section 4. We have to assume (2.4) and pay a little more for the operator norm such as 2^{εm} (see Lemma 5.10). The uniform estimate might be true but 2^{εm} for a small ε > 0 is good enough for our application.

As we did in Section 4, we set up a trilinear form first. Let us ignore the condition (2.4) for a while. If 2^{L_{2}j+M_{2}} < 2^{L_{1}j+M_{1}}/8, let

ω′_{3,j} = {ξ : 2^{L_{1}j+M_{1}}/8 ≤ |ξ| ≤ 19 · 2^{L_{1}j+M_{1}}/8}

and let Φ_{3,j} be a Schwartz function whose Fourier transform is a bump function adapted to ω′_{3,j} such that Φ̂_{3,j}(ξ) = 1 for all 2^{L_{1}j+M_{1}}/4 ≤ |ξ| ≤ 9 · 2^{L_{1}j+M_{1}}/4.

If 2^{L_{1}j+M_{1}} < 2^{L_{2}j+M_{2}}/8, let ω′_{3,j} = {ξ : 2^{L_{2}j+M_{2}}/8 ≤ |ξ| ≤ 19·2^{L_{2}j+M_{2}}/8} and Φ_{3,j} be a Schwartz function whose Fourier transform is a bump function adapted to ω′_{3,j} such that Φ̂_{3,j}(ξ) = 1 for all 2^{L_{2}j+M_{2}}/4 ≤ |ξ| ≤ 9·2^{L_{2}j+M_{2}}/4.

If 2^{L_{1}j+M_{1}}/8 ≤ 2^{L_{2}j+M_{2}} ≤ 8 · 2^{L_{1}j+M_{1}}, let

ω′_{3,j} = {ξ : |ξ| ≤ 18 · max{2^{L_{1}j+M_{1}}, 2^{L_{2}j+M_{2}}}}

and Φ_{3,j} be a Schwartz function whose Fourier transform is a bump function adapted to ω′_{3,j} such that Φ̂_{3,j}(ξ) = 1 for all |ξ| ≤ 17·max{2^{L_{1}j+M_{1}}, 2^{L_{2}j+M_{2}}}. Let Φ_{3,j,m} = Φ_{3,j}, f_{3,j,m}(x) = f_{3,j,0}(x) = f_{3} ∗ Φ_{3,j,0}(x). Define a trilinear form Λ_{L_{1},L_{2},M_{1},M_{2},m} by

(5.1)  Λ_{L_{1},L_{2},M_{1},M_{2},m}(f_{1}, f_{2}, f_{3}) = ∫ ∑_{j∈ℤ} ∏_{ℓ=1}^{3} f_{ℓ,j,m}(x)dx.

Clearly Λ_{L_{1},L_{2},M_{1},M_{2},m} = ∫ Π_{L_{1},L_{2},M_{1},M_{2},m}(f_{1}, f_{2})(x)f_{3}(x)dx.

We will prove the following two lemmata.

Lemma 5.1. Let p_{1}, p_{2} > 2 and 1 < r < 2 such that 1/p_{1} + 1/p_{1} = 1/r. Let F_{1}, F_{2}, F_{3} be measurable sets in ℝ. There exists a constant C independent of F_{1}, F_{2}, F_{3}, f_{1}, f_{2}, f_{3}, M_{1}, M_{2}, m such that

(5.2)  |Λ_{L_{1},L_{2},M_{1},M_{2},m}(f_{1}, f_{2}, f_{3})| ≤ Cm|F_{1}|^{1/p_{1}}|F_{2}|^{1/p_{2}}|F_{3}|^{1/r′}

holds for all f_{1} ∈ X(F_{1}), f_{2} ∈ X(F_{2}) and f_{3} ∈ X(F_{3}).

Lemma 5.2. Let ε be any positive number, 1 < p < 2 and F_{1}, F_{2}, F_{3} be measurable sets in ℝ such that |F_{3}| = 1. Suppose (2.4) holds for all j’s.

Then there is a subset F′_{3} ⊂ F_{3} with |F′_{3}| ≥ |F_{3}|/2 such that for all p_{1}, p_{2} ≥ p with 1/p_{1}+1/p_{2} ≥ 1, and all functions f_{1} ∈ X(F_{1}), f_{2} ∈ X(F_{2}), f_{3} ∈ X(F_{3}), the following inequality holds.

(5.3)  |Λ_{L_{1},L_{2},M_{1},M_{2},m}(f_{1}, f_{2}, f_{3})| ≤ C2^{εm}|F_{1}|^{1/p_{1}}|F_{2}|^{1/p_{2}} ,

where C is a constant independent of S, F_{1}, F_{2}, F_{3}, f_{1}, f_{2}, f_{3}, M_{1}, M_{2}, m.

Theorem 2.2 is a consequence of these two lemmas by using interpolation and duality. We also have a corollary from Lemma 5.1 by a simple interpolation.

Corollary 5.3. Let p_{1}, p_{2} > 2 and 1 < r < 2 such that 1/p_{1} + 1/p_{1} = 1/r. There exists a constant C independent of F_{1}, F_{2}, F_{3}, f_{1}, f_{2}, f_{3}, M_{1}, M_{2}, m such that

(5.4)  ∥Π_{L_{1},L_{2},M_{1},M_{2},m}(f_{1}, f_{2})∥_{r} ≤ Cm∥f_{1}∥_{p_{1}}∥f_{2}∥_{p_{2}}

holds for all f_{1} ∈ L^{p_{1}} and f_{2} ∈ L^{p_{2}}.

5.1. Proof of Lemma 5.1. For ℓ ∈ {1, 2, 3}, let Tr_{ℓ,j,m} be a translation function defined by

(5.5)  Tr_{ℓ,j,m}(x) = x + m_{jℓ},

where m_{jℓ} = 2^{m−jL_{ℓ}−M_{ℓ}} if ℓ ∈ {1, 2} and m_{j3} = 0. Notice that f_{ℓ,j,m}(x) = f_{ℓ,j,0}(Tr_{ℓ,j,m}(x)). Write Λ_{L_{1},L_{2},M_{1},M_{2},m} as

Λ_{L_{1},L_{2},M_{1},M_{2},m}(f_{1}, f_{2}, f_{3})

= ∫_{ℝ} ∏_{ℓ=1}^{3} ∑_{(j,n)∈ℤ×ℤ}1_{j,n}^{∗}(Tr_{ℓ,j,m}(x))f_{ℓ,j,0}(Tr_{ℓ,j,m}(x))dx.

For S ⊂ ℤ(γ) × ℤ we define

(5.6)  Λ_{S,m}(f_{1}, f_{2}, f_{3}) = ∫_{ℝ} ∑_{j∈ℤ} ∏_{ℓ=1}^{3} ∑_{n∈S_{j}} F_{ℓ,j,n,m}(x)dx,

where S_{j} = {n : (j, n) ∈ S} and F_{ℓ,j,n,m} is defined by

(5.7)  F_{ℓ,j,n,m}(x) = ((1_{j,n}^{∗}f_{ℓ,j,0}) ∘ Tr_{ℓ,j,m})(x).

Let k_{jℓ} be an integer such that |ω′_{ℓ,j}| ∼ 2^{k_{jℓ}}. For s = (j, n) ∈ S, let k_{s} = k_{j} = min_{ℓ} k_{jℓ}. The time interval of s is defined by I_{s} = [2^{−k_{s}}n, 2^{−k_{s}}(n + 1)]. We then can define a tree in S as in Section 4. To prove Lemma 5.1, it is sufficient to prove the following lemma.

Lemma 5.4. Let p_{1}, p_{2} > 2 and 1 < r < 2 such that 1/p_{1} + 1/p_{1} = 1/r. Let F_{1}, F_{2}, F_{3} be measurable sets in ℝ. There exists a constant C independent of F_{1}, F_{2}, F_{3}, f_{1}, f_{2}, f_{3}, M_{1}, M_{2}, m such that

(5.8)  |Λ_{S,m}(f_{1}, f_{2}, f_{3})| ≤ Cm|F_{1}|^{1/p_{1}}|F_{2}|^{1/p_{2}}|F_{3}|^{1/r′}

holds for all f_{1} ∈ X(F_{1}), f_{2} ∈ X(F_{2}) and f_{3} ∈ X(F_{3}).

Uniform estimates for some paraproducts 177

By scaling invariance, we can assume that |F_{3}| = 1. We partition S into two subsets S^{(1)} and S^{(2)}, where

(5.9)  S^{(1)} = {(j, n) ∈ S : |ω′_{2,j}| ≤ |ω′_{1,j}|/10 or |ω′_{1,j}| ≤ |ω′_{2,j}|/10}
(5.10)  S^{(2)} = S\S^{(1)}.

We should change the definitions of sizes of trees in S. Definition 5.5. Let (j, n) ∈ S and ℓ ∈ {1, 2, 3}. Define a seminorm ∥f_{ℓ}∥_{j,n} by

(5.11) ∥f_{ℓ}∥_{j,n} = \frac{1}{|I_{s}|^{1/2}} ∥1_{j,n}^{∗∗}f_{ℓ,j,0}∥_{2} + \frac{1}{|I_{s}|^{1/2}} ∥2^{−k_{jℓ}}1_{j,n}^{∗∗}Df_{ℓ,j,0}∥_{2},

where Df_{ℓ,j,0} is the derivative of f_{ℓ,j,0}. Definition 5.6. For ℓ ∈ {1, 2, 3} and a tree T, let (j_{T}, n_{T}) be the top of the tree T. And define

(5.12)  Δ_{ℓ}^{∗}(T)(x) = ( ∑_{(j,n)∈T} |1_{j,n}^{∗}f_{ℓ,j,0}(x)|^{2})^{1/2}.

If T is a tree in S^{(1)}, we define

(5.13)  size_{ℓ}(T) = \frac{1}{|I_{T}|^{1/2}} ∥Δ_{ℓ}^{∗}(T)∥_{2} + ∥f_{ℓ}∥_{j_{T},n_{T}},

for all ℓ ∈ {1, 2, 3}.

If T is a tree in S^{(2)}, define size_{ℓ}(T) by (5.13) only for ℓ ∈ {1, 2}. For ℓ = 3, we define the size by

(5.14)  size_{3}(T) = ∥f_{3}∥_{j_{T},n_{T}},

Let P be a subset of S. Define the ℓ-size^{∗} of T by

(5.15)  size_{ℓ}^{∗}(P) = sup_{T:T⊂P} size_{ℓ}(T),

where T ranges over all trees in P.

One should notice that for Λ_{S^{(1)},m} we have a uniform estimate for p_{1}, p_{2} > 2 and 1 < r < 2. We state it as follow Proposition 5.7. Let p_{1}, p_{2} > 2 and 1 < r < 2 with 1/p_{1} + 1/p_{2} = 1/r. Let f_{1} ∈ L^{p_{1}}, f_{2} ∈ L^{p_{2}} and f_{3} ∈ L^{r′}. Then

(5.16) |Λ_{S^{(1)},m}(f_{1}, f_{2}, f_{3})| ≤ C∥f_{1}∥_{p_{1}}∥f_{2}∥_{p_{2}}∥f_{3}∥_{r′},

where C is independent of m, f_{1}, f_{2}, f_{3}.

Proof. We do not need time frequency analysis for this proposition. The key point is that when s ∈ S^{(1)} the support of Fourier transform of f_{3,j,0} is away from the origin so that we can apply the Littlewood–Paley Theorem for the square function generated by f_{3,j,0}’s. Clearly |Λ_{S^{(1)},m}| is estimated by

∫_{ℝ} ∑_{j} ∏_{ℓ=1}^{3} f_{ℓ,j,0}(Tr_{ℓ,j,m}(x))dx.

By Hölder’s inequality, we dominate |Λ_{S^{(1)},m}| by

∥( ∑_{j} |f_{1,j,0} ∘ Tr_{1,j,m}|^{p_{1}})^{1/p_{1}} ∥_{p_{1}} ∥( ∑_{j} |f_{2,j,0} ∘ Tr_{2,j,m}|^{p_{2}})^{1/p_{2}} ∥_{p_{2}}

· ∥( ∑_{j} |f_{3,j,0}|^{r′})^{1/r′} ∥_{r′}.

By a change of variables, it is clear that for ℓ = 1, 2,

∥( ∑_{j} |f_{ℓ,j,0} ∘ Tr_{ℓ,j,m}|^{p_{ℓ}})^{1/p_{ℓ}} ∥_{p_{ℓ}} = ∥( ∑_{j} |f_{ℓ,j,0}|^{p_{ℓ}})^{1/p_{ℓ}} ∥_{p_{ℓ}}.

Notice the elementary inequality

( ∑_{j} |a_{j}|^{q})^{1/q} ≤ ( ∑_{j} |a_{j}|^{2})^{1/2}

holds for q ≥ 2. We thus dominate |Λ_{S^{(1)},m}| by

∥( ∑_{j} |f_{1,j,0}|^{2})^{1/2}∥_{p_{1}} ∥( ∑_{j} |f_{2,j,0}|^{2})^{1/2}∥_{p_{2}} ∥( ∑_{j} |f_{3,j,0}|^{2})^{1/2}∥_{r′}.

Now the Littlewood–Paley theorem yields the desired estimate (5.16). This proves the proposition. □

We now use time frequency analysis to prove Lemma 5.4. Although we only need to estimate Λ_{S^{(2)},m} due to Proposition 5.7, we still write a proof for both of Λ_{S^{(1)},m} and Λ_{S^{(2)},m}.

We first prove the size estimate for a single tree, that is,

(5.17)  |Λ_{T,m}(f_{1}, f_{2}, f_{3})| ≤ C ∏_{ℓ=1}^{3} size_{ℓ}^{∗}(T)|I_{T}|.

We only prove the case when T is a tree in S^{(2)} for (5.17) since the other case is similar. In this case 2^{k_{jℓ}} ∼ 2^{k_{j}} for all ℓ in {1, 2, 3}. We thus dominate |Λ_{T,m}| by

∫_{ℝ} sup_{(j,n)∈T} |(1_{j,n}^{∗∗}f_{3,j,0})∘Tr_{ℓ,j,m}(x)| ∏_{ℓ≠3} ( ∑_{(j,n)∈T} |(1_{j,n}^{∗∗}f_{ℓ,j,0})∘Tr_{ℓ,j,m}(x)|^{2})^{1/2}dx.

By the definition of Δ_{ℓ} and Hölder’s inequality, we estimate |Λ_{T,m}| by

sup_{(j,n)∈T} ∥F_{3,j,0}^{∗}∥_{∞}∥Δ_{1}^{∗}(T)∥_{2}∥Δ_{2}^{∗}(T)∥_{2},

where F_{3,j,0}^{∗} = 1_{j,n}^{∗∗}f_{3,j,0}. Notice that Lemma 4.13 holds for the seminorm. Thus we have

∥F_{3,j,0}^{∗}∥_{∞} ≤ size_{3}^{∗}(T).

Clearly the definition of size yields

∥Δ_{ℓ}(T)∥_{2} ≤ size_{ℓ}^{∗}(T)|I_{T}|^{1/2}

for ℓ ∈ {1, 2}. Putting all of them together, we obtain (5.17).

Lemma 5.8. Let κ ∈ {1, 2}, T be a tree in S^{(κ)} and P be a subset of S^{(κ)}. Suppose that P ∩ T = ∅ and T is a maximal tree in P ∪ T. Then we have

(5.18) |Λ_{P∪T,m}(f_{1}, f_{2}, f_{3}) − Λ_{P,m}(f_{1}, f_{2}, f_{3})| ≤ Cm ∏_{ℓ=1}^{3} size_{ℓ}^{∗}(T ∪ P)|I_{T}|,

where C is independent of f_{1}, f_{2}, f_{3}, P, T.

Proof. Clearly the difference |Λ_{P∪T,m} − Λ_{P,m}| is dominated by a sum of CΛ_{T,m} and at most finite many following trilinear forms

| ∫ ∑_{j∈scl(T)} ( ∑_{n∈T_{j}} F_{ℓ_{1},j,n,m}(x))( ∑_{n∈P_{j}} F_{ℓ_{2},j,n,m}(x))

· ( ∑_{n∈(P∪T)_{j}} F_{ℓ_{3},j,n,m}(x))dx|,

where (ℓ_{1}, ℓ_{2}, ℓ_{3}) is a permutation of (1, 2, 3). By (5.17), it sufficient to show that this trilinear form can be estimated by the right-hand side of (5.8). We only handle the most difficult case ℓ_{1} = 1, ℓ_{2} = 2. Other cases are similar. We estimate the trilinear form by (5.19)

∑_{j∈scl(T)} ∑_{I:|I|=2^{−k_{j}}} ∥( ∑_{n∈T_{j}} F_{1,j,n,m})( ∑_{n∈P_{j}} F_{2,j,n,m})( ∑_{n∈(P∪T)_{j}} F_{3,j,n,m})∥_{L^{1}(I)}.

There is at least one index ℓ ∈ {1, 2} satisfying k_{jℓ} = k_{j}. Without loss of generality, assume k_{j1} = k_{j}. We have that for any positive integer N,

∥ ∑_{n∈T_{j}} F_{1,j,n,m}∥_{L^{∞}(I)} ≤ \frac{C_{N}}{(1 + 2^{k_{j}}dist(I(m_{j1}), I_{T}))^{N}} ∥1_{j,n′}^{∗∗}f_{ℓ,j,0}∥_{∞},

where I(m_{j1}) = I + m_{j1} is an interval generated by shifting I to the right by m_{j1} and n′ ∈ (P ∪ T)_{j} which minimizes the distance between I_{j,n} and I(m_{j1}). Since Lemma 4.13 holds for the seminorm, we get

∥ ∑_{n∈T_{j}} F_{1,j,n,m}∥_{L^{∞}(I)} ≤ \frac{C_{N}size_{1}^{∗}(P ∪ T)}{(1 + 2^{k_{j}}dist(I(m_{j1}), I_{T}))^{N}} .

And since P ∩ T = ∅ and T is a maximal tree in P ∪ T, we have

∥ ∑_{n∈P_{j}} F_{2,j,n,m}∥_{L^{2}(I)} ≤ \frac{C_{N}}{(1 + 2^{k_{j}}dist(I(m_{j2}), (I_{T})^{c}))^{N}} ∥1_{j,n′}^{∗∗}f_{ℓ,j,0}∥_{2},

which is obviously bounded by

\frac{C_{N}size_{2}^{∗}(P ∪ T)|I|^{1/2}}{(1 + 2^{k_{j}}dist(I(m_{j2}), (I_{T})^{c}))^{N}} .

Similarly, we also have

∥ ∑_{n∈(P∪T)_{j}} F_{3,j,n,m}∥_{L^{2}(I)} ≤ Csize_{3}^{∗}(P ∪ T)|I|^{1/2}.

Thus we estimate (5.19) by

∑_{j∈scl(T)} ∑_{I:|I|=2^{−k_{j}}} \frac{C_{N}size_{1}^{∗}(P ∪ T)size_{2}^{∗}(P ∪ T)size_{3}^{∗}(P ∪ T)|I|}{(1 + 2^{k_{j}}dist(I(m_{j1}), I_{T}))^{N}(1 + 2^{k_{j}}dist(I(m_{j2}), (I_{T})^{c}))^{N}} .

Let j_{T} be the index for the top of T. If j_{T} + 10m ≥ j ≥ j_{T}, we only have at most 10m different values for j. Notice that if I(m_{j1}) ⊂ (I_{T})^{c}, then we can replace dist(I(m_{j1}), I_{T}) by dist(I(m_{j1}), ∂I_{T}). Thus if we only sum j from j_{T} to j_{T} + 10m we get that (5.19) is dominated by

Cm ∏_{ℓ=1}^{3} size_{ℓ}^{∗}(P ∪ T)|I_{T}|.

The remaining thing we need to deal with is to sum all j > j_{T} + 10m. The main difficulty is the case I(m_{j1}) ⊈ (I_{T})^{c} and I(m_{j2}) ⊈ I_{T}, because in other cases we gain (1 + 2^{k_{j}}dist(I(m_{jℓ}), ∂I_{T}))^{−100} in the estimate for at least one of ℓ ∈ {1, 2}, which trivializes the estimate. We also know from the definition of m_{jℓ} that dist(I(m_{j1}), I(m_{j2})) ≤ 2^{m}|I|. To make the difficult case happen, the interval I must satisfy dist(I(m_{jℓ}), ∂I_{T}) ≤ 10 · 2^{m}|I| for both ℓ = 1, 2. Sum |I(m_{jℓ})| for all such I’s to get a upper bound C2^{m}2^{−k_{j}}. Then summing these upper bounds for all j > j_{T} + 10m we get a bound C2^{−8m}|I_{T}|. Therefore we estimate (5.19) by Cm ∏_{ℓ=1}^{3} size_{ℓ}^{∗}(P∪T)|I_{T}|. This proves the lemma. □

Lemma 4.15 still holds for the sizes of trees defined in Subsection 5.1. Let κ ∈ {1, 2}. Applying this organization lemma inductively for S^{(κ)}, we decompose

(5.20)  S^{(κ)} = ⋃_{σ} S_{σ}^{(κ)},

where σ ranges over all possible dyadic numbers, S_{σ}^{(κ)} = ∪_{T∈F_{σ}^{(κ)}}T such that F_{σ}^{κ} is a collection of maximal trees with

(5.21)  count(S_{σ}^{(κ)}) ≤ Cσ^{−2},

and

(5.22)  size_{ℓ}^{∗}(S_{σ}^{(κ)}) ≤ σ|F_{ℓ}|^{1/2}

holds for all ℓ ∈ {1, 2, 3}.

Notice that Lemma 4.12 holds for the new sizes of trees defined in Subsection 5.1. We thus can also sharpen the upper bound in the size estimate for S_{σ}^{(κ)} by

(5.23)  size_{ℓ}^{∗}(S_{σ}^{(κ)}) ≤ min{1, σ|F_{ℓ}|^{1/2}}.

Hence by Lemma 5.8 we estimate Λ_{S,m} by

∑_{κ=1}^{2} ∑_{σ} ∑_{T∈F_{σ}^{(κ)}} m ∏_{ℓ=1}^{3} size_{ℓ}^{∗}(S_{σ}^{(κ)})|I_{T}|.

Applying (5.23) and (5.21), we thus obtain

(5.24) |Λ_{S,m}(f_{1}, f_{2}, f_{3})|

≤ Cm ∑_{σ} σ^{−2} min{1, σ|F_{1}|^{1/2}} min{1, σ|F_{2}|^{1/2}} min{1, σ},

which clearly implies (5.8). Therefore we have proven Lemma 5.4.

5.2. A truncated trilinear form. First by a change of variable, we write Λ_{L_{1},L_{2},M_{1},M_{2},m} as

(5.25)  Λ_{L_{1},L_{2},M_{1}M_{2},m}(f_{1}, f_{2}, f_{3}) = ∫ ∑_{j} ∏_{ℓ=1}^{3} f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))dx,

where T̃r_{1,j,m}(x) = Tr_{1,j,m} − m_{j2}, T̃r_{2,j,m}(x) = x, T̃r_{3,j,m}(x) = x − m_{j2}.

To prove Lemma 5.2, we have to set up our time-frequency decomposition in a slightly different way for technical reasons. Recall that ψ is a nonnegative Schwartz function such that ψ̂ is supported in [−1/100, 1/100] and ψ̂(0) = 1. And ψ_{k}(x) = 2^{k}ψ(2^{k}x). Let Ω be the set defined as in (4.6).

As before, k_{jℓ} is an integer such that 2^{k_{jℓ}} ∼ |ω′_{ℓ,j}| for for ℓ ∈ {1, 2, 3} and k_{j} = min{k_{jℓ}}. For a very small positive number ε, we define

(5.26)  Ω_{j} = {x ∈ Ω : dist(x, Ω^{c}) ≥ 2^{ε^{2}m}2^{−k_{j}}}.
(5.27)  ψ_{j1} = ψ_{j2} = ψ_{j3} = 1_{(Ω_{j})^{c}} ∗ ψ_{k_{j}}(x).

Ω_{j}, ψ_{jℓ} depend on m, ε but this dependence is suppressed for notational convenience. A truncated trilinear form is defined by

(5.28)  Λ_{Ω,m}(f_{1}, f_{2}, f_{3}) = ∫ ∑_{j∈ℤ} ∏_{ℓ=1}^{3} ψ_{jℓ}(x)f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))dx.

Heuristically, ψ_{jℓ} can be considered as 1_{(Ω_{j})^{c}} since it is a smooth approximation of 1_{(Ω_{j})^{c}}. In time space, Ω_{j} is an exceptional set which can be removed. we can handle it well. The technical details about this can be found in Section 4. In order to get 2^{εm} instead of 2^{m} in the estimates, we have to remove only a smaller set. Here is the lemma which allows us to do so.

Lemma 5.9. Let F_{1}, F_{2}, F_{3} be measurable sets. Let F′_{3} = F_{3}\Ω. Then

(5.29) |(Λ_{L_{1},L_{2},M_{1},M_{2},m} − Λ_{Ω,m})(f_{1}, f_{2}, f_{3})|

≤ C2^{−100m} min {1, |F_{1}|^{1/p}} min {1, |F_{2}|^{1/p}}

holds for all functions f_{1} ∈ X(F_{1}), f_{2} ∈ X(F_{2}), f_{3} ∈ X(F′_{3}), where C is a constant independent of L_{1}, L_{2}, M_{1}, M_{2}, m, f_{1}, f_{2}, f_{3}, F_{1}, F_{2}, F_{3}.

Proof. The difference |Λ_{L_{1},L_{2},M_{1},M_{2},m} − Λ_{Ω,m}| is dominated by

∫ ∑_{j}|1 − ∏_{ℓ=1}^{3} ψ_{jℓ}(x)||∏_{ℓ=1}^{3} f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))|dx.

Clearly,

|1 − ∏_{ℓ=1}^{3} ψ_{jℓ}(x)| ≤ 3 ∑_{ℓ=1}^{3} |1 − ψ_{jℓ}(x)|

For ℓ = {1, 2}, by the definition of Ω, we have for any positive integer N,

|f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))| ≤ ∫ \frac{C_{N}|f_{ℓ}(y)|2^{k_{jℓ}}}{(1 + 2^{k_{jℓ}}|T̃r_{ℓ,j,m}(x) − y|)^{N}} dy

≤ C2^{2m}(1 + 2^{k_{jℓ}}dist(T̃r_{ℓ,j,m}(x), Ω^{c}))^{2} min{1, |F_{ℓ}|^{1/p}}.

Since f_{3} ∈ X(F′_{3}), we obtain that

(5.30)  |f_{3,j,0}(x)| ≤ \frac{C_{N}}{(1 + 2^{k_{j3}}dist(T̃r_{3,j,m}(x), Ω^{c}))^{N}} .

Thus by the fact that 2^{k_{j3}} ∼ max{2^{k_{jℓ}}}, k_{j2} > k_{j1} + m and the definition of Ω_{j}, the difference in the left-hand side of (5.29) is estimated by

∑_{j} ∫ ∫_{Ω_{j}} \frac{2^{k_{j}}}{(1 + 2^{k_{j}}|x − y|)^{N}} dy \frac{C_{N}2^{4m} min\{1, |F_{1}|^{1/p}\} min\{1, |F_{2}|^{1/p}\}}{(1 + 2^{k_{j3}}dist(T̃r_{3,j,m}(x), Ω^{c}))^{N}}dx

≤ ∑_{j} ∫_{Ω_{j}} \frac{C_{N}2^{4m} min\{1, |F_{1}|^{1/p}\} min\{1, |F_{2}|^{1/p}\}}{(1 + 2^{k_{j}}dist(y, Ω^{c}))^{N}}dy

≤ C2^{−100m} min{1, |F_{1}|^{1/p}} min{1, |F_{2}|^{1/p}}.

Therefore we have finished the proof. □

By this lemma, we only need to consider Λ_{Ω,m}. For S ⊂ ℤ(γ) × ℤ we define

(5.31)  Λ_{S,Ω,m}(f_{1}, f_{2}, f_{3}) = ∫_{ℝ} ∑_{j∈ℤ} ∏_{ℓ=1}^{3} ∑_{n∈S_{j}} F̃_{ℓ,j,n,m}(x)dx,

where F̃_{ℓ,j,n,m} is defined by

(5.32)  F̃_{ℓ,j,n,m}(x) = ψ_{jℓ}(x)1_{j,n}^{∗}(T̃r_{ℓ,j,m}(x))f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x)).

As before we only need to consider the trilinear form (5.31). To prove Lemma 5.2, it is sufficient to show the following lemma due to Lemma 5.9.

Lemma 5.10. Let ε be any positive number, 1 < p < 2 and F_{1}, F_{2}, F_{3} be measurable sets in ℝ such that |F_{3}| = 1. There is a subset F′_{3} ⊂ F_{3} with |F′_{3}| ≥ |F_{3}|/2 such that for all p_{1}, p_{2} ≥ p with 1/p_{1} + 1/p_{2} ≥ 1, and all functions f_{1} ∈ X(F_{1}), f_{2} ∈ X(F_{2}), f_{3} ∈ X(F_{3}), the following inequality holds.

(5.33)  |Λ_{S,Ω,m}(f_{1}, f_{2}, f_{3})| ≤ C2^{εm}|F_{1}|^{1/p_{1}}|F_{2}|^{1/p_{2}} ,

where C is a constant independent of S, F_{1}, F_{2}, F_{3}, f_{1}, f_{2}, f_{3}, L_{1}, L_{2}, M_{1}, M_{2}, m.

5.3. Preliminary lemmata. To prove Lemma 5.10, we should change the definitions of size of a tree in S and set up some lemmata first.

Definition 5.11. Let (j, n) ∈ S and ℓ ∈ {1, 2, 3}. Let ψ_{jℓ}^{∗} be the function

(5.34)  ψ_{jℓ}^{∗}(x) = ∫_{(Ω_{j})^{c}} \frac{2^{k_{j}}}{(1 + 2^{2k_{j}}|x − y|^{2})^{200}} dy

Define a seminorm ∥f_{ℓ}∥_{j,n,m} by

(5.35)  ∥f_{ℓ}∥_{j,n,m} = \frac{1}{|I_{s}|^{1/p}} ∥1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f_{ℓ,j,0}∥_{p}

+ \frac{1}{|I_{s}|^{1/2}} ∥2^{−k_{jℓ}}1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})Df_{ℓ,j,0}∥_{p},

where T̃r_{ℓ,j,m}^{−1} is the inverse of T̃r_{ℓ,j,m} and Df_{ℓ,j,0} is the derivative of f_{ℓ,j,0}.

Definition 5.12. For ℓ ∈ {1, 2} and a tree T, let (j_{T}, n_{T}) be the top of the tree T. And let Δ_{ℓ,m}^{∗}(T) be defined by

(5.36) Δ_{ℓ,m}^{∗}(T)(x) = ( ∑_{(j,n)∈T} |1_{j,n}^{∗}(x)(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})(x)f_{ℓ,j,0}(x)|^{2})^{1/2}.

If T is a tree in S, we define

(5.37)  size_{ℓ,m}(T) = \frac{1}{|I_{T}|^{1/2}} ∥Δ_{ℓ,m}^{∗}(T)∥_{p} + ∥f_{ℓ}∥_{j_{T},n_{T},m},

for all ℓ ∈ {1, 2}.

Let P be a subset of S. Define the (ℓ, m)-size^{∗} of T by

(5.38)  size_{ℓ,m}^{∗}(P) = sup_{T:T⊂P} size_{ℓ,m}(T),

where T ranges over all trees in P.

In the definition of ψ_{jℓ}^{∗}, we can replace the exponent 200 by a larger number 2^{100} to define a new function. Denote this function by ψ̃_{jℓ}^{∗}. If 1_{j,n}^{∗} and ψ_{jℓ}^{∗} are replaced by ˜1_{j,n}^{∗} and ψ̃_{jℓ}^{∗} respectively in the definition Δ_{ℓ,m}^{∗}(T), we denote the corresponding function by Δ_{ℓ,m}(T).

Lemma 5.13. Let 1 < q < ∞, ℓ ∈ {1, 2, 3} and T be a tree in S. Then

(5.39)  ∥Δ_{ℓ,m}^{∗}(T)∥_{q} ≤ C inf_{x∈I_{T}} M_{q}(Mf_{ℓ})(x)|I_{T}|^{1/q},

(5.40)  size_{ℓ,m}(T) ≤ C min{2^{β_{ℓ}m}|F_{ℓ}|^{1/p}, inf_{x∈I_{T}} M_{p}(Mf_{ℓ})(x)},

where β_{ℓ} = 1 if ℓ = 1, β_{ℓ} = ε^{2} if ℓ = 2, and C is a constant independent of f_{ℓ}, T, S, L_{1}, L_{2}, M_{1}, M_{2}.

Proof. Repeating an argument similar to that used in the proof of (4.25) and (4.26), we easily obtain (5.39) and part of (5.40). The only thing we need to prove is

(5.41)  size_{ℓ,m}(T) ≤ C2^{β_{ℓ}m}|F_{ℓ}|^{1/p}.

Assume 2^{β_{ℓ}m+10}I_{T} ⊂ Ω: otherwise the upper bound inf_{x∈I_{T}} M_{p}(Mf_{ℓ})(x) implies (5.41). Let T_{L} be a collection of all s = (j, n) ∈ T such that 2^{L}I_{s} ⊂ Ω but 2^{L+1}I_{s} ⊈ Ω. Then

T = ⋃_{L=[β_{ℓ}m+10]}^{∞} T_{L}

Let 𝕁_{L} be the set of all time intervals I_{s}’s for s ∈ T_{L}. Clearly, 𝕁_{L} is a set of disjoint intervals and ∑_{J∈𝕁_{L}} |J| ≤ min{|I_{T}|, 1}. Thus it is sufficient to show that for any J ∈ 𝕁_{L} and any (j, n) ∈ T such that I_{s} = J,

(5.42)  ∥1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f̃_{ℓ,j,0}∥_{p}^{p} ≤ C_{N}( inf_{x∈J} M_{p}(Mf_{ℓ})(x))^{p}L^{−N}|J|

holds for a large integer N, where f̃_{ℓ,j,0} is f_{ℓ,j,0} or 2^{−k_{jℓ}}Df_{ℓ,j,0}, since the desired estimate follows by summing up all L’s and J’s. By the definition of ψ_{jℓ}^{∗}, we have

|1_{j,n}^{∗∗}(x)ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1}(x)|

≤ \frac{C}{(1 + 2^{k_{j}}dist(x, J))^{200}(1 + 2^{k_{j}}dist(T̃r_{ℓ,j,m}^{−1}(x), (Ω_{j})^{c}))^{200}} ,

which is clearly dominated by

\frac{C}{(1 + 2^{k_{j}}dist(x, J))^{100}(1 + 2^{k_{j}}dist(J_{j,m}, (Ω_{j})^{c}))^{100}} ,

where J_{j,m} is the interval {T̃r_{ℓ,j,m}(x) : x ∈ J}. Since L ≥ β_{ℓ}m + 9, by the

definition of T̃r_{ℓ,j,m} we thus dominate |1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})| by

\frac{C}{(1 + 2^{k_{j}}dist(x, J))^{100}(1 + 2^{k_{j}}dist(J, (Ω)^{c}))^{100}} .

Thus we have

∥1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f̃_{ℓ,j,0}∥_{p}^{p} ≤ C( inf_{x∈J} M_{p}(Mf_{ℓ})(x))^{p}L^{−100p}|J|,

which yields (5.42). □

Lemma 5.14. Suppose that s = (j, n) ∈ S. If 2^{k_{jℓ}} ∼ 2^{k_{j}}, then

(5.43)  ∥1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f_{ℓ,j,0}∥_{∞} ≤ C∥f_{ℓ}∥_{j,n,m}

holds for ℓ ∈ {1, 2, 3}, where C is a constant independent of s, f_{ℓ}, m, L_{1}, L_{2}, M_{1}, M_{2}.

Proof. Let μ = ∥f_{ℓ}∥_{j,n,m}. By the definition of the seminorm, we have

(5.44) ∥1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f_{ℓ,j,0}∥_{p} + ∥|I_{s}|1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})Df_{ℓ,j,0}∥_{p}

≤ μ|I_{s}|^{1/p}.

First we prove the BMO estimate for the function, that is

(5.45)  ∥1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f_{ℓ,j,0}∥_{BMO} ≤ Cμ.

186 Xiaochun Li If |I_{s}| ≤ |J|, by (5.44) we have

inf_{c} ∫_{J} |1_{j,n}^{∗∗}(x)(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})(x)f_{ℓ,j,0}(x) − c|dx

≤ ∥1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f_{ℓ,j,0}∥_{p}|J|^{1− \frac{1}{p}} ≤ μ|I_{s}|^{\frac{1}{p}} |J|^{1− \frac{1}{p}} ≤ μ|J|.

If |I_{s}| ≥ |J|, by (5.44) we obtain that

inf_{c} ∫_{J} |1_{j,n}^{∗∗}(x)(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})(x)f_{ℓ,j,n_{ℓ}}(x) − c|dx

≤ |J| ∫_{J} |(1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f_{ℓ,j,n_{ℓ}})′(x)|dx

≤ C|J||I_{s}|^{−1} ∫_{J} |1_{j,n}^{∗∗}(x)(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})(x)f_{ℓ,j,n_{ℓ}}(x)|dx + |J| ∫_{J} |1_{j,n}^{∗∗}(x)(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})(x)Df_{ℓ,j,n_{ℓ}}(x)|dx

≤ C|J||I_{s}|^{−1}∥1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f_{ℓ,j,n_{ℓ}} ∥_{p}|J|^{1− \frac{1}{p}} + |J|∥1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})Df_{ℓ,j,n_{ℓ}} ∥_{p}|J|^{1− \frac{1}{p}}

≤ Cμ|J|^{2− \frac{1}{p}} |I_{s}|^{\frac{1}{p} −1} ≤ Cμ|J|.

Thus we get the BMO estimate (5.45). Interpolating (5.45) and (5.44), we have for any p ≤ q < ∞,

∥1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f_{ℓ,j,n_{ℓ}} ∥_{q} ≤ Cμ|I_{s}|^{1/q}.

Notice that an integration by parts and Hölder’s inequality yield that

∥1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f_{ℓ,j,n_{ℓ}} ∥_{∞}

≤ ∥1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f_{ℓ,j,n_{ℓ}} ∥_{p′}^{1/2}∥(1_{j,n}^{∗∗}(ψ_{jℓ}^{∗} ∘ T̃r_{ℓ,j,m}^{−1})f_{ℓ,j,n_{ℓ}})′∥_{p}^{1/2},

where 1/p + 1/p′ = 1. Hence the desired estimate (5.43) follows by (5.44) and the L^{p′} estimate for the functions. □ Lemma 5.15. For any tree T in S, let (5.46) Δ̃_{ℓ,m}(T)(x)

= ( ∑_{(j,n)∈T} |˜1_{j,n}^{∗}(T̃r_{ℓ,j,m}(x))ψ̃_{jℓ}^{∗}(x)f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x)|^{2})^{1/2}.

Then for ℓ = 1 we have

(5.47)  ∥Δ_{ℓ,m}(T)∥_{BMO} ≤ Csize_{ℓ}^{∗}(T),
(5.48)  ∥Δ̃_{ℓ,m}(T)∥_{BMO} ≤ Cm size_{ℓ}^{∗}(T),

(5.49)  ∥Δ̃_{ℓ,m}(T)∥_{q} ≤ Cm^{1−2/q}size_{ℓ}^{∗}(T)|I_{T}|^{1/q},

where q ≥ 2 and C is a constant independent of T, S, L_{1}, L_{2}, M_{1}, M_{2}, f_{ℓ}, n_{ℓ}.

Proof. (5.47) can be obtained routinely as we did for Lemma 4.14. We omit the details. We need only prove (5.48). (5.49) is a simple consequence of (5.47), (5.48) and an interpolation argument.

Clearly by a change of variables ∥Δ_{ℓ,m}(T)∥_{2} = ∥Δ̃_{ℓ,m}(T)∥_{2}. Thus (5.47) and an interpolation yield

(5.50)  ∥Δ̃_{ℓ,m}(T)∥_{2} ≤ Csize_{ℓ}^{∗}(T)|I_{T}|^{1/2}.

Let μ = size_{ℓ}^{∗}(T). Let J be a dyadic interval and T_{J} = {s ∈ T : I_{s} ⊆ 3J}. We then dominate inf_{c} ∫_{J} |Δ_{ℓ}(T)(x) − c|dx by a sum of the following three parts.

∫_{J} ( ∑_{s∈T_{J}} |˜1_{j,n}^{∗}(T̃r_{ℓ,j,m}(x))ψ̃_{jℓ}^{∗}(x)f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))|^{2})^{1/2}dx,

∫_{J} ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|≤|J| \end{matrix}} |˜1_{j,n}^{∗}(T̃r_{ℓ,j,m}(x))ψ̃_{jℓ}^{∗}(x)f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))|^{2})^{1/2} dx,

and

inf_{c} ∫_{J} |( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} |˜1_{j,n}^{∗}(T̃r_{ℓ,j,m}(x))ψ̃_{jℓ}^{∗}(x)f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))|^{2})^{1/2} − c| dx.

T_{J} can be decomposed to a union of trees T_{J,k}’s such that the time intervals I_{T_{J,k}}’s are disjoint and all of them are contained in 3J. Using the Cauchy–Schwarz inequality, the first part is estimated by

(∑_{k} ∥Δ̃_{ℓ,m}(T_{J,k})∥_{2}^{2})^{1/2}|J|^{1/2}.

Appying (5.50), we dominated the first part by Cμ|J|.

Since p ≤ 2 we estimate the second part by

∥( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|≤|J| \end{matrix}} |(˜1_{j,n}^{∗} ∘ T̃r_{ℓ,j,m})ψ̃_{jℓ}^{∗}(f_{ℓ,j,n_{ℓ}} ∘ T̃r_{ℓ,j,m})|^{2})^{1/2}∥_{L^{p}(J)} |J|^{1− \frac{1}{p}}

≤ ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|≤|J| \end{matrix}} ∥(˜1_{j,n}^{∗} ∘ T̃r_{ℓ,j,m})ψ̃_{jℓ}^{∗}(f_{ℓ,j,n_{ℓ}} ∘ T̃r_{ℓ,j,m})∥_{L^{p}(J)}^{p})^{1/p} |J|^{1− \frac{1}{p}}

≤ ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|≤|J| \end{matrix}} \frac{C∥(1_{j,n}^{∗∗} ∘ T̃r_{ℓ,j,m})ψ_{jℓ}^{∗}(f_{ℓ,j,n_{ℓ}} ∘ T̃r_{ℓ,j,m})∥_{p}^{p}}{(1 + |I_{s}|^{−1}dist(J, T̃r_{ℓ,j,m}^{−1}(I_{s})))^{100}} )^{1/p} |J|^{1− \frac{1}{p}}

≤ μ ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|≤|J| \end{matrix}} \frac{C|I_{s}|}{(1 + |I_{s}|^{−1}dist(J, T̃r_{ℓ,j,m}^{−1}(I_{s})))^{100}} )^{1/p} |J|^{1− \frac{1}{p}} ,

where T̃r_{ℓ,j,m}^{−1}(I_{s}) is the interval {T̃r_{ℓ,j,m}^{−1}(x) : x ∈ I_{s}}. Observe that if |I_{s}| ≤ 2^{−m−10}|J| and s ∈ T\T_{J}, then dist(J, T̃r_{ℓ,j,m}^{−1}(I_{s})) ∼ dist(J, I_{s}). Thus summing for all s in this case, we get the desired estimate Cμ|J|. In the remaining case, there are only 10m different scales for |I_{s}|’s since s’s satisfy 2^{−m−10}|J| < |I_{s}| ≤ |J|. The worst situation is that when T̃r_{ℓ,j,m}^{−1}(I_{s})∩J ≠ ∅, because otherwise dist(J, T̃r_{ℓ,j,m}^{−1}(I_{s})) can be replaced by dist(∂J, T̃r_{ℓ,j,m}^{−1}(I_{s})) and thus the desired estimate follows. But in this situation, T̃r_{ℓ,j,m}^{−1}(I_{s}) must be a subset of 3J since |I_{s}| ≤ |J|. For all T̃r_{ℓ,j,m}^{−1}(I_{s}) ⊂ 3J with a fixed scale, the sum of |I_{s}|’s is no more than 3|J|. Summing for at most 10m different scales, we thus get the upper bound Cmμ|J|. Hence the second part is dominated by Cmμ|J|.

The third part is estimated by

(inf_{c} ∫_{J} |( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}}|˜1_{j,n}^{∗}(T̃r_{ℓ,j,m}(x))ψ̃_{jℓ}^{∗}(x)f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))|^{2})^{1/2}− c|^{2}dx)^{1/2}
· |J|^{1/2}

≤ (inf_{c} ∫_{J} | ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} |˜1_{j,n}^{∗}(T̃r_{ℓ,j,m}(x))ψ̃_{jℓ}^{∗}(x)f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))|^{2} − c| dx)^{1/2}

· |J|^{1/2}

Uniform estimates for some paraproducts 189

≤ C (∫_{J} ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} |(|˜1_{j,n}^{∗}(T̃r_{ℓ,j,m}(x))ψ̃_{jℓ}^{∗}(x)f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))|^{2})′|dx)^{1/2} |J|,

which is dominated by a sum of following two terms, R_{1} =

C (∫_{J} ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} |I_{s}|^{−1}|˜1_{j,n}^{∗}(T̃r_{ℓ,j,m}(x))ψ̃_{jℓ}^{∗}(x)f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))|^{2}dx)^{1/2} |J|,

and R_{2} =

C (∫_{J} ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} |˜1_{j,n}^{∗}(T̃r_{ℓ,j,m}(x))ψ̃_{jℓ}^{∗}(x)f_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))||G_{ℓ,j,m}(x)|dx)^{1/2} |J|,

where G_{ℓ,j,m} is the function defined by

G_{ℓ,j,m}(x) = ˜1_{j,n}^{∗}(T̃r_{ℓ,j,m}(x))ψ̃_{jℓ}^{∗}(x)Df_{ℓ,j,0}(T̃r_{ℓ,j,m}(x))

By Lemma 5.14, we see that for any q ≥ p,

∥(1_{j,n}^{∗∗} ∘ T̃r_{ℓ,j,m})ψ_{jℓ}^{∗}(f_{ℓ,j,n_{ℓ}} ∘ T̃r_{ℓ,j,m})∥_{q} ≤ Cμ|I_{s}|^{1/q}.

Thus, by Hölder’s inequality, the first term R_{1} is estimated by

C ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} \frac{|I_{s}|^{−1}∥(1_{j,n}^{∗∗} ∘ T̃r_{ℓ,j,m})ψ_{jℓ}^{∗}(f_{ℓ,j,n_{ℓ}} ∘ T̃r_{ℓ,j,m})∥_{4}^{2}|J|^{1/2}}{(1 + |I_{s}|^{−1}dist(J, T̃r_{ℓ,j,m}^{−1}(I_{s})))^{100}} )^{1/2} |J|

≤ Cμ ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} \frac{|I_{s}|^{−1/2}|J|^{1/2}}{(1 + |I_{s}|^{−1}dist(J, T̃r_{ℓ,j,m}^{−1}(I_{s})))^{100}} )^{1/2} |J| ≤ Cμ|J|.

It is obvious by the fact 2^{k_{jℓ}} ∼ 2^{k_{j}} when ℓ = 1 and the definition of the seminorm that

(5.51)  ∥G_{ℓ,j,m}∥_{p} ≤ ∥f_{ℓ}∥_{j,n,m}|I_{s}|^{1/p−1}.

Thus the second term R_{2} is estimated by

C ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} ∥(˜1_{j,n}^{∗} ∘ T̃r_{ℓ,j,m})ψ̃_{jℓ}^{∗}(f_{ℓ,j,0} ∘ T̃r_{ℓ,j,m})∥_{L^{p′}(J)}∥G_{ℓ,j,m}∥_{p})^{1/2} |J|

≤ C (μ ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}}\frac{|I_{s}|^{\frac{1}{p} −1}∥(1_{j,n}^{∗∗} ∘ T̃r_{ℓ,j,m})ψ_{jℓ}^{∗}(f_{ℓ,j,n_{ℓ}} ∘ T̃r_{ℓ,j,m})∥_{p′+1}|J|^{\frac{1}{p′(p′+1)}}}{(1 + |I_{s}|^{−1}dist(J, T̃r_{ℓ,j,m}^{−1}(I_{s})))^{100}} )^{1/2}

· |J|

≤ Cμ ( ∑_{\begin{matrix}s∈T\T_{J} \\ |I_{s}|>|J| \end{matrix}} \frac{|I_{s}|^{− \frac{1}{p′(p′+1)}} |J|^{\frac{1}{p′(p′+1)}}}{(1 + |I_{s}|^{−1}dist(J, T̃r_{ℓ,j,m}^{−1}(I_{s})))^{100}} )^{1/2} |J| ≤ Cμ|J|.

This completes the proof of (5.48). □

Lemma 5.16. Let T be a tree in S and P be a subset of S. Suppose that P ∩ T = ∅ and T is a maximal tree in P ∪ T. Then we have

(5.52) |Λ_{P∪T,Ω,m}(f_{1}, f_{2}, f_{3}) − Λ_{P,Ω,m}(f_{1}, f_{2}, f_{3})|

≤ |Λ_{T,Ω,m}(f_{1}, f_{2}, f_{3})| + Cm ∏_{ℓ=1}^{2} size_{ℓ}^{∗}(T ∪ P)|I_{T}|,

where C is independent of f_{1}, f_{2}, f_{3}, L_{1}, L_{2}, M_{1}, M_{2}, P, T.

The proof is similar to the proof of Lemma 5.16. We omit the details and leave it as an exercise to the readers.

5.4. Proof of Lemma 5.10. It is easy to prove a size estimate for the trilinear form on a single tree, that is, for any tree T,

(5.53)  |Λ_{T,Ω,m}(f_{1}, f_{2}, f_{3})| ≤ Cm^{2/p−1} ∏_{ℓ=1}^{2} size_{ℓ}^{∗}(T)|I_{T}|,

where C is independent of L_{1}, L_{2}, M_{1}, M_{2}, m, f_{1}, f_{2}, f_{3}, T.

In fact, by Hölder’s inequality, we estimate |Λ_{T,Ω,m}| by

∥Δ̃_{1,m}^{∗}(T)∥_{p′} ∥Δ_{2,m}^{∗}(T)∥_{p}.

By (5.49) and the definition of size, we obtain (5.53) immediately.

Lemma 4.15 still holds for our new sizes of trees and S. Applying this organization lemma inductively for S, we decompose

(5.54)  S = ⋃_{σ} S_{σ},

Uniform estimates for some paraproducts 191

where σ ranges over all possible dyadic numbers, S_{σ} = ∪_{T∈F_{σ}}T such that F_{σ} is a collection of maximal trees with

(5.55)  count(S_{σ}) ≤ Cσ^{−p},

and

(5.56)  size_{ℓ}^{∗}(S_{σ}) ≤ σ|F_{ℓ}|^{1/p}

holds for all ℓ ∈ {1, 2}.

By (5.40), the upper bound in the size estimates for S_{σ} can be sharpened by

(5.57)  size_{ℓ}^{∗}(S_{σ}) ≤ min{1, 2^{β_{ℓ}m}|F_{ℓ}|^{1/p}, σ|F_{ℓ}|^{1/p}}.

Hence by Lemma 5.16 and (5.53) we estimate Λ_{S,Ω,m} by

∑_{σ} ∑_{T∈F_{σ}} m ∏_{ℓ=1}^{2} size_{ℓ}^{∗}(S_{σ})|I_{T}|.

Applying (5.57) and (5.55), we thus dominate |Λ_{S,Ω,m}(f_{1}, f_{2}, f_{3})| by (5.58)

Cm ∑_{σ} σ^{−p} min{1, 2^{m}|F_{1}|^{1/p}, σ|F_{1}|^{1/p}} min{1, 2^{ε^{2}m}|F_{2}|^{1/p}, σ|F_{2}|^{1/p}},

which clearly implies (5.33). Therefore we have completed the proof of Lemma 5.10.

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Xiaochun Li, Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA xcli@math.uiuc.edu

This paper is available via http://nyjm.albany.edu/j/2008/14-6.html.