Tài liệu Đề tài " Uniform bounds for the bilinear Hilbert transforms, I " ppt

Annals of Mathematics Uniform bounds for the bilinear Hilbert transforms, I By Loukas Grafakos and Xiaochun Li... Uniform bounds for the bilinear Hilbert transforms, IBy Loukas Grafako

Annals of Mathematics

Uniform bounds for the

bilinear Hilbert transforms, I

By Loukas Grafakos and Xiaochun Li

Uniform bounds for the bilinear Hilbert transforms, I

By Loukas Grafakos and Xiaochun Li*

p1+p2 < 2 Combining this result with the main

result in [9], we deduce that the operators H 1,α map L2(R)×L ∞(R)→ L2(R)

uniformly in the real parameter α ∈ [0, 1] This completes a program initiated

by A Calder´on

1 Introduction

The study of the Cauchy integral along Lipschitz curves during the period1965–1995 has provided a formidable impetus and a powerful driving force forsignificant developments in euclidean harmonic analysis during that period andlater The Cauchy integral along a Lipschitz curve Γ is given by

where h is a function on Γ, which is taken to be the graph of a Lipschitz

function A : R → R Calder´on [2] wrote CΓ(h)(z) as the infinite sum

reducing the boundedness of CΓ(h) to that of the operators C m (f ; A) with constants having suitable growth in m The operators C m (f ; A) are called the commutators of f with A and they are archetypes of nonconvolution singular

integrals whose action on the function 1 has inspired the fundamental work

on the T 1 theorem [5] and its subsequent ramifications The family of bilinear

com-with derivative A  in L ∞ In fact, in the mid 1960’s Calder´on observed that

the linear operator f → C1(f ; A) can be written as the average

via a different approach, the issue of the uniform boundedness of the operators

H 1,α from L2(R)× L ∞ (R) into L2(R) remained open up to now The purpose

of this article and its subsequent, part II, is to obtain exactly this, i.e the

uniform boundedness (in α) of the operators H 1,αfor a range of exponents thatcompletes in particular the above program initiated by A Calder´on about 40years ago This is achieved in two steps In this article we obtain bounds for

H 1,α from L p1(R)×L p2(R) into L p (R) uniformly in the real parameter α when

2 < p1, p2 < ∞ and 1 < p = p1p2

p1+p2 < 2 In part II of this work, the second

author obtains bounds for H 1,α from L p1(R)× L p2(R) into L p(R), uniformly

in α satisfying |α − 1| ≥ c > 0 when 1 < p1, p2 < 2 and 23 < p = p1p2

p1+p2 < 1.

Interpolation between these two results yields the uniform boundedness of H 1,α

from L p(R)×L ∞ (R) into L p(R) for 43 < p < 4 when α lies in a compact subset

of R This in particular implies the boundedness of the commutator C1(· ; A)

on L p(R) for 43 < p < 4 via the Calder´on method described above but also hasother applications See [9] for details We note that the restriction to compact

subsets of R is necessary, as uniform L p × L ∞ → L p bounds for H 1,α cannot

Boundedness for the operators H 1,α was first obtained by M Lacey and

C Thiele in [7] and [8] Their proof, though extraordinary and pioneering,

gives bounds that depend on the parameter α, in particular that blow up polynomially as α tends to 0, 1 and ±∞ The approach taken in this work is

based on powerful ideas of C Thiele ([10], [11]) who obtained that the H 1,α’s

map L2(R)× L2(R)→ L 1, ∞ (R) uniformly in α satisfying |α − 1| ≥ δ > 0.

The theorem below is the main result of this article.

Theorem Let 2 < p1, p2 < ∞ and 1 < p = p1p2

p1+p2 < 2 Then there is a constant C = C(p1, p2) such that for all f1, f2 Schwartz functions on R,

sup

α1,α2∈R H α1,α2(f1, f2) p ≤ C f1 p1f2 p2.

By dilations we may take α1 = 1 It is easy to see that the boundedness

of the operator H 1, −α on any product of Lebesgue spaces is equivalent to that

considerations, it suffices to obtain uniform bounds near only one of the three

‘bad’ directions α = −1, 0, ∞ of H 1, −α In this article we choose to work

with the ‘bad’ direction 0 This direction corresponds to bilinear multipliers

whose symbols are characteristic functions of planes of the form η < α1ξ For

simplicity we will only consider the case where α1 = 2m , m ∈ Z+ The ments here can be suitably adjusted to cover the more general situation where

uniformly in m ≥ 2200 where p1, p2, p are as in the statement of the theorem.

The rest of the paper is devoted to the proof of (1.2) In the following

sections, L = 2100 will be a fixed large integer We will use the notation|S| for

the Lebesgue measure of set S and S c for its complement By c(J ) we denote the center of an interval J and by AJ the interval with length A |J| (A > 0)

and center c(J ) For J, J  sets we will use the notation

J < J  ⇐⇒ sup

x ∈J x ≤ inf

x ∈J
x.

The Hardy-Littlewood maximal operator of g is denoted by M g and M p g will

be (M |g| p)1/p The derivative of order α of a function f will be denoted by

D α f When L p norms or limits of integration are not specified, they are to

be taken as the whole real line Also C will be used for any constant that

de-pends only on the exponents p1, p2 and is independent of any other parameter,

in particular of the parameter m Finally N will denote a large (but fixed)

integer whose value may be chosen appropriately at different times

Acknowledgments. The authors would like to thank M Lacey for manyhelpful discussions during a visit at the Georgia Institute of Technology Theyare grateful to C Thiele for his inspirational work [10] and for some help-ful remarks They also thank the referee for pointing out an oversight in aconstruction in the first version of this article

2 The decomposition of the bilinear operator T m

We begin with a decomposition of the half plane η < 2 m ξ on the ξ-η plane.

We can write the characteristic function of the half plane η < 2 m ξ as a union

of rectangles of size 2−k × 2 −k+m as in Figure 1 Precisely, for k, l ∈ Z we set

which provides a (nonsmooth) partition of unity of the half-plane η < 2 m ξ.

Next we pick a smooth partition of unity {Ψ (r)

k,l (ξ, η) } k,l,r of the half-plane

η < 2 m ξ with each Ψ (r) k,l supported only in a small enlargement of the rectangle

J1(r) (k, l) × J (r)

2 (k, l) and satisfying standard derivative estimates Since the

functions Ψ(r) k,l (ξ, η) are not of tensor type, (i.e products of functions of ξ and functions of η) we apply the Fourier series method of Coifman and Meyer [4,

rectangle size

2−k × 2 −k+m

ξ

rectangle t

of ype 2

rectangle t

of ype 1

rectangle t

convenience, we will drop the dependence of these functions on r and we will concentrate on the case n = (n1, n2) = (0, 0) In the cases n

mial appearance of |n| in the estimates will be controlled by the rapid decay

of C(n), while the exponential functions in (2.1) and (2.2) can be thought of

as almost “constant” locally (such as when n1 = n2 = 0), and thus a small

adjustment of the case n = (0, 0) will yield the case for general n in Z2.Based on these remarks, we may set Φj,k,l = Φj,k,l,0and it will be sufficient

to prove the uniform (in m) boundedness of the operator T m0 defined by

The representation of T m0 into a sum of products of functions of ξ and η will be

crucial in its study If follows from (2.1) and (2.2) that there exist the followingsize estimates for the functions Φ1,k,l and Φ2,k,l

|Φ 1,k,l (x) | ≤ C N2−k(1 + 2−k |x|) −N ,

(2.4)

|Φ 2,k,l (x) | ≤ C N2−k+m(1 + 2−k+m |x|) −N

(2.5)

for any N ∈ Z+ The next lemma is also a consequence of (2.1) and (2.2)

Lemma 1 For all N ∈ Z+, there exists C N > 0 such that for all f ∈

(2.7)

where C N is independent of m.

Proof. To prove the lemma we first observe that whenever Φl ∈ S

has Fourier transform supported in the interval [2l − 3, 2l + 3] and satisfies

supl D αΦl  ∞ ≤ C α for all sufficiently large integers α, then we have

Once (2.8) is established, we apply it to the function Φl (x) = 2 kΦ1,k,l(2k x),

which by (2.1) satisfies |D αΦl (ξ) | ≤ C α, to obtain (2.6) Similarly, applying(2.8) to the function Φl (x) = 2 k −mΦ2,k,l(2k −m x), which by (2.2) also satisfies

|D αΦl (ξ) | ≤ C α, we obtain (2.7).

By a simple translation, it will suffice to prove (2.8) when x = 0 Then

3 The truncated trilinear form

Let ψ be a nonnegative Schwartz function such that  ψ is supported in

[−1, 1] and satisfies  ψ(0) = 1 Let ψ k (x) = 2 −k ψ(2 −k x) For E ⊂ R and k ∈ Z

Note that ψ 1,k , ψ 2,k , and ψ 3,k depend on the set E but we will suppress this

dependence for notational convenience, since we will be working with a fixed

set E Also note that the functions ψ 2,k and ψ 3,k depend on m, but this

dependence will also be suppressed in our notation The crucial thing is that

all of our estimates will be independent of m Define

rectangle in which the Fourier transforms of Φ1,k,l and Φ2,k,l are supported.)One easily obtains the size estimate

|Φ 3,k,l (x) | ≤ C2 −k+m(1 + 2−k+m |x|) −N .

(3.5)

Because of the assumption on the indices p1, p2, there exists a 2 < p3< ∞

Lemma 2 will be proved in the next sections Now, we have

Proof To prove (1.2), it will be sufficient to prove that for all λ > 0,

m (f1, f2)| > 2}, and assuming |G| ≥ 1 (otherwise there is

nothing to prove) choose f3∈ S with f3 L ∞ (E c)≤ 1, supp f3 ⊂ E c , and

{x ∈ R : M p3(M f3)(x) > 2 } is empty Now define

Therefore, to prove (3.7), we only need to show that

k∈Zl ∈Z

(1−

for all N > 0 Therefore, (3.11) can be estimated by

Similar reasoning works for (3.12) This completes the proof of (3.9) andtherefore of Lemma 3

We now set up some notation For k, n ∈ Z, define I k,n = [2k n, 2 k (n + 1)]

For an integer r with 0 ≤ r < L, let Z r ={ ∈ Z : = κL+r for some κ ∈ Z}.

Also for S ⊂ Z r × Z × Z r we let S k,l ={n ∈ Z : (k, n, l) ∈ S} and define

For simplicity we will only consider the case where m ∈ Z0 The argument

below can be suitably adjusted to the case where m has a different remainder when divided by L We will therefore concentrate on proving Lemma 2 for the

expression ΛE,S (f1, f2, f3) when m ∈ Z0 To achieve this goal, we introducethe grid structure

Definition 1 A set of intervalsG is called a grid if the condition below

then it will be called a central grid.

Given S ⊂ Z r × Z × Z r and s = (k, n, l) ∈ S we set I s = I k,n For eachfunction Φj,k,l and each n ∈ Z we define a family of intervals ω j,s , s = (k, n, l)

∈ S so that conditions (3.19)–(3.25) below hold: Say that
Φ1,k,l (ξ)
Φ2,k,l (η) is supported in a small neighborhood of a rectangle of type r = 1 Then we define ω j,ssuch that

supp
Φ3,k,l ⊂ [−(1 + 2 −m )a, −(1 + 2 −m )b), where [a, b) = ω 3,s ,

These properties are trivially adjusted when
Φ1,k,l (ξ)
Φ2,k,l (η) is supported in

a small neighborhood, a rectangle of type r = 2 or r = 3.

As in [7], we prove the existence of ω j,s by induction If S is nonempty, pick

s0 = (k, n, l) ∈ S such that k is minimal and define S  = S \{s0} By induction,

we may assume that for any s  ∈ S  there exists a ω j,s
so that the collection

of all such intervals satisfies (3.19)–(3.25) Now we try to define ω j,s0 so that

(3.19)–(3.25) still hold Let [a1, b1) be an interval with length (1 + 2−2L)2−kwhich contains supp
Φ1,k,l And for j = 2, 3, let [a j , b j) be an interval withlength (1 + 2· 2 −2L)2−k+m which contains supp
Φ

j,k,l By (3.22), we know thatsupp
Φ3,k,l ⊂ [−(1+2 −m )a3, −(1+2 −m )b3) Define ω j,s

0,1 as the union of [a j , b j)

and all intervals 5ω j,s
with s  ∈ S  which satisfy dist(ω j,s
, [a j , b j)) ≤ 2|ω j,s
|

and ω j,s
be the next smaller interval in S Inductively we define ω j,s0,l for

l ≥ 1 Let ω j,s0 =

l ≥1 ω j,s0,l It is easy to verify conditions (3.19)–(3.25) for

ω j,s0 This completes the proof of the existence of a grid structure

Furthermore, we have the following geometric picture for ω j,s

Lemma 4 For s, s  ∈ S and ω j,s j,s
, the following properties hold :

(1) If ω 1,s ⊂ ω 1,s
, then ω j,s
< ω j,s and 12|ω j,s
| < dist(ω j,s , ω j,s
) < 2 |ω 2,s
| for j = 2, 3.

(2) If ω j,s ⊂ ω j,s
for j = 2, 3, then ω 1,s < ω 1,s
and 18|ω 1,s
| < dist(ω 1,s , ω 1,s
)

As in [10] we give the following definition.

Definition 2 A subset S of Z r ×Z×Z r is called convex if for all s, s  ∈ S,

s  ∈ Z r × Z × Z r , j ∈ {1, 2} with I s ⊂ I s
⊂ I s

and ω j,s

⊂ ω j,s
⊂ ω j,s, we

have s  ∈ S.

It is sufficient to prove bounds on ΛE,S for all finite convex sets S of triples

of integers, provided the bound is independent of S and of course m.

4 The selection of the trees

Definition 3 Fix T ⊂ S and t ∈ T If for any s ∈ T , we have I s ⊂ I t

and ω j,s ⊃ ω j,t , then we call T a tree of type j with top t Now, T is called a

maximal tree of type j ∈ {1, 2} with top t in S if there does not exist a larger

tree of type j with the same top strictly containing T Let T be a maximal tree

of type j

tree of type i with top t in S by  T

Lemma 5 Let S ⊂ Z r × Z × Z r be a convex set and T ⊂ S be a maximal tree of type j ∈ {1, 2} with top t in S Then T is a convex set.

Proof Let s, s  ∈ T , s  ∈ Z r × Z × Z r , i ∈ {1, 2} with I s ⊂ I s
⊂ I s

and

from Lemma 4 that i = j Using that I s
⊂ I s

⊂ I t , ω j,t ⊂ ω j,s

⊂ ω j,s
, and

the maximality of T , we obtain that s  ∈ T , hence the convexity of T follows.

Lemma 6 Let S ⊂ Z r × Z × Z r be a convex set and T be a maximal tree

of type j ∈ {1, 2} with top t in S Then S\(T T ) is convex.

Proof Assume that S\(T T ) is not convex Then there exist s, s  ∈ S\(T T ), s  ∈ T T , i ∈ {1, 2} with I s ⊂ I s
⊂ I s

and ω i,s

⊂ ω i,s
⊂ ω i,s

If s  ∈ T , then I s
⊂ I t and ω j,t ⊂ ω j,s
Since s is not in T , we have i

By Lemma 4, we have dist(ω i,t , ω i,s
) < 2 |ω j,s
| Since 5ω i,s
⊂ ω i,s we have

ω i,t ⊂ ω i,s Thus s ∈  T , which is a contradiction.

For a given subset T of S we define T k,l to be the set

{n ∈ Z : (k, n, l) ∈ T }.

If T is a tree of type j for j ∈ {1, 2, 3} and k ∈ Z r, then there is at most

Φj,k,T = Φj,k,l Otherwise, let T k = ∅ and Φ j,k,T = 0 For brevity, we write

(k, n) ∈ T if and only if there exists an l ∈ Z r with (k, n, l) ∈ T Thus

identifying trees with sets of pairs of integers, we will use this identificationthroughout.

Therefore, if (k, n, l) ∈ T , we can write ω j,k,n,l = ω j,k,l = ω j,k,T, and

Let t = (k T , n T , l T ) be the top of T We write I T = I k T ,n T and ω j,T = ω j,k T ,T

For a tree T of type 2 (or 3) with top t and k ∈ Z r , define θ j,k,T+ and θ j,k,T −

accordance with the definitions of φ j,k,n and ψ j,k we define the functions

Let ∆k be the set of all connected components of E k \E k+L Obviously ∆k is

a set of intervals Observe that if J ∈ ∆ k, then 2k ≤ |J| < 2 k+L, and 

We now describe a procedure for selecting a collection of trees T µ,i,j,l ν and

where T µ,i,j,l ν ,  T µ,i,j,l ν are defined as follows:

Let l ≥ 0 be an integer and assume that we have already defined T ν

µ,i,j,λ,



T µ,i,j,λ ν for λ < l If one of the sets T µ,i,j,λ ν ,  T µ,i,j,λ ν with λ < l is empty, then let

T µ,i,j,l ν = T µ,i,j,l ν =∅ Otherwise, let F denote the set of all trees T of type i

which satisfy conditions (1)–(8) below:

and T is a maximal tree of type i in S µ −1 \ 

λ<l

(T ν µ,i,j,λ T ν

µ,i,j,λ)

(2) If (i, j, ν) = (1, 1, 1), then for (k, n) ∈ T , one of the following inequalities

holds:

φ ∗ 1,k,n ψ 1,k ∗ (f1∗ Φ 1,k,l)

(5) If i = 2 or 3, j = 2 or 3, ν = 2, then there exists ˜ k ∈ {−L, 0, L, 2L, 3L, 4L}

such that, for (k, n) ∈ T , one of the following inequalities holds:

j,k,T)2 2

T µ,i,j,l ν =∅ Otherwise, we select T ν

µ,i,j,l and T µ,i,j,l ν as follows

(9) If (i, j, ν) ∈ {(1, 2, 1), (1, 3, 1), (2, 2, 4), (2, 3, 4), (3, 2, 4), (3, 3, 4)}, then

se-lect T µ,i,j,l ν ∈ F such that for any T ∈ F,

ω j,T ν µ,i,j,l ≯ ω j,T

(10) If (i, j, ν) ∈ {(2, 1, 1), (3, 1, 1), (2, 2, 3), (2, 3, 3), (3, 2, 3), (3, 3, 3)}, then

se-lect T µ,i,j,l ν ∈ F such that for any T ∈ F,

ω j,T ν ≮ ω j,T

(4.17)

Let T µ,i,j,l ν be the maximal tree of type i  with top t in

Lemma 5 and Lemma 6, we have that S µ , T µ,i,j,l ν and T µ,i,j,l ν are convex

Lemma 7 For µ ≥ 0, (i, j, ν) ∈ H,



l

|I T ν µ,i,j,l | ≤ C2 10ηpj µ

2µ(4.18)

where C, C q1 are independent of m.

The core of the proof consists of the proofs of these lemmata These will

be given in the next section We now state and prove one more lemma whichwill allow us to conclude the proof of (1.2), assuming the validity of Lemmas 7and 8

Lemma 9 Let µ ≥ 0, T ⊂ S µ −1 be a tree of type j ∈ {1, 2, 3}, P ⊂ S µ −1,

Proof Notice there exists at most one l such that T k,l

We now deduce the proof of (1.2) using assumed Lemmas 7 and 8, andLemma 9.

5 Some technical material

In this section we prove a variety of technical facts that will be used inthe proofs of Lemmas 7 and 8 presented in the next sections

Lemma 10 For any (k, n, l) ∈ S there exists the following:

φ ∗ 1,k,n (f1∗ Φ 1,k,l)

2 ≤ C

inf

Φ1,k,l(·)e −2πic(ω 1,k,l)(·) (x) ≤ C 2 −2k(1 + 2−k |x|) −N .

Using this estimate and a similar argument as before we obtain (5.2)

We now prove (5.3) assuming that I k,n ⊂ E; otherwise (5.3) follows

immediately from (5.1) and from the fact that M f1(x) ≤ M p1f1(x) Pick

x ∈2AI k,n

M p1f1(x)

2

|I k,n | ≤ C|I k,n |.

This completes the proof of (5.3) The proof of (5.4) is similar

Applying the same idea and the fact that the Littlewood-Paley square

function is bounded from L2 to L2, we can prove the following

Lemma 11 For any tree T of type 1 and any j ∈ {2, 3},

 

(k,n) ∈T

φ ∗ j,k,n (f j ∗ Φ j,k,T)2

2

1 2

2

1 2

≤ C|I k T ,n T |1

2.

(5.6)

Similarly we obtain the following lemmas whose proofs we omit

Lemma 12 For any tree T of type j, j ∈ {2, 3},

 

(k,n) ∈T

φ ∗ 1,k,n (f1∗ Φ 1,k,T)2

2

1 2

Lemma 13 For (k, n, l) ∈ S, ˜k ∈ {−L, 0, L, 2L, 3L, 4L}, and j ∈ {2, 3},

φ ∗ j,k+˜ k,n (f j ∗ Φ j,k+˜ k,l)

2≤ C|I k,n |1

2,

(5.14)

φ ∗ 1,k,n ψ ∗ 2,k+m+˜ k (f j ∗ Φ j,k+m+˜ k,l)

j,k,T)2 2

1 2

2

1 2

j,k,T)2 2

1 2

2

1 2

≤ C|I k T ,n T |1

.

(5.22)

Proof We prove (5.22) first Since

ρ k −m,J (f j ∗ Φ j,k+˜ k,T)2

2≤

1(1 + 2−k+m dist(x, J )) N

(1+2−k+m |z0− x|)2

x ∈I T

M2f j (x)

2

,

which proves (5.21) and thus completes the proof of Lemma 15

The following lemma is just a version of the boundedness of the

Littlewood-Paley square function from L ∞ to BMO Its proof follows standard argumentsand is also omitted

Lemma 16 Let j ∈ {2, 3} and T ⊂ S be a convex tree of type j Then

ψ ∗ j,k (f j ∗ Φ j,k,l)

where C is independent of m and BMO denotes dyadic BMO.

6 The size estimate for the trees

Having proved all these preliminary lemmas we now concentrate on theproof of Lemma 8 This section is entirely devoted to its proof

We begin by showing (4.19) For a tree T of type 1 and T ⊂ S µ,

≤ C sup (k,n) ∈T

φ ∗ 1,k,n ψ 1,k ∗ (f1∗ Φ 1,k,T)

∞2−

µ p22−

µ p3 |I T |.

This completes the proof of (4.19) for trees of type 1 We now turn our tion to the proof of (4.20) Let

Definition A set of intervals G is called a grid if the condition below< /i>

then it will be called... r with (k, n, l) ∈ T Thus< /i>