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 Annals of Mathematics, 159 (2004), 889–933 Uniform bounds for the bilinear Hilbert transforms, I By Loukas Grafakos and Xiaochun Li* Abstract It is shown that the bilinear Hilbert transforms H α,β (f,g)(x) = p.v.  R f(x − αt)g(x − βt) dt t map L p 1 (R) × L p 2 (R) → L p (R) uniformly in the real parameters α, β when 2 <p 1 ,p 2 < ∞ and 1 <p= p 1 p 2 p 1 +p 2 < 2. Combining this result with the main result in [9], we deduce that the operators H 1,α map L 2 (R)×L ∞ (R) → L 2 (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 period 1965–1995 has provided a formidable impetus and a powerful driving force for significant developments in euclidean harmonic analysis during that period and later. The Cauchy integral along a Lipschitz curve Γ is given by C Γ (h)(z)= 1 2πi p.v.  Γ h(ζ) ζ − z dζ , 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 C Γ (h)(z) as the infinite sum 1 2πi ∞  m=0 (−i) m C m (f; A)(x) , where z = x + iA(x), f(y)=h(y + iA(y))(1 + iA  (y)), and C m (f; A)(x) = p.v.  R  A(x) − A(y) x − y  m f(y) x − y dy , *Research of both authors was partially supported by the NSF. 890 LOUKAS GRAFAKOS AND XIAOCHUN LI reducing the boundedness of C Γ (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 Hilbert transforms H α 1 ,α 2 (f 1 ,f 2 )(x) = p.v.  R f 1 (x − α 1 t)f 2 (x − α 2 t) dt t ,α 1 ,α 2 ,x∈ R, was also introduced by Calder´on in one of his attempts to show that the com- mutator C 1 (f; A) is bounded on L 2 (R) when A(t) is a function on the line with derivative A  in L ∞ . In fact, in the mid 1960’s Calder´on observed that the linear operator f →C 1 (f; A) can be written as the average C 1 (f; A)(x)=  1 0 H 1,α (f,A  )(x) dα , and the boundedness of C 1 (f; A) can be therefore reduced to the uniform (in α) boundedness of H 1,α . Although the boundedness of C 1 (f; A) was settled in [1] via a different approach, the issue of the uniform boundedness of the operators H 1,α from L 2 (R) × L ∞ (R)intoL 2 (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,α for a range of exponents that completes in particular the above program initiated by A. Calder´on about 40 years ago. This is achieved in two steps. In this article we obtain bounds for H 1,α from L p 1 (R)×L p 2 (R)intoL p (R) uniformly in the real parameter α when 2 <p 1 ,p 2 < ∞ and 1 <p= p 1 p 2 p 1 +p 2 < 2. In part II of this work, the second author obtains bounds for H 1,α from L p 1 (R) × L p 2 (R)intoL p (R), uniformly in α satisfying |α − 1|≥c>0 when 1 <p 1 ,p 2 < 2 and 2 3 <p= p 1 p 2 p 1 +p 2 < 1. Interpolation between these two results yields the uniform boundedness of H 1,α from L p (R)×L ∞ (R)intoL p (R) for 4 3 <p<4 when α lies in a compact subset of R. This in particular implies the boundedness of the commutator C 1 ( · ; A) on L p (R) for 4 3 <p<4 via the Calder´on method described above but also has other 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 hold as α →±∞. 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 L 2 (R) × L 2 (R) → L 1,∞ (R) uniformly in α satisfying |α − 1|≥δ>0. UNIFORM BOUNDS FOR THE BILINEAR HILBERT TRANSFORMS, I 891 The theorem below is the main result of this article. Theorem. Let 2 <p 1 , p 2 < ∞ and 1 <p= p 1 p 2 p 1 +p 2 < 2. Then there is a constant C = C(p 1 ,p 2 ) such that for all f 1 , f 2 Schwartz functions on R, sup α 1 ,α 2 ∈R H α 1 ,α 2 (f 1 ,f 2 ) p ≤ C f 1  p 1 f 2  p 2 . By dilations we may take α 1 = 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 of the operator (f 1 ,f 2 ) →  R  R  f 1 (ξ)  f 2 (η)e 2πi(ξ+η)x 1 {η<α −1 ξ} (ξ,η) dξ dη , where 1 A denotes the characteristic (indicator) function of the set A. More- over in the range 2 <p 1 , p 2 < ∞ and 1 <p= p 1 p 2 p 1 +p 2 < 2, in view of duality 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< 1 α ξ.For simplicity we will only consider the case where 1 α =2 m , m ∈ Z + . The argu- ments here can be suitably adjusted to cover the more general situation where 2 m ≤ 1 α < 2 m+1 as well. For a positive integer m, we consider the following pseudodifferential op- erator T m (f 1 ,f 2 )(x)=  R  R  f 1 (ξ)  f 2 (η)e 2πi(ξ+η)x 1 {η<2 m ξ} (ξ,η) dξ dη,(1.1) and prove that it satisfies T m (f 1 ,f 2 ) p ≤ Cf 1  p 1 f 2  p 2 (1.2) uniformly in m ≥ 2 200 where p 1 ,p 2 ,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 =2 100 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 Mg 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- 892 LOUKAS GRAFAKOS AND XIAOCHUN LI pends only on the exponents p 1 ,p 2 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 many helpful discussions during a visit at the Georgia Institute of Technology. They are 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 a construction 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 J (1) 1 (k, l)=[2 −k (2l), 2 −k (2l + 1)],J (1) 2 (k, l)=[2 −k+m (2l − 2), 2 −k+m (2l − 1)], J (2) 1 (k, l)=[2 −k (2l +1), 2 −k (2l + 2)],J (2) 2 (k, l)=[2 −k+m (2l − 2), 2 −k+m (2l − 1)], J (3) 1 (k, l)=[2 −k (2l +1), 2 −k (2l + 2)],J (3) 2 (k, l)=[2 −k+m (2l − 1), 2 −k+m (2l)]. We call the rectangles J (r) 1 (k, l) × J (r) 2 (k, l) of type r, r ∈{1, 2, 3}.Itis easy to see that 1 η<2 m ξ =  k∈Z  l∈Z  1 J (1) 1 (k,l) (ξ)1 J (1) 2 (k,l) (η) +1 J (2) 1 (k,l) (ξ)1 J (2) 2 (k,l) (η)+1 J (3) 1 (k,l) (ξ)1 J (3) 2 (k,l) (η)  , 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 J (r) 1 (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, pp. 55–57] to write Ψ (r) k,l (ξ,η)=  n∈Z 2 C(n)(Φ (r) 1,k,l,n )(ξ)(Φ (r) 2,k,l,n )(η) where |C(n)|≤C M (1 + |n| 2 ) −M for all M>0(n =(n 1 ,n 2 ), |n| 2 = n 2 1 + n 2 2 ) and the functions Φ (r) 1,k,l,n and Φ (r) 2,k,l,n are Schwartz and satisfy: UNIFORM BOUNDS FOR THE BILINEAR HILBERT TRANSFORMS, I 893 rectangle size 2 −k × 2 −k+m ξ rectangle tof ype 2 rectangle tof ype 1 rectangle tof ype 3 η η =2 m ξ − 2 m−k 1 η =2 m ξ θ = arctan(2 −m ) + Figure 1: The decomposition of the plane η<2 m ξ. (2.1) |D α ((Φ (r) 1,k,l,n ))|≤C α (1 + |n|) α 2 αk , supp (Φ (r) 1,k,l,n )⊂(1+2 −2L )J (r) 1 (k, l), (Φ (r) 1,k,l,n )(ξ)=e 2πin 1 2 k (ξ−c(J (r) 1 (k,l))) on (1 − 2 −2L )J (r) 1 (k, l), (2.2) |D α ((Φ (r) 2,k,l,n ))|≤C α (1 + |n|) α 2 α(k−m) , supp (Φ (r) 2,k,l,n )⊂(1+2 −2L )J (r) 2 (k, l), (Φ (r) 2,k,l,n )(η)=e 2πin 2 2 k−m (η−c(J (r) 2 (k,l))) on (1 − 2 −2L )J (r) 2 (k, l), for all nonnegative integers α and all r ∈{1, 2, 3}. In the sequel, for notational convenience, we will drop the dependence of these functions on r and we will concentrate on the case n =(n 1 ,n 2 )=(0, 0). In the cases n = 0, the polyno- mial appearance of |n| in the estimates will be controlled by the rapid decay 894 LOUKAS GRAFAKOS AND XIAOCHUN LI of C(n), while the exponential functions in (2.1) and (2.2) can be thought of as almost “constant” locally (such as when n 1 = n 2 = 0), and thus a small adjustment of the case n =(0, 0) will yield the case for general n in Z 2 . Based on these remarks, we may set Φ j,k,l =Φ j,k,l,0 and it will be sufficient to prove the uniform (in m) boundedness of the operator T 0 m defined by T 0 m (f 1 ,f 2 )(x)=  k∈Z  l∈Z  R  R  f 1 (ξ)  f 2 (η)e 2πi(ξ+η)x  Φ 1,k,l (ξ)  Φ 2,k,l (η)dξdη . (2.3) The representation of T 0 m 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 following size estimates for the functions Φ 1,k,l and Φ 2,k,l . |Φ 1,k,l (x)|≤C N 2 −k (1+2 −k |x|) −N ,(2.4) |Φ 2,k,l (x)|≤C N 2 −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 al l N ∈ Z + , there exists C N > 0 such that for all f ∈ S(R),  l∈Z |(f ∗ Φ 1,k,l )(x)| 2 ≤ C N  |f(y)| 2 2 −k (1+2 −k |x − y|) N dy,(2.6)  l∈Z |(f ∗ Φ 2,k,l )(x)| 2 ≤ C N  |f(y)| 2 2 −k+m (1+2 −k+m |x − y|) N dy,(2.7) where C N is independent of m. Proof. To prove the lemma we first observe that whenever Φ l ∈S has Fourier transform supported in the interval [2l − 3, 2l + 3] and satisfies sup l D α  Φ l  ∞ ≤ C α for all sufficiently large integers α, then we have  l∈Z |(f ∗ Φ l )(x)| 2 ≤ C N  R |f(y)| 2 (1 + |x − y|) N dy.(2.8) Once (2.8) is established, we apply it to the function Φ l (x)=2 k Φ 1,k,l (2 k x), which by (2.1) satisfies |D α  Φ l (ξ)|≤C α , to obtain (2.6). Similarly, applying (2.8) to the function Φ l (x)=2 k−m Φ 2,k,l (2 k−m x), which by (2.2) also satisfies |D α  Φ l (ξ)|≤C α , we obtain (2.7). UNIFORM BOUNDS FOR THE BILINEAR HILBERT TRANSFORMS, I 895 By a simple translation, it will suffice to prove (2.8) when x = 0. Then we have  l∈Z |(f ∗ Φ l )(0)| 2 =  l∈Z       [2l−3,2l+3]  f(−·) (1+4π 2 |·| 2 ) N  ∧ (y)  (I − ∆) N  Φ l  (y) dy      2 ≤  l∈Z  [2l−3,2l+3]      f(−·) (1+4π 2 |·| 2 ) N  ∧ (y)     2 dy  |(I − ∆) N  Φ l (y)| 2 dy ≤ C N  R |f(−y)| 2 (1+4π 2 |y| 2 ) N dy ≤ C N  R |f(y)| 2 (1 + |y|) N dy. 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 define E k = {x ∈ E : dist(x, E c ) ≥ 2 k },(3.1) ψ 1,k (x)=(1 (E k ) c ∗ ψ k )(x), and ψ 2,k (x)=ψ 3,k (x)=ψ 1,k−m (x).(3.2) 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 Λ E (f 1 ,f 2 ,f 3 )=  k∈Z  l∈Z  3  j=1 ψ j,k (x)(f j ∗ Φ j,k,l )(x)dx(3.3) where for any α ≥ 0, Φ 3,k,l depends on Φ 1,k,l and Φ 2,k,l and is chosen so that it satisfies |D α  Φ 3,k,l |≤C2 α(k−m) , supp  Φ 3,k,l ⊂ (1+2 −2L )J (r) 3 (k, l), and(3.4)  Φ 3,k,l =1 on J (r) 3 (k, l)=−(1 + 2 −2L )J (r) 1 (k, l) − (1+2 −2L )J (r) 2 (k, l), for all nonnegative integers α. (The number r in (3.4) is the type of the rectangle in which the Fourier transforms of Φ 1,k,l and Φ 2,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) 896 LOUKAS GRAFAKOS AND XIAOCHUN LI Because of the assumption on the indices p 1 ,p 2 , there exists a 2 <p 3 < ∞ such that 1 p 1 + 1 p 2 + 1 p 3 > 1. Fix such a p 3 throughout the rest of the paper. The following two lemmas reduce matters to the truncated trilinear form (3.3). Lemma 2. Let 2 <p 1 ,p 2 ,p 3 < ∞, 1 p 1 + 1 p 2 + 1 p 3 > 1, and f j  p j =1for f j ∈S and j ∈{1, 2, 3}. Define E = 3  j=1 {x ∈ R : M p j (Mf j )(x) > 2}. Then for some constant C independent of m and f 1 ,f 2 ,f 3 , |Λ E (f 1 ,f 2 ,f 3 )|≤C. Lemma 2 will be proved in the next sections. Now, we have Lemma 3. Lemma 2 implies (1.2). Proof. To prove (1.2), it will be sufficient to prove that for all λ>0, |{x : |T 0 m (f 1 ,f 2 )(x)| >λ}| ≤ Cλ − p 1 p 2 p 1 +p 2 whenever f 1  p 1 = f 2  p 2 = 1. By linearity and scaling invariance, it suffices to show that |{x : |T 0 m (f 1 ,f 2 )(x)| > 2}| ≤ C.(3.6) Let E =  2 j=1 {x ∈ R : M p j (Mf j )(x) > 1}. Since |E|≤C, it will be enough to show that |{x ∈ E c : |T 0 m (f 1 ,f 2 )(x)| > 2}| ≤ C.(3.7) Let G = E c  {|T 0 m (f 1 ,f 2 )| > 2}, and assuming |G|≥1 (otherwise there is nothing to prove) choose f 3 ∈S with f 3  L ∞ (E c ) ≤ 1, supp f 3 ⊂ E c , and     f 3 − 1 G |G| 1/p 3 T 0 m (f 1 ,f 2 ) |T 0 m (f 1 ,f 2 )|     p 3 ≤ min{1, T 0 m (f 1 ,f 2 ) −1 p  3 }. Note that for the f 3 chosen we have f 3  p 3 ≤ 2 and thus the set {x ∈ R : M p 3 (Mf 3 )(x) > 2} is empty. Now define Λ(f 1 ,f 2 ,f 3 )=  k∈Z  l∈Z  3  j=1 (f j ∗ Φ j,k,l )(x)dx.(3.8) Then by Lemma 2 it follows that |G| 1/p  3 ≤  T 0 m (f 1 ,f 2 ), 1 G |G| 1/p 3 T 0 m (f 1 ,f 2 ) |T 0 m (f 1 ,f 2 )|  ≤|Λ(f 1 ,f 2 ,f 3 )−Λ E (f 1 ,f 2 ,f 3 )|+C. UNIFORM BOUNDS FOR THE BILINEAR HILBERT TRANSFORMS, I 897 Therefore, to prove (3.7), we only need to show that |Λ(f 1 ,f 2 ,f 3 ) − Λ E (f 1 ,f 2 ,f 3 )|≤C(3.9) whenever f 3  L ∞ (E c ) ≤ 1 and supp f 3 ⊂ E c . To prove (3.9) note that |Λ(f 1 ,f 2 ,f 3 ) − Λ E (f 1 ,f 2 ,f 3 )|≤        k∈Z  l∈Z  (1 − 3  j=1 ψ j,k (x)) 3  j=1 (f j ∗ Φ j,k,l )(x)dx       . (3.10) But recall that ψ 2,k = ψ 3,k , hence |1 − 3  j=1 ψ j,k (x)|≤|1 − ψ 1,k (x)| +2|1 − ψ 2,k (x)|. Thus the expression on the right in (3.10) is at most equal to the sum of the following two quantities  k∈Z  |1 − ψ 1,k (x)| 2  j=1   l |f j ∗ Φ j,k,l (x)| 2  1 2 sup l |f 3 ∗ Φ 3,k,l (x)|dx,(3.11) 2  k∈Z  |1 − ψ 2,k (x)| 2  j=1   l |f j ∗ Φ j,k,l (x)| 2  1 2 sup l |f 3 ∗ Φ 3,k,l (x)|dx.(3.12) Using (2.6) and the fact that p 1 > 2, for any point z 0 ∈ E c , we obtain (  l∈Z |f 1 ∗ Φ 1,k,l (x)| 2 ) 1 2 ≤ C   |f 1 (y)| p 1 2 −k (1+2 −k |x − y|) N dy  1 p 1 ≤ C  1+2 −k dist(x, E c )  . Similarly, using (2.7) and the fact that p 2 > 2 we obtain (  l∈Z |f 2 ∗ Φ 2,k,l (x)| 2 ) 1 2 ≤ C  1+2 −k+m dist(x, E c )  . By (3.5) and the facts that f 3  L ∞ (E c ) ≤ 1 and supp f 3 ⊂ E c , |f 3 ∗ Φ 3,k,l (x)|≤C N  1+2 −k+m dist(x, E c )  −N (3.13) for all N>0. Therefore, (3.11) can be estimated by C  k  E k 2 −k (1+2 −k |x − y|) N dy 1 (1+2 −k+m dist(x, E c )) N−2 dx ≤ C  E  k∈Z 2 k ≤dist(y,E c ) 1 (1+2 −k dist(y, E c )) N−2 dy ≤ C|E|≤C. Similar reasoning works for (3.12). This completes the proof of (3.9) and therefore of Lemma 3. [...]... (k, n) ∈ T but there exists / (k + L, n ) ∈ T such that Ik,n ⊂ Ik+L,n Wk+L = + C2 φ∗ j,k+L,n , |Ik,n | + kkJ 913 UNIFORM BOUNDS FOR THE BILINEAR HILBERT TRANSFORMS, I Since TJ is a union of trees of type 2 or 3, we have −1 2 ∗ φ∗ ψ1,k (f1 1,k,n J1 ≤ |J| ∗ Φ1,k,T ) 2 2 1 2 ≤ C2 µ −p 1 , (k,n)∈TJ which proves the required estimate for J1 For J2 , we use (4.6) (which fails for µ − 1) to obtain 1 J2 ≤ C2 µ −p −1 2 |J| 1 (k,n)∈T \TJ k≤kJ |Ik,n | 2 −µ ≤ C2 p1 (1 + 2−k dist(J, . 889–933 Uniform bounds for the bilinear Hilbert transforms, I By Loukas Grafakos and Xiaochun Li* Abstract It is shown that the bilinear Hilbert transforms H α,β (f,g)(x). Z r with (k, n, l) ∈ T.Thus UNIFORM BOUNDS FOR THE BILINEAR HILBERT TRANSFORMS, I 901 identifying trees with sets of pairs of integers, we will use this identification throughout. Therefore,