Continuity properties for linear commutators of Calderón-Zygmund operators

Transcription

1 Collect. Math. 49, 1 (1998), c 1998 Universitat de Barcelona Continuity properties for linear commutators of Calderón-Zygmund operators Josefina Alvarez Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM , USA alvarez@nmsu.edu Received Feruary 5, Revised Decemer 4, 1996 Astract It is nown that the linear commutator [, T ] of a Calderón-Zygmund operator T with a BMO function does not satisfy some of the continuity properties typical of a Calderón-Zygmund operator, for instance, continuity from the Hardy space H p into L p for p 1 large enough, and wea type (1, 1). We otain in this paper alternative results. Indeed, we prove in the first part of the paper that [, T ] is continuous from H p into Lp, where H p denotes an atomic space with atoms satisfying an extra cancellation condition involving the function. In the second part of the paper we define a wea version H p, of this atomic space and we prove that [, T ] maps continuously H p, into L p,. Introduction Given a function locally integrale on R n, and given a Calderón-Zygmund operator T, we consider the linear commutator [, T ] defined for smooth, compactly supported functions f as [, T ](f) =T (f) T (f). A classical result of R. Coifman, R. Rocherg and G. Weiss (cf. [5]), states that the commutator [, T ] is continuous on L p for 1 <p<, when is a BMO function. 17

2 18 Alvarez Unlie the theory of Calderón-Zygmund operators, the proof of this result does not rely on a wea type (1, 1) estimate for [, T ]. In fact, it was shown in [9] that, in general, the linear commutator fails to e of wea type (1, 1), when is a BMO function. Instead, an endpoint theory was provided for this operator. Weighted L p estimates were proved in [2], and it was shown in [3] that the linear commutator is continuous on Morrey spaces. It is well nown that Calderón-Zygmund operators map continuously H p into L p for p 1 large enough. However, it was oserved in [8] that the commutator [, T ] with BMO does not map, in general, H 1 into L 1. If H 1 is replaced y a suitale atomic space H 1 as defined in [9], then the commutator is continuous from H 1 into L1. The purpose of this paper is twofold. First, we define a p version H p of the space H 1 and we use it to prove a natural sustitute for the (Hp,L p ) continuity. Next, we define a wea version H p, of this atomic space H p and we prove that [, T ] maps continuously H p, into L p,, for p 1 large enough. The notation used in this paper is standard in the suect. The symols C0, S,D,L p, etc. will indicate the usual spaces of distriutions or functions defined on R n, withcomplex values. Moreover, f p will denote the L p norm of the function f. With χ A we will indicate the characteristic function of a set A. With A we will denote the Leesgue measure of a measurale set A, and Q(z,σ) will e the cue withsides parallel to the coordinate axes, centered at z withside lengthσ. As usual, the letter C will e an asolute constant, proaly different at different occurrences. Other notations will e introduced at the appropriate time. The rest of the paper is divided in two sections, as follows: 1. (H p,lp ) continuity for the linear commutator. 2. (H p,,l p, ) continuity for the linear commutator. All the relevant definitions are included in each section. The paper ends with a list of references. Acnowledgment. The question answered in Proposition 1.7 was raised y David Cruz-Urie at the AMS meeting in Burlington, Vermont, August 1995.

3 Continuity properties for linear commutators of Calderón-Zygmund operators (H p, Lp ) continuity for the linear commutator Definition 1.1. Let T:C 0 D e a linear and continuous operator. We call T a Calderón-Zygmund operator if i) The distriution ernel (x,y) of T is a locally integrale function outside the diagonal, satisfying y z ε (x, y) (x, z) C x z n+ε for some 0 <ε 1, if 2 y z < x z. ii) The operator T extends to a continuous operator from L p 0 to itself, for some 1 <p 0 <. Remar 1.2. Any Calderón-Zygmund operator as defined in [5], p. 78, will satisfy Definition 1.1. However, this definition does not assume that the ernel (x, y) satisfies an smoothness condition in oth, x and y. For instance, let T e the doule layer potential associated to a ounded domain Ω R n+1. Assume that the oundary of Ω is locally the graph of a Lipschitz function. Then, the ernel (x, y) oft is given in local coordinates y (x, y) = 1 ϕ(x) ϕ(y) (x y) ϕ(y) ω ] (n+1)/2 n [ x y 2 +(ϕ(x) ϕ(y)) 2 where ω n denotes the volume of the unit all in R n and (x, ϕ(x)) descries locally the oundary of Ω. Thus, the adoint T, ut not T, satisfies Definition 1.1. As mentioned in the introduction, a Calderón-Zygmund operator T as aove maps continuously the Hardy space H p into L p n for n+ε <p 1. However, M. Paluszyńsi oserved in [8] that the linear commutator [, T ] with BMO does not map, in general, H 1 into L 1. In fact, this oservation can e extended to n n+ε < p 1, as the following counterexample shows. Let T = H e the one dimensional Hilert transform, and let = χ (0, ). Given 1/2 <p 1, let a(x) ethe(p, ) atom χ (0,1/2) χ ( 1/2,0). Then, for x>1we can write, Thus, for p 1. [χ(0, ),H](a)(x) 1/2 dy = 0 x + y x = 1 2x x. [χ (0, ),H](a)(x) p dx 1 dx 3 p x p = 1

4 20 Alvarez Remar 1.3. as We can write the commutator [, T ] acting in the sense of distriutions ([, T ](f),g)=((x, y)((x) (y)),f g). In particular, if supp(f) supp(g) = then we have ([, T ](f),g)= (x, y)((x) (y))f(y)g(x)dydx. The aove integral representation shows that the ernel of the linear commutator does not satisfy in general Definition 1.1. This explains why it is expected that this operator will not have all the typical properties of a Calderón-Zygmund operator. The aove counterexample shows that we need to loo for a space smaller than H p, if we want to otain a continuity result into L p. Following the definition of H 1 introduced in [9], we consider an atomic space, H p, which is a suspace of the Hardy space H p, for <p 1. n n+1 Definition 1.4. Let e a locally integrale function. Given a ounded function a, wesaythatais a (p,, ) atom if i) supp(a) Q = Q(z,σ) ii) a Q 1/p iii) a(x)dx = a(x)(x)dx =0. A temperate distriution f is said to elong to H p if, in the S -sense, it can e written as f = 1 λ a, where a are (p,, ) atoms, λ C, and 1 λ p <. As usual, we define on H p the quasinorm, f p = inf λ a =f λ p 1 1/p. Theorem 1.5 Given a BMO function, and a Calderón-Zygmund operator T, the linear commutator [,T] maps continuously H p into Lp n for n+ε <p 1. Proof. It suffices to show that there exists C>0suchthat for every (p,, ) atom a we have [, T ](a) p dx C. Thus, let us fix a (p,, ) atom a supported on a cue Q = Q(z,σ). Then, [, T ](a) p dx [, T ](a) p dx + [, T ](a) p dx = (1) + (2) Q(z,Nσ) R n \Q(z,Nσ) where N = N(n) is a positive constant to e chosen later.

5 Continuity properties for linear commutators of Calderón-Zygmund operators 21 In (1) we will use Hölder s inequality withexponent q/p and the fact that the operator [, T ] is continuous on all L q spaces for 1 <q<. Thus, ( (1) ( C ) p/q [, T ](a) q 1 p/q dx Q(z,Nσ) ) p/q a q 1 p/q dx Q(z,Nσ) Q(z,σ) 1 Q(z,σ) p/q Q(z,Nσ) 1 p/q = C. To estimate (2) we write the integral as the sum of a series and we use in each term Hölder s inequality withexponent 1/p. (2) = [, T ](a) p dx 0 Q(z,2 +1 Nσ)\Q(z,2 Nσ) C ( p Q(z,2 [, T ](a) dx) Nσ) 1 p. 0 Q(z,2 +1 Nσ)\Q(z,2 Nσ) For revity, let us write Q(z,2 +1 Nσ)\Q(z,2 Nσ)=C(z,2 Nσ). Given any numer c C, we will estimate the integral aove as (x) c T (a)(x) dx + T (( c)a)(x) dx. C(z,2 Nσ) C(z,2 Nσ) Let us consider the first term aove. We choose N so that 2 y z < x z for y Q(z,σ), x C(z,2 Nσ). Then, (x) c T (a)(x) dx C(z,2 Nσ) (x) c (x, y) (x, z) a(y) dydx C(z,2 Nσ) y z ε C (x) c n+ε a(y) dydx C(z,2 Nσ) x z C Q ε/n Q(z,2 Nσ) 1 ε/n Q 1/p +1 Q(z,2 +1 Nσ) C Q ε/n 1/p +1 Q(z,2 Nσ) ε/n 1 Q(z,2 +1 Nσ) (x) c dx Q(z,2 +1 Nσ) (x) c dx.

6 22 Alvarez Let us now consider the second term with N fixed as aove. Using the cancellation conditions on a, we have, T (( c)a) dx C(z,2 Nσ) (x, y) (x, z) (y) c a(y) dydx C(z,2 Nσ) dx C C(z,2 Nσ) x z n+ε y z ε (y) c a(y) dy C Q ε/n 1/p t n 1 n ε dt (y) c dy 2 Nσ C Q ε/n 1/p +1 Q(z,2 Nσ) ε/n 1 Q We sustitute these estimates in the series aove to otain, Q Q (y) c dy. C ( (2 n ) 1 p(1+ ε/n) 1 Q(z,2 +1 (x) c dx Nσ) 0 Q(z,2 +1 Nσ) + C ( p 1 0 (2n ) 1 p(1+ ε/n) (x) c dx). Q Taing the inf c C and the sup Q, we have [, T ](a) p dx C p BMO (2 n ) 1 p(1+ ε/n). The condition p> n n+ε Q 0 implies that the series converges. So we otain [, T ](a) p dx C p BMO ) p where the positive constant C does not depend on a or. This completes the proof of Theorem 1.5. Remar 1.6. The case p = 1 in Theorem 1.5 was proved in [9]. A modification of the counterexample shown aove, will prove that [, T ] is not continuous from H p into Lp, when p = n n+ε.

7 Continuity properties for linear commutators of Calderón-Zygmund operators 23 In fact, let T = H e the one dimensional Hilert transform, and let = χ (0, ). The function a = χ (0,1/2) χ (1/2,1) is a (p,, ) atom for each p 1. For x<0we can write [, T ](a)(x) = T (a)(x) = (x 1/2)2 ln x(x 1) = (1 ln 1 (2x 1) 2 ) = 1 If x< 1 then 8/9 ( 2x +1) 2 ( 2x +1) 2 t ( 2x +1) 2. Thus, given p>0 we have [, T ](a)(x) p dx 1 0 dt 1 (2x 1) 2 t = dt ( 2x +1) 2 t. [, T ](a)(x) p dx 1 0 dx ( 2x +1) 2p. And this integral diverges if p 1/2. In the case of a Calderón-Zygmund operator T, it is nown (cf. [1]), that T maps continuously H p into L p, when p = n n+ε. For the linear commutator, the natural version of this result would e to prove the continuity from H p into Lp,. Whether this is the case, is not nown in general. We have the following partial result. Proposition 1.7 Given an L function and a Calderón-Zygmund operator T, the linear commutator [, T ] maps continuously H p into Lp, for p = n n+ε, 0 <ε<1. In the proof of Proposition 1.7 we will use the following result, which was proved independently y N.J. Kalton [6], and y E.M. Stein, M. Taileson, and G. Weiss [10]. Lemma 1.8 Let {g } e a sequence of measurale functions and let 0 <p<1. Assume that {x : g (x) >λ} C λ with C not depending on and λ. Then, for everynumerical p sequence {c } p summale we have { } x : c g (x) >λ 2 p C 1 p λ p c p.

8 24 Alvarez Proof of Proposition 1.7. According to Definition 1.4 and Lemma 1.8, it suffices to show that there exists a constant C>0 not depending on, so that for each (p,, ) atom a and each λ>0wehave We can write Let us first estimate (1). λ p {x R n : [, T ](a)(x) >λ} C p. λ p {x R n : [, T ](a)(x) >λ} λ p {x R n : (x)t (a)(x) >λ/2} + λ p {x R n : T (a)(x) >λ/2} = (1) + (2). (1) λ p {x R n : T (a)(x) >λ/2 }. We oserve that a isa(p, ) atom in the space H p. According to Theorem 2.1 in [1], p. 21, the operator T maps continuously H p into L p, when p = n n+ε. Thus, we can write (1) C p for some C>0 not depending on a,. Now we estimate (2). According to Definition 1.4, the function a is also a (p, ) atom in the space H p. Thus, using again Theorem 2.1 in [1], p. 21, we have ( ) (2) = λ {x p R n : T a (x) >λ/2 This completes the proof of Proposition 1.7. } C p. 2. (H p,, L p, ) continuity for the linear commutator It was oserved in [9] that the commutator of a BMO function witha Calderón- Zygmund operator is not of type (L 1,L 1, ). Furthermore, it was also proved that it satisfies an estimate of type L log L. The example presented at the eginning of Section 1 shows that this linear commutator is not of type (H p,l p, ), for p<1. Indeed, λ p {x R n : [χ (0, ),H](a)(x) >λ} λ p 1( 1 ) 3 λ for 0 <λ<1/3.

9 Continuity properties for linear commutators of Calderón-Zygmund operators 25 We will now introduce an atomic space that provides wea type estimates for p 1. This atomic space comines features from the space H p, withthe type of atomic decomposition otained in [7] and [1] for the space H p,. Definition 2.1. Let e a locally integrale function. We say that a temperate distriution f elongs to the space H p, if there exists a sequence {f } << L suchthat 1. f = << f,inthes sense. 2. Eachfunction f can e decomposed as f = 1 in L H p, where the functions satisfy the following properties. 2i) supp( ) is contained in a cue Q, with sup χ Q <, sup 2 p Q < 1 1 2ii) there exists a constant C = C(n, p) > 0 suchthat C2, for every,, 2iii) (x)dx = (x)(x)dx =0. We define on the space H p, the following quasinorm Q. f p H p, = inf << 1 =f sup Z 2 p 1 For revity, we will sometimes denote C 1 = sup Z 2 p 1 Q. Let us oserve that the sequence { } almost provides an atomic decomposition of f into (p,, ) atoms. In fact, if we write a = 2 Q 1/p then a has the required cancellation, supp(a ) Q and a 2 Q 1/p C Q C 1/p. Moreover, 2 Q 1/p p =2 p Q < 1 1 uniformly in. Theorem 2.2 Let e a BMO function and let T e a Calderón-Zygmund operator. The linear commutator [,T] maps continuously H p, into L p, n for n+ε <p 1. C

10 26 Alvarez Proof. Given f H p,, let f = f as descried in Definition 2.1. Fix N =0, 1, 2,..., and consider N N 1 e an atomic decomposition for f. By a limiting argument it is enoughto show that there exists C = C(, T ) > 0 suchthat } sup λ {x p R n N : λ>0 [, T ]( f )(x) >λ CC 1 for every N =0, 1, 2,... Given λ>0, let 0 Z e suchthat 2 0 λ Then, write N 0 f = f + N N N N f = F 1 + F Let us first oserve that F 1 elongs to L q, for any 1 <q<. In fact, Thus, F 1 (x) 0 = N 1 F 1 q C C (x) 0 C 0 = N 0 = N 2 1 Q 2 1 = N 2 1 1/q χ Q 1/q Q CC 1/q 1 C 0 = N 0 = N 2 χ 1 Q. 2 2 p/q CC 1/q 1 2 0(1 p/q) CC 1/q 1 λ 1 p/q. Since the commutator [, T ] is nown to e continuous on L q for any 1 <q< we can write λ p {x R n : [, T ](F 1 )(x) λ} Cλ p q F 1 q q CC 1 λ p q λ q p = CC 1. Now, let A Q e the cue with the same center as Q and sides A times longer, A eing a positive numer 2 n+1, possily depending on, to e chosen later. For revity, let B 0,N = 0 +1 N, 1A Q. We have, λ p {x R n : [, T ](F 2 )(x) >λ} = λ p {x B 0,N : [, T ](F 2 )(x) >λ} + λ p {x R n \B 0,N : [, T ](F 2 )(x) >λ} = (1) + (2).

11 Continuity properties for linear commutators of Calderón-Zygmund operators 27 Let us first estimate (2). If c is any complex numer, we can write, (2) λ p {x R n \B 0,N : (x) c T (F 2 )(x) >λ/2} + λ p {x R n \B 0,N : T (( c)f 2 )(x) >λ/2} = (2) 1 + (2) 2. We have, (2) 1 (x) c p T (F 2 )(x) p dx, R n \B 0,N R n \A Q (x) c p T ( )(x) p dx. Let us now fix,. (x) c p T ( )(x) p dx R n \A Q = (x) c p T ( )(x) p dx l 0 2 l+1 A Q \2l A Q ( ) 1 p ( (x) c p/(1 p) dx T ( )(x) ) p dx l 0 2 l+1 A Q 2 l+1 A Q \2l A Q = [ ] 1 p 2 l+1 A Q 1 p 1 l 0 2 l+1 A Q (x) c p/(1 p) dx 2 l+1 A Q [ T ( )(x) ] p dx. 2 l+1 A Q \2l A Q p Oserve that 1 p 1 for p> n n+ε. ( If we denote for revity B 1 = sup Q aove can e maorized y CB p 1 l 0 2 l+1 A Q 1 p { 2 l+1 A Q \2l A Q 1 Q Q (x) c p/(1 p) dx) (1 p)/p, th e (x, y) (x, x ) (y) dydx } p where x is the center of the cue Q.

12 28 Alvarez Let us consider first the integral on x. IfA is large enough, we can write, (x, y) (x, x ) dx 2 l+1 A Q \2l A Q C y x ε dx 2 l+1 A Q \2l A Q x x n+ε = C y x ε 2 l A C Q tn 1 n ε dt C Q ε/n 1/n 2 εl A ε Replacing this estimate in the sum over l aove, we otain, 2 l(1 p)n A n(1 p) Q 1 p ( 2 εl A ε ) p 2 p Q p CB p 1 l 0 = CB p 1 An p(n+ε) Q Q ε/n. 2 p 2 l(n p(n+ε)) CB p 1 2p A n p(n+ε) l 0 Q. Let us sustitute this estimate ac in the sum over and. We can write (2) 1 CB p 1 N = p A n p(n+ε) Q CC1 B p 1 1 N = 0 +1 A n p(n+ε). We now select an appropriate dilation A, for instance, A = A2 ( 0)/(n+ε), A fixed large enough so that the ernel estimate used aove is true. Thus, we get, (2) 1 CC 1 B p 1 N = ( 0)(n/(n+ε) p) = CC 1 B p 1. This completes the estimate of (2) 1. Let us now consider (2) 2. N (2) 2 λ p x Rn \B 0,N : T (( c) )(x) >λ/2. = Using the cancellation properties of, we can write, T (( c) )(x) (x, y) (x, x ) (y) c (y) dy C 2 Q ε/n +1 ( ) 1 x x n+ε Q (y) c dy Q 2 Q CB 2 x x ε/n +1 n+ε

13 Continuity properties for linear commutators of Calderón-Zygmund operators 29 where for revity we have denoted B 2 = sup Q ( ) 1 (y) c dy. Q Q Thus, (2) 2 λ p x Rn \B 0,N : N CB 2 2 Q 1 x x = 0 +1 ε/n +1 n+ε >λ/2. Now, { } x R n 1 \B 0,N : x x n+ε >λ/2 { = x R n \B 0,N : x x < (λ/2) 1/(n+ε)} { x R n : } x x 1/(n+ε) < (λ/2) = Cλ n/(n+ε). Moreover, N (CB 2 2 ) Q (n+ε)/n n/(n+ε) = = CB n/(n+ε) 2 CC 1 B n/(n+ε) 2 N 2 n/(n+ε) = N = 0 +1 So, using Lemma 1.8, we have, Q 2 (n/(n+ε) p) CC 1 B n/(n+ε) 2 2 0(n/(n+ε) p). (2) 2 λ p CC 1 B n/(n+ε) 2 2 0(n/(n+ε) p) λ n/(n+ε) = CC 1 B n/(n+ε) 2. This completes the estimate of (2) 2. Then, with A = A2 ( 0)/(n+ε),wehave proved that ( 1 (2) CC 1 [sup Q Q + CC 1 [sup Q Q ( 1 Q ) ] (1 p)/p p (x) c p/(1 p) dx Q (x) c dx)] n/(n+ε).

14 30 Alvarez Thus, taing inf c C in each term on the right hand side of the aove inequality, we otain that [ ] (2) CC 1 p BMO + n/(n+ε). Now we estimate (1) with A as aove. (1) λ p 0 +1 N, 1A Q λ p λ p AC 1 = λ p A2 0p N BMO 2 n( 0)/(n+ε) 2 p A n Q ( 0)(n/(n+ε) -p) CC 1. Thus, we have proved that } λ {x p R n N : [, T ]( ( ) f )(x) >λ CC 1 1+ p BMO + n/(n+ε) BMO. Since p > n/(n + ε), this inequality implies that } λ {x p R n N : [, T ]( f )(x) >λ CC 1 (1 + p BMO ). N Finally, taing the limit as N and taing the infimum of C 1 over all possile representations = f, we complete the proof of Theorem 2.2. References 1. J. Alvarez, H p and wea H p continuity of Calderón-Zygmund type operators, Fourier Analysis: Analytic and Geometric Aspects, Marcel Deer, 17 34, J. Alvarez, R. Bagy, D. Kurtz and C. Pérez, Weighted estimates for commutators of linear operators, Studia Math. 104 (1993), J. Alvarez and C. Pérez, Estimates with A weights for various singular integral operators, Boll. Un. Mat. Ital. A (7) 8 (1994), R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque 57 (1979), Soc. Math. France. 5. R. Coifman, R. Rocherg and G. Weiss, Factorization theorems for Hardy spaces in several variales, Ann. of Math. 103 (1976),

15 Continuity properties for linear commutators of Calderón-Zygmund operators N.J. Kalton, Linear operators on L p for 0 <p<1, Trans. Amer. Math. Soc. 259 (1980), Heping Liu, The wea H p spaces on homogeneous groups, Lectures Notes in Math (1991), M. Paluszyńsi, Characterization of Lipschitz spaces via commutator of Coifman, Rocherg and Weiss; a multiplier theorem for the semigroup of contractions, Ph. D. Thesis, Washington Univ., C. Pérez, End point estimates for commutators of singular integral operators, J. Funct. Anal. 128 (1995), E.M. Stein, M. Taileson and G. Weiss, Wea type estimates for maximal operators on certain H p spaces, Rend. Circ. Mat. Palermo (2) Suppl. 1 (1981),