Sharp bounds for general commutators on weighted Lebesgue spaces

arXiv:1002.2396v2 [math.CA] 8 Mar 2011 SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES DAEWON CHUNG, MAR´ IA CRISTINA PEREYRA AND CARLOS PEREZ. Abstract. We show that if a linear operator Tis bounded on weighted Lebesgue space L2(w) and obeys a linear bound with respect to the A2constant of the weight, then its commutator [b, T ] with a function bin BMO will obey a quadratic bound with respect to the A2constant of the weight. We also prove that the kth-order commutator Tk b= [b, T k−1 b] will obey a bound that is a power (k+ 1) of the A2 constant of the weight. Sharp extrapolation provides corresponding Lp(w) estimates. In particular these estimates hold for Tany Calder´on-Zygmund singular integral operator. The results are sharp in terms of the growth of the operator norm with respect to the Apconstant of the weight for all 1 < p < ∞, all k, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show. 1. Introduction Singular integral operators are known to be bounded in weighted Lebesgue spaces Lp(w) if the weight belongs to the Apclass of Muckenhoupt. Recently there has been renewed interest in understanding the dependence of the operator norm in terms of the Apconstant of the weight, more precisely one seeks estimates of the type, kTkLp(w)≤ϕp([w]Ap) 1 < p < ∞, where the function ϕp: [1,∞)→[0,∞) is optimal in terms of growth. The first result of this type was obtained by S. Buckley [4] who showed that the maximal function obeyed such estimates with ϕp(t) = cpt1 p−1for 1 < p < ∞, and this is optimal (see [21] for another recent proof). This problem has attracted renewed attention because of the work of Astala, Iwaniec and Saksman [2]. They proved sharp regularity results for solutions to the Beltrami equation, assuming that the operator norm of the BeurlingAhlfors transform grows linearly in terms of the Apconstant for p≥2. This linear growth was proved by S. Petermichl and A. Volberg [32] and by Petermichl [30,31] for the Hilbert transform and the Riesz Transforms. In these papers it has been shown that if Tis any of these operators, then (1.1) kTkLp(w)≤cp,n [w]max{1,1 p−1} Ap1< p < ∞, 2010 Mathematics Subject Classification. Primary 42B20, 42B25. Secondary 46B70, 47B38. Key words and phrases. commutators, singular integrals, BMO, A2,Ap. The third author would like to acknowledge the support of Spanish Ministry of Science and Innovation via grant MTM2009-08934. 1 2 DAEWON CHUNG, MAR´ IA CRISTINA PEREYRA AND CARLOS PEREZ. and the exponent max n1,1 p−1ois best possible. It has been conjectured, and very recently proved [16], that the same estimate holds for any Calder´on-Zygmund operator T. By the sharp version of the Rubio de Francia extrapolation theorem [12], it suffices to prove this inequality for p= 2, namely (1.2) kTkL2(w)≤cn[w]A2. We remit the reader to [10] for a new proof and for generalizations and applications. The linear growth in L2(w) has been shown to hold for dyadic operators (martingale transform [34], dyadic square function [15], dyadic paraproduct [3]), or for operators who have lots of symmetries and can be written as averages of dyadic shift operators (such us the Hilbert transform [30], Riesz transforms [31], Beurling transform [32], [13]). All these estimates were obtained using Bellman functions. Recently all the above results have been recovered using different sets of techniques, and obtaining linear bounds for alarger class of Haar shift operators [20], [8]. In particular, in the latter paper [8], no Bellman function techniques nor any two weight results are used, and the methods can be extended to other important operators in Harmonic Analysis such as dyadic square functions and paraproducts, maximal singular integrals and the vector-valued maximal function as can be found in [9]. The sharp bound (1.2) for any Calder´on-Zygmund operator Thas been proved in [16] by T. Hyt¨onen. Hyt¨onen’s proof is based on approximating Tby generalized dyadic Haar shift operators with good bounds combined with the key fact that to prove (1.2) it is enough to prove the corresponding weak type (2,2) estimate with the same linear bound as proved in [29]. A direct proof avoiding this weak (2,2) reduction can be found in [17]. A bit earlier, in [22], the sharp Lp(w) bound for Twas obtained for values of poutside the interval (3/2,3) and the proof is based on the corresponding estimates for the intrinsic square functions. It should be mentioned that until the A2-conjecture was proved, only the following special case (1.3) kTkLp(w)≤Cp[w]A11< p < ∞, had been shown to be true for any Calder´on-Zygmund operator. Observe that the condition imposed on the weight is the A1weight condition which is stronger than Ap but there is a gain in the exponent since it is linear for any 1 < p < ∞(compare with (1.1)). This has been shown in [23,24] and we remit the reader to [28] for a survey on this topic. The main purpose of this paper is to prove estimates similar to (1.1) for commutators of appropriate linear operators Twith BMO functions b. These operators are defined formally by the expression [b, T]f=bT(f)−T(b f). When Tis a singular integral operator, these operators were considered by Coifman, Rochberg and Weiss in [7]. Although the original interest in the study of such operators SHARP BOUNDS FOR GENERAL COMMUTATORS 3 was related to generalizations of the classical factorization theorem for Hardy spaces many other applications have been found. The main result from [7] states that [b, T] is a bounded operator on Lp(Rn), 1 < p < ∞, when bis a BMO function and Tis a singular integral operator. In fact, the BMO condition of bis also a necessary condition for the Lp-boundedness of the commutator when Tis the Hilbert transform. Later on a different proof was given by J. O. Str¨omberg (cf. [33] p.417) with the advantage that it allows to show that these commutators are also bounded on weighted Lp(w), when w∈Ap. This approach is based on the use of the classical C. Fefferman-Stein maximal function and it is not precise enough for further developments. Indeed, we may think that these operators behave as Calder´on-Zygmund operators, however there are some differences. For instance, an interesting fact is that, unlike what it is done with singular integral operators, the proof of the Lp-boundedness of the commutator does not rely on a weak type (1,1) inequality. In fact, simple examples show that in general [b, T] fails to be of weak type (1,1) when b∈BMO. This was observed by the third author in [26] where it is also shown that there is an appropriate weak-L(log L) type estimate replacement. This shows that the operator cannot be a Calder´on-Zygmund singular integral operator. To stress this point of view it is also shown by the third author [27] that the right operator controlling [b, T] is M2=M◦M, instead of the Hardy-Littlewood maximal function M. In the present paper we pursue this point of view by showing that commutators have an extra “bad” behavior from the point of view of the Aptheory of weights that it is not reflected in the classical situation. Our argument will be based on the second proof for the Lp-boundedness of the commutator presented in [7]. This proof is interesting because there is no need to assume that Tis a singular integral operator, to show the boundedness of the commutator it is enough to assume that the operator Tis linear and bounded on Lp(w) for any w∈Ap. These ideas were exploited in [1]. The first author showed in [5] that the commutator with the Hilbert transform obeys an estimate of the following type, (1.4) k[b, H]kL2(w)≤C[w]2 A2kbkBMO, and he also showed that the quadratic growth with respect to the A2constant of the weight is sharp. The techniques used in that paper rely very much in dyadic considerations and the use of Bellman function arguments. Using recent results on Haar shifts operators, he also deduced the quadratic growth for commutators of Haar shift operators and operators in their convex hull, including the Riesz transforms and the Beurling-Ahlfors operator, see [5] for the Also, it should be mentioned that there is a corresponding version of (1.3) for commutators of any Calder´on-Zygmund operator with quadratic growth as in (1.4). This is proved in [25] where an endpoint estimate can also be found. By completely different methods, we show in this paper that if an operator obeys an initial linear bound in L2(w), then its commutator will obey a quadratic bound in L2(w). In light of the positive resolution of the A2-conjecture, we conclude that the 4 DAEWON CHUNG, MAR´ IA CRISTINA PEREYRA AND CARLOS PEREZ. commutator of any Calder´on-Zygmund singular integral operator and a BMO function obeys a quadratic bound in L2(w). In fact we show that if an operator Tobeys a bound in L2(w) of the form ϕ([w]A2), then its k-th order commutator with b∈BMO, Tk b:= [b, Tk−1 b], will obey a bound of the form ck nk!ϕ(γn[w]A2)[w]k A2kbkk BMO. Observe that if we consider the special case of Calder´on-Zygmund operators with kernel Kthen Tk b(f)(x)ZRn (b(x)−b(y))kK(x, y)f(y)dy, and the larger kis, the more singular the operator will be, because the exponent in [w]k A2becomes larger. Corresponding estimates in Lp(w) are deduced by the sharp version of the Rubio de Francia extrapolation theorem found in [12], and are shown to be sharp in the case of the Hilbert and Riesz transforms (in any dimension) for all 1 < p < ∞, and for all k≥1. It will be interesting to recover the result for the higher order commutators with the Haar shift operators (and hence for the Hilbert, Riesz and Beurling transforms) using the dyadic methods, but so far we do not know how to do this. Recently extensions of our result to two weight settings, fractional integrals and more have been obtained by D. Cruz-Uribe and Kabe Moen, see [11]. The remainder of this paper is organized as follows. In Section 2we gather some basic results. In Section 3we give the proof of the main result. In Section 4we show with examples that the main theorem in the paper, and its corollaries are sharp. Finally, the last section is an appendix where we show a result that it is claimed but never proved in the literature, a sharp reverse H¨older’s inequality for A2weights. 2. Preliminary results 2.1. A Sharp John-Nirenberg. For a locally integrable b:Rn→Rwe define kbkBMO = sup Q 1 |Q|ZQ|b(y)−bQ|dy < ∞, where the supremum is taken over all cubes Q∈Rnwith sides parallel to the axes, and bQ=1 |Q|ZQ b(y)dy. The main relevance of BMO is because of its exponential self-improving property, recorded in the celebrated John-Nirenberg Theorem [18]. We need a very precise version of it, as follows: Theorem 2.1. [Sharp John-Nirenberg] There are dimensional constants 0≤αn<1< βnsuch that (2.1) sup Q 1 |Q|ZQ exp αn kbkBMO |b(y)−bQ|dy ≤βn. In fact we can take αn=1 2n+2 . SHARP BOUNDS FOR GENERAL COMMUTATORS 5 For the proof of this we remit to p. 31-32 of [19] where a proof different from the standard one can be found. We derive from Theorem 2.1 the following Lemma 2.2. that will be used in the proof of the main theorem. First recall that a weight wsatisfies the A2condition if [w]A2= sup Q1 |Q|ZQ w 1 |Q|ZQ w−1<∞, where the supremum is taken over all cubes Q∈Rnwith sides parallel to the axes. Notice that [w]A2≥1. It is well known that if w∈A2then b= log w∈BMO. A partial converse also holds, if b∈BMO there is an s0>0 such that w=esb ∈Ap,|s| ≤ s0. As a consequence of the Sharp John-Nirenberg Theorem we can get a more precise version of this partial converse. Lemma 2.2. Let b∈BMO and let αn<1< βnbe the dimensional constants from (2.1). Then s∈R,|s| ≤ αn kbkBMO =⇒es b ∈A2and [es b]A2≤β2 n. Proof. By Theorem 2.1, if |s| ≤ αn kbkBMO and if Qis fixed 1 |Q|ZQ exp(|s||b(y)−bQ|)dy ≤1 |Q|ZQ exp( αn kbkBMO |b(y)−bQ|)dy ≤βn, thus 1 |Q|ZQ exp(s(b(y)−bQ)) dy ≤βn, and 1 |Q|ZQ exp(−s(b(y)−bQ)) dy ≤βn. If we multiply the inequalities, the bQparts cancel out: 1 |Q|ZQ exp(s(b(y)−bQ)) dy 1 |Q|ZQ exp(s(bQ−b(y))) dy =1 |Q|ZQ exp(sb(y)) dy 1 |Q|ZQ exp(−sb(y)) dy≤β2 n namely es b ∈A2with [es b]A2≤β2 n.  We remark that it follows easily from minor modifications to the proof of Lemma 2.2 that if 1 < p < ∞ s∈R,|s| ≤ αn kbkBMO min 1,1 p−1=⇒es b ∈Apand [es b]Ap≤βp n, 6 DAEWON CHUNG, MAR´ IA CRISTINA PEREYRA AND CARLOS PEREZ. where as usual [w]Ap= sup Q1 |Q|ZQ w(x)dx 1 |Q|ZQ w(x)−1/(p−1)dxp−1 <∞. 2.2. Sharp reverse H¨older inequality for the A2class of weights. Recall that if w∈A2then wsatisfies a reverse H¨older condition, namely, there are constants r > 1 and c≥1 such that for any cube Q (2.2) 1 |Q|ZQ wrdx1 r ≤c |Q|ZQ w In the usual proofs, both constants, cand r, depend upon the A2constant of the weight. There is a more precise version of (2.2). Lemma 2.3. Let w∈A2and let rw= 1 + 1 2n+5[w]A2. Then 1 |Q|ZQ wrwdx1 rw≤2 |Q|ZQ w This result was stated and used in [4] but no proof was given. The author mentioned instead the celebrated work [6] where no explicit statement can be found. We supply in Section 5a proof taken from [28], where a more general version can be found as well as more information. 3. Main result Theorem 3.1. Let Tbe a linear operator bounded on L2(w)for any w∈A2. Suppose further that there is an increasing function ϕ: [1,∞)→[0,∞)such that (3.1) kTkL2(w)≤ϕ([w]A2). then there are constants γnand cnindependent of [w]A2such that (3.2) k[b, T]kL2(w)≤cnϕ(γn[w]A2) [w]A2kbkBMO. For the particular case ϕ(t) = c0tr, where r > 0, and c0>0, a simple induction argument shows that if kTkL2(w)≤a0[w]r A2, then for each integer k≥1 there is a constant akdepending on k, the initial value a0, and the parameters γnand cnin the theorem, such that the kth-order commutator Tk b defined recursively by Tk b:= [b, Tk−1 b], obeys the following weighted estimates kTk bkL2(w)≤ak[w]r+k A2kbkk BMO. More precisely, the sequence {ak}k≥0obeys the following recurrence equation that can be solved easily, ak=cnak−1γr+k−1 n=ck na0γkr+(k−2)(k−1) 2 n. SHARP BOUNDS FOR GENERAL COMMUTATORS 7 Using the method of proof of Theorem 3.1 we can obtain a weighted estimate for the kth-order commutator that works for general increasing function ϕ: [1,∞)→[0,∞). Notice the difference in the constants with what we just argued by induction for the particular case ϕ(t) = a0tr: in the corollary the constant is ck na0γr nk!, whereas in the induction argument the constant is ck na0γkr+(k−2)(k−1) 2 n. Corollary 3.2. Let Tbe a bounded linear operator on L2(w)with w∈A2and (3.3) kTkL2(w)≤ϕ([w]A2). then there are constants γnand cnindependent of [w]A2such that (3.4) kTk bkL2(w)≤ck nk!ϕ(γn[w]A2) [w]k A2kbkk BMO. The constants γnand cnthat appear in Corollary 3.2 are the same that appeared in Theorem 3.1. We first present the proof of Theorem 3.1, and afterwards we discuss the necessary modifications to obtain Corollary 3.2. As an easy consequence of Corollary 3.2 and the Rubio de Francia extrapolation theorem with sharp constants [12], we have the following. Corollary 3.3. Let Tbe a linear operator bounded on L2(w)with w∈A2and (3.5) kTkL2(w)≤ϕ([w]A2). Then, for 1< p < ∞, there are constants γn,p, and cn,p, which only depend on p, and the dimension n, such that for all weights w∈Ap (3.6) kTk bkLp(w)≤√2ck n,p k!ϕγn,p [w]max{1,1 p−1} Ap[w]kmax{1,1 p−1} Apkbkk BMO. In the particular case ϕ(t) = a0trthe extrapolated estimate looks like (3.7) kTk bkLp(w)≤√2a0ck nk!γr ncr+k p[w](r+k) max{1,1 p−1} Apkbkk BMO. where cpdepends only on p,γnand cnare the constants that appeared in Theorem 3.1. We will show in Section 4that for r= 1, ϕ(t) = a0tthe power (1 + k) max{1,1 p−1} cannot be decreased for T=Hand T=Rjthe Hilbert and Riesz transforms, for all k≥1 and p > 1, therefore the theorem is optimal in terms of the rate of the dependence on [w]Ap. In [5], examples for k= 1, for all p > 1, and for Tthe Hilbert, Beurling and Riesz transforms, were presented. Proof of Theorem 3.1.We “conjugate” the operator as follows: if zis any complex number we define Tz(f) = ezbT(e−zbf). Then, a computation gives (for instance for ”nice” functions), [b, T](f) = d dzTz(f)|z=0 =1 2πi Z|z|=ǫ Tz(f) z2dz , ǫ > 0 by the Cauchy integral theorem, see [7], [1]. 8 DAEWON CHUNG, MAR´ IA CRISTINA PEREYRA AND CARLOS PEREZ. Now, by Minkowski’s inequality (3.8) k[b, T](f)kL2(w)≤1 2π ǫ2Z|z|=ǫkTz(f)kL2(w)|dz|ǫ > 0. The key point is to find the appropriate radius ǫ. To do this we look at the inner norm kTz(f)kL2(w) kTz(f)kL2(w)=kT(e−zbf)kL2(we2Rez b), and try to find appropriate bounds on z. To do this we use the main hypothesis, namely that Tis bounded on L2(w) if w∈A2with kTkL2(w)≤ϕ([w]A2). Hence we should compute [we2Rez b]A2= sup Q1 |Q|ZQ we2Rez b(x)dx 1 |Q|ZQ w−1e−2Rez b(x)dx. Now, since w∈A2we use Lemma 2.3: if r=rw= 1 + 1 2n+5[w]A2<2 then 1 |Q|ZQ wrdx1 r ≤2 |Q|ZQ w , and similarly for w−1since rw=rw−1, 1 |Q|ZQ w−rdx1 r ≤2 |Q|ZQ w−1. Using this and Holder’s inequality we have for an arbitrary Q 1 |Q|ZQ w(x)e2Rez b(x)dx 1 |Q|ZQ w(x)−1e−2Rez b(x)dx≤ 1 |Q|ZQ wrdx1 r1 |Q|ZQ e2Rez r′b(x)dx1 r′1 |Q|ZQ w−rdx1 r1 |Q|ZQ e−2Rez r′b(x)dx1 r′ ≤41 |Q|ZQ w dx 1 |Q|ZQ w−1dx 1 |Q|ZQ e2Rez r′b(x)dx1 r′1 |Q|ZQ e−2Rez r′b(x)dx1 r′ ≤4 [w]A2[e2Rez r′b] 1 r′ A2 Now, since b∈BMO we are in a position to apply Lemma 2.2, if |2Rez r′| ≤ αn kbkBMO then [e2Rez r′b]A2≤β2 n. Hence for these z, and since 1 < r < 2, [we2Rez b]A2≤4 [w]A2β 2 r′ n≤4 [w]A2βn. Using this estimate for these z, and observing that ke−zbfkL2(we2Rezb)=kfkL2(w), kTz(f)kL2(w)=kT(e−zbf)kL2(we2Rez b)≤ϕ([we2Rez b]A2)kfkL2(w)≤ϕ(4[w]A2βn)kfkL2(w). SHARP BOUNDS FOR GENERAL COMMUTATORS 9 Choosing now the radius ǫ=αn 2r′kbkBMO , we can continue estimating the norm in (3.8) k[b, T](f)kL2(w)≤1 2π ǫ2Z|z|=ǫkTz(f)kL2(w)|dz| ≤1 2π ǫ2Z|z|=ǫ ϕ(4[w]A2βn)kfkL2(w)|dz|=1 ǫϕ(4[w]A2βn)kfkL2(w), since |2Rez r′| ≤ 2|z|r′= 2ǫ r′=αn kbkBMO . Finally, for this ǫ, k[b, T](f)kL2(w)≤C22nϕ(4[w]A2βn) [w]A2kbkBMO, because r′= 1 + 2n+5[w]2≈2n[w]2, and αn=1 2n+2 . Observe that the optimal radius is essentially the inverse of [w]2kbkBMO. This proves the theorem with cn∼22nand γn= 4 βn.  Proof of Corollary 3.2.In this case a computation gives (for instance for ”nice” functions), see [1] for example or the original paper [7], Tk b(f) = dk dzkTz(f)|z=0 =k! 2πi Z|z|=ǫ Tz(f) zk+1 dz ǫ > 0 by the Cauchy integral theorem. The same calculation as in the case k= 1 gives the required estimate, with ck n≈22nk, and the same γn= 4βn.  4. Examples In this section, we show that one can not have estimates better than Theorem 3.1, Corollary 3.2, and Corollary 3.3. We present examples which return the same growth with respect to the Apconstant of the weight that appears in our results. First, we discuss the simpler case in dimension one. The following example shows that the quadratic estimate for the first commutator of the Hilbert transform is sharp for p= 2. 4.1. Sharp example for the commutator of the Hilbert transform. Consider the Hilbert transform Hf(x) = p.v. ZR f(y) x−ydy, and consider the BMO function b(x) = log |x|. We know that there is a constant c such that (4.1) k[b, H]kL2(w)≤c[w]2 A2 16 DAEWON CHUNG, MAR´ IA CRISTINA PEREYRA AND CARLOS PEREZ. [34] J. Wittwer, A sharp estimates on the norm of Martingale transform, Math. Res. Lett. 7(2000) 1-12. Daewon Chung, Department of Mathematics and Statistics MSC01 1115, 1 University of New Mexico, Albuquerque, NM 87131-0001 E-mail address:[email protected]du Mar´ ıa Cristina Pereyra, Department of Mathematics and Statistics, MSC01 1115, 1 University of New Mexico, Albuquerque, NM 87131-0001 E-mail address:[email protected] Carlos P´ erez, Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad De Sevilla, 41080 Sevilla, Spain. E-mail address:[email protected]