SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES. date: Feb

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1 SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES DAEWON CHUNG, CRISTINA PEREYRA AND CARLOS PEREZ. Abstract. We show that if an operator T is bounded on weighted Lebesgue space L 2 (w) and obeys a linear bound with respect to the A 2 constant of the weight, then its commutator [b, T ] with a function b in BMO will obey a quadratic bound with respect to the A 2 constant of the weight. We also prove that the kth-order commutator Tb k = [b, T k b ] will obey a bound that is a power (k + ) of the A 2 constant of the weight. Sharp extrapolation provides corresponding L p (w) estimates. The results are sharp in terms of the growth of the operator norm with respect to the A p constant of the weight for all < p <, all k, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show. date: Feb-8-2. Introduction Singular integral operators are known to be bounded in weighted Lebesgue spaces L p (w) if the weight belongs to the A p class of Muckenhoupt. Recently there has been renewed interest in understanding the dependence of the operator norm in terms of the A p constant of the weight, more precisely one seeks estimates of the type, T L p (w) ϕ p ([w] Ap ) < p <, where the function ϕ p : [, ) [, ) 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 ϕ p (t) = c p t p for < p <, and this is optimal (see [7] 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 Beurling- Ahlfors transform grows linearly in terms of the A p constant for p 2. This linear growth was proved by S. Petermichl and A. Volberg [27] and by Petermichl [25, 26] for the Hilbert transform and the Riesz Transforms. In these papers it has been shown that if T is any of these operators, then (.) T L p (w) c p,n [w] max{, p } A p < p <, 99 Mathematics Subject Classification. Primary 42B2, 42B25. Secondary 46B7, 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 MTM

2 2 DAEWON CHUNG, CRISTINA PEREYRA AND CARLOS PEREZ. { } and the exponent max, is best possible. It has been conjectured that the p same estimate holds for any Calderón-Zygmund operator T. By the sharp version of the Rubio de Francia extrapolation theorem [9], it suffices to prove this inequality for p = 2, namely (.2) T L 2 (w) c n [w] A2. So far the linear growth in L 2 (w) has been shown to hold for dyadic operators (martingale transform [29], dyadic square function [2], 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 [25], Riesz transforms [26], Beurling transform [27], []). 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 the larger class of Haar shift operators [5], [6], [6]. In particular, in the latter paper [6], 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. It should be mentioned that only the following special case T L p (w) C p [w] A < p <, has been shown to be true for any Calderón-Zygmund operator. Observe that the condition imposed on the weight is the A weight condition which is stronger than A p but there is a gain in the exponent since it is linear for any < p < (compare with (.)). This has been shown in [8, 9] and we remit the reader to [23] for a survey on this topic. The main purpose of this paper is to prove estimates similar to (.) for commutators of appropriate linear operators T with BM O functions b. These operators are defined formally by the expression [b, T ]f = bt (f) T (b f). When T is a singular integral operator, these operators were considered by Coifman, Rochberg and Weiss in [8]. Although the original interest in the study of such operators was related to generalizations of the classical factorization theorem for Hardy spaces many other applications have been found. The main result from [8] states that [b, T ] is a bounded operator on L p (R n ), < p <, when b is a BMO function and T is a singular integral operator. In fact, the BMO condition of b is also a necessary condition for the L p -boundedness of the commutator when T is the Hilbert transform. Later on a different proof was given by J. O. Strömberg (cf. [28] p.47) with the advantage that it allows to show that these commutators are also bounded on weighted L p (w), when w A p. 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 Calderón-Zygmund operators, however there are some differences. For instance, an

3 SHARP BOUNDS FOR GENERAL COMMUTATORS 3 interesting fact is that, unlike what it is done with singular integral operators, the proof of the L p -boundedness of the commutator does not rely on a weak type (, ) inequality. In fact, simple examples show that in general [b, T ] fails to be of weak type (, ) when b BMO. This was observed by the third author in [2] 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 Calderón-Zygmund singular integral operator. To stress this point of view it is also shown by the third author [22] that the right operator controlling [b, T ] is M 2 = 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 A p theory of weights that it is not reflected in the classical situation. Our argument will be based on the second proof for the L p -boundedness of the commutator presented in [8]. This proof is interesting because there is no need to assume that T is a singular integral operator, to show the boundedness of the commutator it is enough to assume that the operator T is linear and bounded on L p (w) for any w A p. These ideas were exploited in []. The first author showed in [5] that the commutator with the Hilbert transform obeys an estimate of the following type, [b, H] L 2 (w) C [w] 2 A 2 b BMO, and he also showed that the quadratic growth with respect to the A 2 constant 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 details. By completely different methods, we show in this paper that if an operator obeys an initial linear bound in L 2 (w), then its commutator will obey a quadratic bound in L 2 (w). In fact we show that if an operator T obeys a bound in L 2 (w) of the form ϕ([w] A2 ), then its k-th order commutator with b BMO, Tb k k := [b, Tb ], will obey a bound of the form c k nk!ϕ(γ n [w] A2 )[w] k A 2 b k BMO. Observe that if we consider the special case of Calderón-Zygmund operators with kernel K then Tb k (f)(x) (b(x) b(y)) k K(x, y)f(y) dy, R n and the larger k is, the more singular the operator will be, because the exponent in [w] k A 2 becomes larger. Corresponding estimates in L p (w) are deduced by the sharp version of the Rubio de Francia extrapolation theorem found in [9], and are shown to be sharp in the case of the Hilbert and Riesz transforms (in any dimension) for all < p <, and for all k.

4 4 DAEWON CHUNG, CRISTINA PEREYRA AND CARLOS PEREZ. 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. The remainder of this paper is organized as follows. In Section 2 we gather some basic results. In Section 3 we give the proof of the main result. In Section 4 we 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 Hölder s inequality for A 2 weights. 2. Preliminary results 2.. A Sharp John-Nirenberg. For a locally integrable b : R n R we define b BMO = sup b(y) b dy <, where the supremum is taken over all cubes R n with sides parallel to the axes, and b = b(y) dy. The main relevance of BM O is because of its exponential self-improving property, recorded in the celebrated John-Nirenberg Theorem [3]. We need a very precise version of it, as follows: Theorem 2.. [Sharp John-Nirenberg] There are dimensional constants α n < < β n such that ( ) αn (2.) sup exp b(y) b dy β n. b BMO In fact we can take α n = 2 n+2. For the proof of this we remit to p of [4] where a proof different from the standard one can be found. We derive from Theorem 2. the following Lemma 2.2. that will be used in the proof of the main theorem. First recall that a weight w satisfies the A 2 condition if [w] A2 = sup ( ) ( w w ) <, where the supremum is taken over all cubes R n with sides parallel to the axes. Notice that [w] A2. It is well known that if w A 2 then b = log w BMO. A partial converse also holds, if b BMO there is an s > such that w = e sb A p, s s. As a consequence of the Sharp John-Nirenberg Theorem we can get a more precise version of this partial converse.

5 SHARP BOUNDS FOR GENERAL COMMUTATORS 5 Lemma 2.2. Let b BMO and let α n < < β n be the dimensional constants from (2.). Then s R, s α n b BMO = e s b A 2 and [e s b ] A2 β 2 n. Proof. By Theorem 2., if s αn b BMO exp( s b(y) b ) dy and if is fixed α n exp( b(y) b ) dy β n, b BMO thus exp(s(b(y) b )) dy β n, and exp( s(b(y) b )) dy β n. If we multiply the inequalities, the b parts cancel out: ( ) ( exp(s(b(y) b )) dy exp(s(b b(y))) dy = ( namely e s b A 2 with ) ( exp(sb(y)) dy ) exp( sb(y)) dy βn 2 [e s b ] A2 β 2 n. We remark that it follows easily from minor modifications to the proof of Lemma 2.2 that if < p < s R, s α { } n min, = e s b A p and [e s b ] Ap β p b BMO p n, where as usual [w] Ap = sup ( ) ( p w(x)dx w(x) dx) /(p ) < Sharp reverse Hölder inequality for the A 2 class of weights. Recall that if w A 2 then w satisfies a reverse Hölder condition, namely, there are constants r > and c such that for any cube (2.2) ( ) w r r c dx w In the usual proofs, both constants, c and r, depend upon the A 2 constant of the weight. There is a more precise version of (2.2). )

6 6 DAEWON CHUNG, CRISTINA PEREYRA AND CARLOS PEREZ. Lemma 2.3. Let w A 2 and let r w = + 2 n+5 [w] A2. Then ( w rw dx ) rw 2 This result was stated and used in [4] but no proof was given. The author mentioned instead the celebrated work [7] where no explicit statement can be found. We supply in Section 5 a proof taken from [24], where a more general version can be found as well as more information. 3. Main result Theorem 3.. Let T be a linear operator bounded on L 2 (w) for any w A 2. Suppose further that there is an increasing function ϕ : [, ) [, ) such that (3.) T L 2 (w) ϕ([w] A2 ). then there are constants γ n and c n independent of [w] A2 w such that (3.2) [b, T ] L 2 (w) c n ϕ(γ n [w] A2 ) [w] A2 b BMO. For the particular case ϕ(t) = c t r, where r >, and c >, a simple induction argument shows that if T L 2 (w) a [w] r A 2, then for each integer k there is a constant a k depending on k, the initial value a, and the parameters γ n and c n in the theorem, such that the kth-order commutator Tb k defined recursively by Tb k k := [b, T ], obeys the following weighted estimates b T k b L 2 (w) a k [w] r+k A 2 b k BMO. More precisely, the sequence {a k } k obeys the following recurrence equation that can be solved easily, a k = c n a k γ r+k n = c k n a γ (k 2)(k ) kr+ 2 n. Using the method of proof of Theorem 3. we can obtain a weighted estimate for the kth-order commutator that works for general increasing function ϕ : [, ) [, ). Notice the difference in the constants with what we just argued by induction for the particular case ϕ(t) = a t r : in the corollary the constant is c k n a γn r k!, whereas in the induction argument the constant is c k n a γ (k 2)(k ) kr+ 2 n. Corollary 3.2. Let T be a bounded linear operator on L 2 (w) with w A 2 and (3.3) T L 2 (w) ϕ([w] A2 ). then there are constants γ n and c n independent of [w] A2 such that (3.4) T k b L 2 (w) c k n k! ϕ(γ n [w] A2 ) [w] k A 2 b k BMO.

7 SHARP BOUNDS FOR GENERAL COMMUTATORS 7 The constants γ n and c n that appear in Corollary 3.2 are the same that appeared in Theorem 3.. We first present the proof of Theorem 3., 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 [9], we have the following. Corollary 3.3. Let T be a linear operator bounded on L 2 (w) with w A 2 and (3.5) T L 2 (w) ϕ([w] A2 ). Then, for < p <, there are constants γ n,p, and c n,p, which only depend on p, and the dimension n, such that for all weights w A p (3.6) Tb k L p (w) ( 2 c k n,p k! ϕ γ n,p [w] max{, p } [w] k max{, p } b k BMO. A p ) In the particular case ϕ(t) = a t r the extrapolated estimate looks like (3.7) T k b L p (w) 2 a c k n k! γ r n c r+k A p p [w] (r+k) max{, p } A p b k BMO. where c p depends only on p, γ n and c n are the constants that appeared in Theorem 3.. We will show in Section 4 that for r =, ϕ(t) = a t the power ( + k) max{, } p cannot be decreased for T = H and T = R j the Hilbert and Riesz transforms, for all k and p >, therefore the theorem is optimal in terms of the rate of the dependence on [w] Ap. In [5], examples for k =, for all p >, and for T the Hilbert, Beurling and Riesz transforms, were presented. Proof of Theorem 3.. We conjugate the operator as follows: if z is any complex number we define T z (f) = e zb T (e zb f). Then, a computation gives (for instance for nice functions), [b, T ](f) = d dz T z(f) z= = T z (f) dz, ɛ > 2πi z 2 z =ɛ by the Cauchy integral theorem, see [8], []. Now, by Minkowski s inequality (3.8) [b, T ](f) L 2 (w) T 2π ɛ 2 z (f) L 2 (w) dz ɛ >. z =ɛ The key point is to find the appropriate radius ɛ. To do this we look at the inner norm T z (f) L 2 (w) T z (f) L 2 (w) = T (e zb f) L 2 (we 2Rez b ), and try to find appropriate bounds on z. To do this we use the main hypothesis, namely that T is bounded on L 2 (w) if w A 2 with T L 2 (w) ϕ([w] A2 ).

8 8 DAEWON CHUNG, CRISTINA PEREYRA AND CARLOS PEREZ. Hence we should compute [we 2Rez b ] A2 = sup ( ) ( ) we 2Rez b(x) dx w e 2Rez b(x) dx. Now, since w A 2 we use Lemma 2.3: if r = r w = + 2 n+5 [w] A2 < 2 then ( ) w r r 2 dx w, and similarly for w since r w = r w, ( w r dx ) r 2 w. Using this and Holder s inequality we have for an arbitrary ( ) ( ) w(x)e 2Rez b(x) dx w(x) e 2Rez b(x) dx ( ) ( ) w r r ( ) dx e 2Rez r b(x) r ( dx w r r dx ( ) ( ) ( ) ( 4 w dx w dx e 2Rez r b(x) r dx 4 [w] A2 [e 2Rez r b ] r A 2 Now, since b BMO we are in a position to apply Lemma 2.2, if 2Rez r α n b BMO then [e 2Rez r b ] A2 β 2 n. Hence for these z, and since < r < 2, [we 2Rez b ] A2 4 [w] A2 β 2 r n 4 [w] A2 β n. ) e 2Rez r b(x) r dx ) e 2Rez r b(x) r dx Using this estimate for these z, and observing that e zb f L 2 (we 2Rezb ) = f L 2 (w), T z (f) L 2 (w) = T (e zb f) L 2 (we 2Rez b ) ϕ([we 2Rez b ] A2 ) f L 2 (w) ϕ(4[w] A2 β n ) f L 2 (w). Choosing now the radius α n ɛ =, 2r b BMO we can continue estimating the norm in (3.8) [b, T ](f) L 2 (w) T 2π ɛ 2 z (f) L 2 (w) dz 2π ɛ 2 z =ɛ z =ɛ ϕ(4[w] A2 β n ) f L 2 (w) dz = ɛ ϕ(4[w] A 2 β n ) f L 2 (w),

9 since Finally, for this ɛ, SHARP BOUNDS FOR GENERAL COMMUTATORS 9 2Rez r 2 z r = 2ɛ r = α n b BMO. [b, T ](f) L 2 (w) C2 2n ϕ(4[w] A2 β n ) [w] A2 b BMO, because r = + 2 n+5 [w] 2 2 n [w] 2, and α n = 2 n+2. Observe that the optimal radius is essentially the inverse of [w] 2 b BMO. This proves the theorem with c n 2 2n and γ n = 4 β n. Proof of Corollary 3.2. In this case a computation gives (for instance for nice functions), see [] for example or the original paper [8], Tb k (f) = dk dz T z(f) k z= = k! T z (f) dz ɛ > 2πi zk+ by the Cauchy integral theorem. The same calculation as in the case k = gives the required estimate, with c k n 2 2nk, and the same γ n = 4β n. z =ɛ 4. Examples In this section, we show that one can not have estimates better than Theorem 3., Corollary 3.2, and Corollary 3.3. We present examples which return the same growth with respect to the A p constant 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 = Sharp example for the commutator of the Hilbert transform. Consider the Hilbert transform f(y) Hf(x) = p.v. R x y dy, and consider the BMO function b(x) = log x. We know that there is a constant c such that (4.) [b, H] L 2 (w) c [w] 2 A 2 and we show that the result is sharp. More precisely, for any increasing function t 2 φ : [, ) [, ) such that lim t φ(t) = then (4.2) sup w A 2 φ([w] 2 ) [b, T ] L 2 (w) =. In particular if φ(t) = t 2 ɛ for any ɛ >, then (4.2) must hold. For < δ <, we let w(x) = x δ and it is easy to see that [w] A2 /δ. We now consider the function f(x) = x +δ χ (,) (x)

10 DAEWON CHUNG, CRISTINA PEREYRA AND CARLOS PEREZ. and observe that f is in L 2 (w) with f L 2 (w) = / δ. To estimate the L 2 (w)-norm of [b, H]f, we claim and hence from which the sharpness (4.2) will follow. We now prove the claim: if < x <, Now, [b, H]f(x) = /x [b, H]f(x) δ 2 f(x) [b, H]f L 2 (w) δ 2 f L 2 (w) log(x) log(y) x y = x +δ /x y +δ dy = log( t ) t t +δ dt log( x y ) x y y +δ dy log( ) log( t t t +δ dt = ) /x log( t t t +δ dt + ) t t t +δ dt and since log( t ) is positive for (, ) (, ) we have for < x < t But since and the claim [b, H]f(x) > x +δ log( t ) t t +δ dt > log( t ) t +δ dt = [b, H]f(x) δ 2 f(x) log( t ) t t +δ dt. s e sδ ds = δ 2 follows. One can find this example and similar examples which show the commutators with the Beurling-Ahlfors operator and the Riesz transforms obey the quadratic growth in [5] Sharp example for the kth-order commutator with the Riesz transforms. We now consider the higher order commutators with degree k in the case of the j-th directional Riesz transform on R n : Rj,bf(x) k := p.v. R n (x j y j )(b(x) b(y)) k f(y) x y n+ dy. To demonstrate the sharpness of the estimate for the higher order commutator with the Riesz transforms, we show sharpness for < p 2. Then we can extend the sharpness for all < p < by using a duality argument, because R j = R j, so the higher order commutators of the Riesz transforms are almost self-adjoint operators. For < p 2, we consider weights w(x) = x (n δ)(p ), f(x) = x δ n χ E (x) where

11 SHARP BOUNDS FOR GENERAL COMMUTATORS E = {y y (, ) n B(, )}, b(x) = log x, and evaluate L p (w)-norm over Ω = {x B(, ) c x i < for all i =, 2,..., n}. Note that, for all y E and x Ω, x j y j x j and x y y + x. Then for x Ω, Rj,bf(x) k = (x j y j )(log x log y ) k y δ n dy E x y n+ x j y j (log( x / y )) k y δ n = dy x E x y n+ j (log( x /r)) k r δ n r n = x j drdσ (r + x ) n+ = c x j E S n / x c x j x n+ δ (log(/t)) k (t x ) δ x dt = c x j ( x + t x ) n+ ) n+ / x (log(/t)) k t δ dt. ( x x + E x n+ δ (log( x / y )) k y δ n ( y + x ) n+ / x dy (log(/t)) k t δ ( + t) n+ dt Note that the constant c = c(n) is the surface measure of E S n, depends on the dimension only. On the other hand, / x (log(/t)) k t δ dt = log x s k e δs ds = δ e δs s k = δ e δ log x (log x ) k + k δ = (log x )k δ x δ + k δ log x log x e δs s k ds. log x + k δ e δs s k ds log x e δs s k ds For x Ω, (log x ) k /δ x δ is positive, therefore after neglecting some positive terms and applying the integration by parts k times, we get / x (log(/t)) k t δ dt k δ log x k! δ k e δs s k ds log x e δs s ds k! δ k+ x δ. Combining the previous computations, we have that for x Ω, and recalling that f(x) = x δ n χ E (x), R k j,b f(x) k! c x j δ k+ ( x + ) n+. Thus, we estimate R k j,b f p L p (w) Ω ( k! c x j ) p x (n δ)(p ) dx δ k+ ( x + ) n+

12 2 DAEWON CHUNG, CRISTINA PEREYRA AND CARLOS PEREZ. ( ) p k! c x j p x (n δ)(p ) = dx δ k+ Ω ( x + ) p(n+) ( ) p k! c γ p j rp r (n δ)(p ) r n drdσ(γ) δ k+ Ω S n (r + ) ( ) p(n+) p k! cn,p = r δ( p) dr = (k!) p cp n,p δ k+ p δ p(k+). Since f p L p (w) = /δ, and [w] A p /δ p, we conclude that R k j,b f L p (w) k! c n,p (p ) /p δ (k+) /p k! [w] k+ p A p f L p (w). This shows that Theorem 3., Corollary 3.2, and Corollary 3.3 are sharp in the multidimensional case. The constant c n,p c n, > as p, therefore the estimate blows up as p, as it should since the operators are not bounded in L (w). 5. Appendix: the sharp reverse Hölder s inequality for A 2 weights In this section we give a proof of Lemma 2.3, namely if w A 2, then ( ) rw 2 (5.) w rw dx w, where r w = + Proof. Let w = = δ w 2 5+n [w] A2. w(x) +δ dx = w and δ > w(x) δ w(x)dx = λ δ w({x : w(x) > λ}) dλ λ + δ δ λ δ w({x : w(x) > λ}) dλ λ w λ δ w({x : w(x) > λ}) dλ λ = I+II. Observe that I (w ) δ+, where w = w(). To estimate II we make two claims. The first is the following observation: if we let then E = {x : w(x) 2[w] A2 w }, (5.2) E 2. Indeed, by (Cauchy-Schwartz) we have for any f ( 2 f(y) dy) w() [w] A2 f(y) 2 w(y)dy,

13 SHARP BOUNDS FOR GENERAL COMMUTATORS 3 and hence if E, setting f = χ E, ( ) 2 E w(e) [w] A2 w() and in particular, by definition of E, ( ) 2 E w(e ) [w] A2 w() [w] A 2 w w() E = 2[w] A2 2 from which the claim follows. In particular, this implies that (5.3) 2 \E = 2 {x : w(x) > 2[w] A2 w }. The second claim is the following E, (5.4) w({x : w(x) > λ}) 2 n+ λ {x : w(x) > λ 2[w] Ap w } λ > w. Indeed, since λ > w to prove this claim we consider the standard (local) Calderón- Zygmund decomposition of w at level λ. The there is a family of disjoint cubes { i } contained in satisfying λ < w i 2 n λ for each i. Now, observe that except for a null set we have {x : w(x) > λ} {x : M d w(x) > λ} = i i, where M d is the dyadic maximal operator restricted to a cube. This together with (5.3) yields 2 n λ i w({x : w(x) > λ}) i i 2 n+ λ i w( i ) {x i : w(x) > 2[w] A2 w i } 2 n+ λ {x : w(x) > λ} 2[w] A2 since w i > λ. This proves the second claim (5.4). Now, combining II = 2n+ δ δ (2[w] A2 ) +δ 2 n+ δ w λ δ w({x : w(x) > λ}) dλ λ λ δ+ {x : w(x) > λ} dλ w 2[w] A2 λ w 2[w] A2 λ δ+ {x : w(x) > λ} dλ λ

14 4 DAEWON CHUNG, CRISTINA PEREYRA AND CARLOS PEREZ. (2[w] A2 ) +δ 2 n+ δ + δ w +δ dx. Setting here δ = 2 5+n [w] A2, we obtain using that t /t 2, t II w δ+ dx 2 and finally which proves (5.). w δ+ dx 2(w ) δ+, References [] J. Alvarez, R. Bagby, D. Kurtz, C. Perez, Weighted estimates for commutators of linear operators, Studia Mat. 4 (2) (994) [2] K. Astala, T. Iwaniec, E. Saksman, Beltrami operators in the plane, Duke Mathematical Journal. 7, no., (2), [3] O. Beznosova, Linear bound for dyadic paraproduct on weighted Lebesgue space L 2 (w), J. Func. Anal (28) [4] S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc., 34 (993), no., [5] D. Chung, Sharp estimates for the commutators of the Hilbert, Riesz and Beurling transforms on weighted Lebesgue spaces, Preprint (2) available at [6] D. Cruz-Uribe, J. Martell, C. Perez, Sharp weighted estimates for classical operators Preprint 29. [7] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 5, (974), [8] R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 3 (976), [9] O. Dragi cević, L. Grafakos, C. Pereyra, and S. Petermichl Extrapolation and sharp norm estimates for classical operators on weighted Lebegue spaces Publ. Math 49 (25), no., [] O. Dragi cević, A. Volberg, Sharp estimate of the Ahlfor-Beurling operator via averaging Martingale transforms, Michigan Math J. 5 (23) [] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North Holland, Amsterdam, 985. [2] S. Hukovic, S. Treil, A. Volberg, The Bellman functions and the sharp weighted inequalities for square function, Operator Theory: Advances and Applications, the volume in memory of S. A. Vinogradov 3 Birkhauser Verlag (2) [3] F. John, L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math 4 (96) [4] J. L. Journé, Calderón Zygmund operators, pseudo differential operators and the Cauchy integral of Calderón, Lect. Notes Math. 994, Springer Verlag, (983). [5] M. Lacey, S. Petermichl, M. C. Reguera, Sharp A 2 inequality for Haar shift operators Math. Ann. to appear, Preprint (29) available at [6] M. Lacey, E. Sawyer, I. Uriarte-Tuero, A characterization of two weight norm inequalities for maximal singular integrals Preprint (29) available at [7] A. K. Lerner, An elementary approach to several results on the Hardy-Littlewood maximal operator, Proc. Amer. Math. Soc. 36 (28) no 8,

15 SHARP BOUNDS FOR GENERAL COMMUTATORS 5 [8] A. Lerner, S. Ombrosi and C. Pérez, Sharp A bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden, International Mathematics Research Notices, 28, no. 6, Art. ID rnm6, pp. 42B2. [9] A. Lerner, C. Pérez and S. Ombrosi, A bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden, Math. Res. Lett. 6 (29), [2] C. Ortiz. uadratic A bounds for commutators of singular integrals with BMO functions, Preprint 2 [2] C. Pérez, Endpoint Estimates for Commutators of Singular Integral Operators, Journal of Functional Analysis, () 28 (995), [22] C. Pérez, Sharp estimates for commutators of singular integrals via iterations of the Hardy- Littlewood maximal function, J. Fourier Anal. Appl. 3 (997), [23] C. Pérez The growth of the A p constant on classical estimates, Revista de la Unión Matemática Argentina, 5, 2, 29, [24] C. Pérez A revised version of the A p theory of weights and applications, Lecture Notes of a course given at the Centre de Recerca Matematica, Barcelona, Spain, In preparation 29. [25] S. Petermichl, The sharp bound for the Hilbert trnasform on weighted Lebesgue spaces in terms of the classical A p characteristic, Amer. J. of Math. 29 (27) [26] S. Petermichl, The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc. 36 (28) [27] S. Petermichl, A. Volberg, Heating of the Ahlfors-Beurling operator: Weakly quasiregular maps on the plane are quasiregular, Duke Math J. 2 (22) [28] A. Torchinsky, Real Variable Methods in Harmonic in Analysis, Academic Press, New York, (986). [29] J. Wittwer, A sharp estimates on the norm of Martingale transform, Math. Res. Lett. 7 (2)- 2 Daewon Chung, Department of Mathematics and Statistics MSC3 25, University of New Mexico, Albuquerque, NM address: midiking@math.unm.edu Cristina Pereyra, Department of Mathematics and Statistics, MSC3 25, University of New Mexico, Albuquerque, NM address: crisp@math.unm.edu Carlos Pérez, Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad De Sevilla, 48 Sevilla, Spain. address: carlosperez@us.es