Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture
Abstract
We prove two-weight norm inequalities for Calderón-Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.
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Advances in Mathematics, 216 (2007) 647-676 SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS, WITH APPLICATIONS TO THE HILBERT TRANSFORM AND THE SARASON CONJECTURE D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´ EREZ Abstract. We prove two-weight norm inequalities for Calder´on-Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces 1. Introduction 1.1. Background. A long-standing problem in harmonic analysis has been to characterize the weights governing strong-type norm inequalities for classical operators. To be precise: given an operator Tand p, 1 < p < ∞, determine sufficient conditions on a pair of weights (i.e., non-negative, measurable functions) (u, v) such that for all f∈Lp(vp), (1.1) ZRn |u(x)Tf(x)|pdx ≤CZRn |v(x)f(x)|pdx. This problem was originally posed in the early 1970’s for the Hardy-Littlewood maximal operator and for the Hilbert transform on the real line, but it was soon expanded to include a variety of operators—singular integrals, fractional integrals, and square functions—on Rn. While a great deal of progress has been made, many questions remain open even for the Hilbert transform. For many of these problems, inequality (1.1) is usually stated in an equivalent form: (1.2) ZRn |Tf(x)|pU(x)dx ≤CZRn |f(x)|pV(x)dx, where U=upand V=vp. But for our purposes (1.1) is a more suitable form as it makes the statement of our main results more elegant. 2000 Mathematics Subject Classification. 42B20,42B25,47B35. Key words and phrases. Weights, Hilbert transform, singular integral operators, Sarason conjecture, Toeplitz operator, maximal functions, commutators. The first author is partially supported by the Stewart faculty development fund of Trinity College; he would also like to thank his advisor, Donald Sarason, for posing the problem discussed in Section 1.4. The second author is partially supported by Spanish Ministerio de Educaci´on y Ciencia “Programa Ram´on y Cajal, 2005” and by Grant MTM2004-00678 from the same institution; the third author is partially supported by Grant MTM2006-05622 from the same institution. 1
2 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´ EREZ The purpose of this paper is to give new two-weight norm inequalities for singular integrals and other operators that are sharp for the Hilbert and Riesz transforms. To put our results into context, we will sketch the outlines of some earlier work. For more information on the history of this problem, we refer the reader to Muckenhoupt [28], Dynkin and Osilenker [15], Garc´ıa-Cuerva and Rubio de Francia [18], and Duoandikoetxea [13]. The earliest weighted norm inequalities were for the one-weight problem (i.e., when u=v). Muckenhoupt [27], and Hunt, Muckenhoupt and Wheeden [20] showed that for the maximal operator and for the Hilbert transform on the real line, (1.1) held if and only if upsatisfied the so-called Apcondition: there exists a finite constant C such that for all intervals Q, (1.3) 1 |Q|ZQ u(x)pdx1 p1 |Q|ZQ u(x)−p0dx1 p0 ≤C. The proof was simplified by Coifman and Fefferman [3] and extended to Calder´onZygmund singular integrals on Rn(with intervals replaced by cubes in (1.3)). It was immediately conjectured that in the two-weight case, the corresponding two-weight Apcondition, (1.4) 1 |Q|ZQ u(x)pdx1 p1 |Q|ZQ v(x)−p0dx1 p0 ≤C < ∞, was necessary and sufficient for these operators to be bounded from Lp(vp) to Lp(up). However, while this condition is necessary for the maximal operator and for the Hilbert transform, it is not sufficient: see Muckenhoput and Wheeden [30]. Sawyer [44] gave a necessary and sufficient condition for the maximal operator which involves the operator itself. Cotlar and Sadosky [4, 5] gave a necessary and sufficient condition for the Hilbert transform which is reminiscent of the Helson-Szeg¨o theorem and is grounded in operator theory. However, their condition is difficult to check and does not readily extend to higher dimensions and general singular integrals. Following these results, a great deal of effort was devoted to finding stronger conditions related to the more geometric two-weight Apcondition and that are sufficient for (1.1) to hold for a variety of operators, especially singular integrals. In passing, we note the work of Muckenhoupt and Wheeden [30], Fujii [17], Katz and Pereyra [22], Leckband [25], Rakotondratsimba [39, 40], Wilson [52], and P´erez [34]. An important result in this direction is due to Neugebauer [32]: he showed that if the pair of weights (u, v) is such that for some r > 1 the pair (ur, vr) satisfies (1.4), then (1.1) holds for singular integrals. He did not prove this directly; rather, by applying the ideas on factorization of weights due to Rubio de Francia, he showed that there exists w∈Apsuch that c1u≤w≤c2vif and only if (ur, vr)∈Apfor some r > 1. Two-weight inequalities for singular integrals and other operators then follow immediately from the one-weight case.
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 3 We can restate Neugebauer’s result as follows. Given a cube Q, write kukp,Q =1 |Q|ZQ |u(x)|pdx1/p for the normalized Lpnorm on Q. The Apcondition is then equivalent to kukp,Qkv−1kp0,Q ≤C < ∞, and the condition that (ur, vr)∈Apcan be rewritten as kukrp,Qkv−1krp0,Q ≤C < ∞. In other words, if we replace the normalized Lpand Lp0norms in the Apcondition by larger norms (in the scale of Lebesgue spaces), then we get a sufficient condition for (1.1) to hold for singular integrals and other operators. We refer to these larger norms as “power bumps.” P´erez [35, 36] first considered the question of whether power bumps could be replaced by other function space norms larger than the Lpnorm but smaller than the Lrp norm. He showed that for the maximal operator and fractional integrals certain norms in the scale of Orlicz spaces, the so-called “Orlicz bumps”, are sufficient. To state his results we need several definitions. Given a Young function B: [0,∞)→[0,∞), and a cube Q, define the normalized Luxemburg norm on Qby kukB,Q = inf λ > 0 : 1 |Q|ZQ B|u(x)| λdx ≤1. If B(t) = tp, then kukB,Q =kukp,Q and the Luxemburg norm reduces to the Lpnorm. When B(t) = tplog(e+t)awe get the norm on the Zygmund spaces Lp(log L)a. When used to define an Aptype condition, this norm is referred to as a “log bump.” Given a Young function B, let ¯ Bdenote its associate function: the Young function with the property that t≤B−1(t)¯ B−1(t)≤2t,t > 0. If B(t) = tp, then ¯ B(t) = tp0; if B(t) = tplog(e+t)a, then ¯ B(t)≈tp0log(e+t)−ap0/p. The following growth condition on Young functions plays an important role in determining suitable Orlicz bumps for generalizing the Apcondition. Definition 1.1. Given p,1< p < ∞, a Young function Bsatisfies the Bpcondition if for some c > 0, (1.5) Z∞ c B(t) tp dt t<∞. If B(t) = tq, 1 < q < p, then it is immediate that B∈Bp. More interesting examples are given by the functions B(t) = tp log(e+t)1+δ, δ > 0, B(t) = tp log(e+t) log log(ee+t)1+δ, δ > 0.
4 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´ EREZ The Bpcondition was introduced in [36] where it was used in to state and prove sharp two-weight norm inequalities for the Hardy-Littlewood maximal function. If Bis a Young function such that ¯ B∈Bp, and the pair of weights (u, v) is such that for every cube Q, (1.6) kukp,Qkv−1kB,Q ≤C < ∞, then (1.1) holds for the Hardy-Littlewood maximal function. Furthermore, the Bp condition is necessary: if (1.1) holds and (u, v) satisfy (1.6), then ¯ B∈Bp. Note that unlike in the original result by Neugebauer, there is no bump on the weight u. Via a discretization argument, the same techniques were applied in [34] to prove weighted norm inequalities for the fractional integral operators Iα, 0 < α < n. Let Aand Bbe Young functions such that ¯ A∈Bp0and ¯ B∈Bp. If (u, v) is a pair of weights such that (1.7) `(Q)αkukA,Qkv−1kB,Q ≤C < ∞, then ZRn |u(x)Iαf(x)|pdx ≤CZRn |v(x)f(x)|pdx. The condition (1.7) can be viewed as a two-weight version of the Chang-Wilson-Wolff condition [2] for Schr¨odinger operators which is an improvement of the well-known Fefferman-Phong condition [16]. This result for fractional integrals immediately suggested the following conjecture: Conjecture. If Aand Bare Young functions such that ¯ A∈Bp0and ¯ B∈Bp, and if the pair of weights (u, v)is such that for every cube Q, (1.8) kukA,Qkv−1kB,Q ≤C < ∞, then (1.1) holds for Calder´on-Zygmund singular integrals. An important special case of this conjecture is when Aand Bare log bumps: A(t) = tplog(e+t)p−1+δ, B(t) = tp0log(e+t)p0−1+δ, δ > 0. Our conjecture is closely connected to an old conjecture of Muckenhoupt and Wheeden [29]: if the pair (u, v) is such that the maximal operator Msatisfies (1.9) M:Lp(vp)→Lp(up) and M:Lp0(u−p0)→Lp0(v−p0), then the Hilbert transform is bounded from Lp(vp) to Lp(up). By the results in [36] described above, (1.8) is sufficient for both inequalities in (1.9) to hold, so our conjecture is a special case of theirs. Our conjecture is known to be true in a number of special cases. When Aand Bare power bumps—i.e., A(t) = trp and B(t) = trp0,r > 1—then our conjecture reduces to the theorem of Neugebauer stated above. His result was improved in [11], where it was shown that it is sufficient to take Aa power bump and Bsuch that ¯ B∈Bp. In [7] it was shown that if Ais a large Orlicz bump, e.g., if A(t)≈tpexp[log(e+tp)r],0< r < 1,
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 5 then the conjecture is true. However, it was also shown in this paper that such functions represent the best that can be gotten using the techniques in [11]; they cannot be used to prove the full conjecture or even the case when Ais a log bump. A related but weaker version of our conjecture was proved by Treil, Volberg and Zheng [48] for the periodic Hilbert transform (i.e., the conjugate function) on the unit circle. For z∈D, let φzbe the M¨obius transform in the closed unit disk, φz(w) = z−w 1−¯zw, w ∈¯ D. If Aand Bare Young functions such that ¯ A∈Bp0and ¯ B∈Bp, and if (u, v) is a pair of weights such that (1.10) sup z∈D ku◦φzkA,∂Dkv−1◦φzkB,∂D<∞, then the periodic Hilbert transform is bounded from Lp(vp, ∂D) to Lp(up, ∂D). Another result closely related to our conjecture was proved in [9]. There it was shown that if Ais the log bump A(t) = tplog(e+t)p−1+δand if the pair of weights (u, v) is such that for every cube Q, (1.11) kukA,Qkv−1kp0,Q ≤C < ∞, then Calder´on-Zygmund singular integrals satisfy the weak (p, p) inequality (1.12) up({x∈Rn:|Tf(x)|> t})≤C tpZRn |v(x)f(x)|pdx. Note that condition (1.11) is a special case of (1.6), and it is natural to conjecture that (1.12) holds if Asuch that ¯ A∈Bp0. This is a special case of another conjecture due to Muckenhoupt and Wheeden [29]: if the maximal operator satisfies M:Lp0(u−p0)→Lp0(v−p0), then the Hilbert transform satisfies (1.12). 1.2. Results for singular integrals. Our main results improve all previous work by allowing us to take Ato be a log bump. Our first theorem is a sharp inequality for the Hilbert transform. Theorem 1.2. Given p,1< p < ∞, suppose the pair of weights (u, v)satisfies (1.13) kukA,Qkv−1kB,Q ≤C < ∞, where A(t) = tplog(e+t)p−1+δ,δ > 0, and ¯ B∈Bp. Then (1.14) ZR |u(x)Hf(x)|pdx ≤CZR |v(x)f(x)|pdx. Further, this inequality is sharp since it does not hold in general if we take δ= 0 in the definition of A. A counter-example showing that (1.14) need not hold if δ= 0 when p= 2 is given in [9]. The example there is a pair of weights for which (1.2) does not hold: (U, MΦU), where Φ(t) = tlog(e+t), and MΦis the Orlicz maximal operator (1.15) MΦf(x) = sup Q3x kfkΦ,Q.
6 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´ EREZ (See Lemma 2.8 below.) By a change of variables in the definition of the Luxemburg norm it is easy to see that the pair of weights u=U1/2, v = (MΦU)1/2satisfies (1.13) with A(t) = t2log(e+t). Theorem 1.2 is a special case of a more general result which holds on Rn, provided p>n. Recall that a Calder´on-Zygmund singular integral Tis a singular convolution operator, Tf(x) = p.v. ZRn K(x−y)f(y)dy, where the kernel Kis continuously differentiable on Rn\ {0}, has zero average on the unit sphere, and for all x6= 0, |K(x)| ≤ C |x|nand |∇K(x)| ≤ C |x|n+1 . More generally, we may assume that Tis a Calder´on-Zygmund operator. For a precise definition see Duoandikoetxea [13]. Theorem 1.3. Let Tbe a Calder´on-Zygmund singular integral. Fix p,n<p<∞. Suppose (u, v)is a pair of weights such that for all cubes Q, (1.16) kukA,Qkv−1kB,Q ≤C < ∞, where A(t) = tplog(e+t)p−1+δ,δ > 0, and ¯ B∈Bp. Then Tsatisfies the strong (p, p) inequality (1.17) ZRn |u(x)Tf(x)|pdx ≤CZRn |v(x)f(x)|pdx. Further, this result is sharp in the sense that there exists a family of pairs of weights (u, v)such that (1.16) holds with δ= 0, but (1.17) does not hold for all of the Riesz transforms. The sharpness of Theorem 1.3 comes from a necessary condition proved in [34]. Translated to our setting (the results there are stated in terms of inequality (1.2)), it shows that if the pairs of weights (u, MAu) (which clearly satisfy (1.16)) are such that (1.17) holds for all nof the Riesz transforms, then δ > 0. By contraposition, if δ= 0 then (1.17) must fail for at least one of the Riesz transforms. The restriction that p > n in Theorem 1.3 seems unnatural, but despite repeated efforts we cannot eliminate it. If n≥2, then by duality we have that (1.17) holds for 1<p<n0or p > n if Aand Bare both log bumps: A(t) = tplog(e+t)p−1+δand B(t) = tp0log(e+t)p0−1+δ,δ > 0. However, this still leaves the gap n0≤p≤n. Our next result shows that we can fill this gap if we replace Aby a larger log bump. Theorem 1.4. Let Tbe a Calder´on-Zygmund singular integral. Given p,1< p < ∞, suppose (u, v)is a pair of weights such that for all cubes Q, (1.18) kukA,Qkv−1kB,Q ≤C < ∞,
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 7 where A(t) = tplog(e+t)2p−1+δ,δ > 0, and ¯ B∈Bp. Then Tsatisfies the strong (p, p)inequality (1.19) ZRn |u(x)Tf(x)|pdx ≤CZRn |v(x)f(x)|pdx. The proofs of both Theorems 1.3 and 1.4 involve careful discretization arguments using the properties of Calder´on-Zygmund cubes. They are very similar in spirit, though not in detail, to the discretization argument used to prove two-weight norm inequalities for fractional integrals in [35]. The problem with this approach is that there does not exist as good a technique for discretizing singular integrals as exists for fractional integrals. Consequently, we need to argue more obliquely using the sharp maximal operator (explicitly in the proof of Theorem 1.4 and in essence in the proof of Theorem 1.3). This leads directly to the technical obstacles which prevent us from proving the full conjecture we described above. In particular, in both proofs we use the following property of log bumps: given A(t) = tplog(e+t)p−1+δ,δ > 0, then ¯ A∈Bp0and there exists q, 0 < q < 1, such that if C(t) = A(t1/q), then ¯ C∈B(p/q)0. This property does not hold for arbitrary Young functions: a counter-example is given by A(t) = tplog(e+t)p−1log log(ee+t)p−1+δ. Details are left to the reader. Key to the proof of Theorem 1.4 is the pointwise inequality [1]: (1.20) M# q(Tf)(x) = M#(|Tf|q)(x)1/q ≤CMf(x), for some 0 < q < 1, where M#is the sharp maximal operator of Fefferman-Stein. Vector-valued singular integrals satisfy essentially the same inequality [38]: if 0 < q < 1 and 1 < r < ∞there exists a constant such that M# qk{Tfj}jk`r(x)≤C Mk{fj}jk`r(x). Therefore, as a corollary to the proof of Theorem 1.4 we get two-weight estimates for vector-valued singular integrals. On the other hand, it is not difficult to observe that the proof of Theorem 1.3 can be carried out for vector-valued singular integrals and thus we get better conditions on (u, v) in the range n < p < ∞. Details are left to the reader. Corollary 1.5. Let Tbe a Calder´on-Zygmund singular integral. Given p,rwith 1< p, r < ∞, suppose (u, v)satisfy (1.18). Then X j |uTfj|r1 r Lp(Rn)≤C X j |vfj|r1 r Lp(Rn). Moreover, the same estimate holds if 1< r < ∞,p>nand (u, v)satisfy (1.16). Remark 1.6.Other operators, including some pseudo-differential operators and square functions, satisfy inequality (1.20), and so similar weighted norm inequalities hold for them. For examples see [1] and [11].
8 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´ EREZ 1.3. Results for other operators. The proofs of Theorems 1.3 and 1.4 can be adapted to give results for other operators. Here we consider two: the dyadic square function and commutators of singular integrals. Dyadic square functions. We first consider the dyadic square function. Let ∆ denote the set of dyadic cubes in Rn, and for each m∈Z, let ∆m={Q∈∆ : `(Q) = 2m}. For each Q∈∆, let b Qdenote the dyadic parent of Q: if Q∈∆m, the unique cube b Q∈∆m+1 such that Q⊂b Q. Given a function f, let fQ=|Q|−1RQf(x)dx. For each f, the dyadic square function, Sdf, is defined by Sdf(x) = X Q∈∆ |fQ−fb Q|2χQ(x)1/2 . Theorem 1.7. Given p,1< p < ∞, suppose (u, v)is a pair of weights such that for all dyadic cubes Q, (1.21) kukA,Qkv−1kB,Q ≤C < ∞, where A(t) = tplog(e+t)p−1+δ,δ > 0, and ¯ B∈Bp. Then the dyadic square function satisfies the strong (p, p)inequality (1.22) ZRnu(x)Sdf(x)pdx ≤CZRn |v(x)f(x)|pdx. The proof of Theorem 1.7 is nearly identical to that of Theorem 1.3; the difference is that the square function is sufficiently localized that we can eliminate the restriction on p. Given the close connection between square functions and singular integrals, we take this result as evidence that the restriction on pin Theorem 1.3 is not necessary. Theorem 1.7 is related to two-weight norm inequalities for the dyadic square function due to Uchiyama [49] and Cruz-Uribe and P´erez [10]. They showed that for any weight u,ZRn Sdf(x)pu(x)dx ≤CZRn |f(x)|pMu(x)dx, 1< p ≤2, ZRn Sdf(x)pu(x)dx ≤CZRn |f(x)|pMCu(x)dx, 2< p < ∞, where C(t) = tlog(e+t)p/2−1+δ,δ > 0, and MCis the Orlicz maximal operator (1.15). Similar but weaker inequalities follow from Theorem 1.7: it is straightforward to see that weights of the form (u1/p,(MDu)1/p), where D(t) = tlog(e+t)p−1+δ, δ > 0, satisfy (1.21). On the other hand, one can also find pairs of weights which satisfy (1.21) which cannot be written in this form. It is tempting to speculate that Theorem 1.7 can be improved to include all of these results as special cases. 1.3.1. Commutators. The second class of operators we consider are commutators of singular integrals. Given a Calder´on-Zygmund singular integral Tand b∈BMO, define the first order commutator, [b, T], by [b, T]f(x) = b(x)Tf(x)−T(bf)(x).
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 9 These operators are more singular than the associated singular integrals, and so a larger log bump is required on both weights. Theorem 1.8. Let Tbe a Calder´on-Zygmund singular integral and let b∈BMO. Given p,1< p < ∞, suppose that for all cubes Qthe pair of weights (u, v)satisfies (1.23) kukA,Qkv−1kB,Q ≤C < ∞, where A(t) = tplog(e+t)3p−1+δ, and B(t) = tp0log(e+t)2p0−1+δ,δ > 0. Then (1.24) ZRnu(x)[b, T]f(x)pdx ≤CZRn |v(x)f(x)|pdx. Theorem 1.8 improves a result in [11], where the same inequality was proved assuming that Ais a power bump: A(t) = trp,r > 1. Remark 1.9.An analogous result holds for higher order commutators Tk b, with k≥2. (For k= 1, T1 b= [b, T].) These are defined inductively by Tk b= [b, Tk−1 b]. In this case the condition imposed on the pair of weights (u, v) is (1.23) with A(t) = tplog(e+ t)(k+2) p−1+δand B(t) = tp0log(e+t)(k+1) p0−1+δ,δ > 0. The proof is essentially the same and some details are given in Remark 5.7 below. Remark 1.10.We conjecture that Theorem 1.8 can be improved by taking A(t) = tplog(e+t)2p−1+δ—the commutator should require one more log term on each weight than the associated singular integral. 1.4. Application to the Sarason conjecture. Theorem 1.2 has an application to an open problem in operator theory on the unit disk. This problem was first posed by Sarason (see Khavin and Nikol’ski˘ı [23]) and is referred to as the Sarason conjecture. To state it we recall some basic facts about operator theory on the unit circle. (For complete information, see Koosis [24].) Given a function f∈L1(∂D), we define the periodic Hilbert transform of f, also known as the conjugate function of f, by ˜ f(eiθ) = ˜ Hf(eiθ) = 1 πZπ 0 f(ei(θ−t))−f(ei(θ+t)) 2 tan(t/2) dt. The periodic Hilbert transform is a Calder´on-Zygmund singular integral and so is a bounded operator on L2(∂D). Define the Riesz projection operator Pby Pf(eiθ) = f(eiθ) + ˜ Hf(eiθ) + ˆ f(0) 2. Then Pis also bounded on L2(∂D), and in fact is the orthogonal projection from L2(∂D) to the Hardy space H2(∂D), the closure of the analytic polynomials in L2(∂D). Given a function h∈L2(∂D), define the Toeplitz operator with symbol hby Thf(eiθ) = P(hf)(eiθ). The Toeplitz operator This densely defined on H2(∂D) and is a bounded operator on H2(∂D) if and only if h∈L∞(∂D). Toeplitz operators have been intensively studied and appear in many problems in operator theory.
16 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´ EREZ Let w=uqh; then w∈L∞ csince h∈L∞ c(Rn) and u∈L∞(Rn). We will now form a kind of atomic decomposition of wthat is due to Lerner [26] and lies at the heart of his proof of Lemma 2.6. Fix a > 2nand m > 0 such that kwkL∞≤am. For each k≤m, let {Qk j}be the Calder´on-Zygmund cubes of wat height ak(Lemma 2.2). Let wQk j=|Qk j|−1RQk jw(x)dx, and for each kdefine the functions bk(x) = X j (w(x)−wQk j)χQk j(x), gk(x) = w(x)−bk(x) = (wQk jx∈Qk j w(x)x∈Rn\Ωak. Again by Lemma 2.2, for all kwe have gk(x)≤2nakand kgkk1=kwk1. Since the set Ωamis empty, bm= 0. Therefore, for every integer l < 0, we have the telescoping sequence w(x) = m−1 X k=lbk(x)−bk+1(x)+gl(x). By (2.1), wQk j≤2nak. Since for each jand k, (3.1) (bk(x)−bk+1(x))χQk j(x) = (w(x)−wQk j)χQk j(x)−X Qk+1 i⊂Qk j (w(x)−wQk+1 i)χQk+1 i(x), it follows immediately that for all x, (3.2) |bk(x)−bk+1(x)| ≤ (1 + a) 2nak. Further, by integrating (3.1) we see that (3.3) ZQk jbk(x)−bk+1(x)dx = 0. We can now estimate as follows: for any l < 0, (3.4) ZRn |Tf(x)|qu(x)qh(x)dx =ZRn |Tf(x)|qw(x)dx = m−1 X k=lZRn |Tf(x)|q(bk(x)−bk+1(x)) dx +ZRn |Tf(x)|qgl(x)dx. We now claim that the last term on the righthand side tends to 0 as l→ −∞. This follows at once from H¨older’s inequality, the fact that Tis bounded on L2(Rn), and that fand ware bounded functions with compact support: 0≤ZRn |Tf(x)|qgl(x)dx ≤ZRn |Tf(x)|2dxq/2ZRn gl(x)(2/q)0dx1/(2/q)0 ≤Ckfkq/2 2(2nal)(q/2)kglk1/(2/q)0 1=Ckfkq/2 2(2nal)(q/2)kwk1/(2/q)0 1.
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 17 As l→ −∞ the last term tends to zero. Therefore, taking the limit in (3.4) we get (3.5) ZRn |Tf(x)|qu(x)qh(x)dx = m−1 X k=−∞ ZRn |Tf(x)|q(bk(x)−bk+1(x)) dx. We estimate the righthand side of (3.5) as follows. For each j, k, let ck jbe a constant whose value will be specified below. Since q < 1, ||a|q− |b|q|≤|a−b|q. Therefore, by (3.3) and (3.2), m−1 X k=−∞ ZRn |Tf(x)|qbk(x)−bk+1(x)dx =X k,j ZQk j |Tf(x)|qbk(x)−bk+1(x)dx =X k,j ZQk j|Tf(x)|q− |ck j|qbk(x)−bk+1(x)dx ≤CX k,j (1 + a) 2nakZQk j|Tf(x)|q− |ck j|qdx ≤CX k,j wQk jZQk j |Tf(x)−ck j|qdx ≤CX k,j wQk jZQk j |T(fχ2Qk j)(x)|qdx +CX k,j wQk jZQk j |T(fχRn\2Qk j)(x)−ck j|qdx =C(I1+I2). We consider each term separately. To estimate I1we use Kolmogorov’s inequality (since q < 1) and Lemmas 2.3 and 2.7: I1≤CX k,j 1 |2Qk j|Z2Qk j w(x)dx1 |2Qk j|Z2Qk j |f(x)|dxq|Qk j| =CX k,j 1 |2Qk j|Z2Qk j u(x)qh(x)dx1 |2Qk j|Z2Qk j v(x)|f(x)|v(x)−1dxq|Qk j| ≤CX k,j kuqkC,2Qk jkhk¯ C,2Qk jkv fkq ¯ B,2Qk j kv−1kq B,2Qk j |e Qk j|, where C(t) = trlog(e+t)r−1+. Let Cq(t) = C(tq) = tplog(e+tq)r−1+≈A(t). Therefore, by a change of variables in the definition of the Orlicz norm, kuqkC,2Qk j=kukq Cq,2Qk j ≈ kukq A,2Qk j . Hence, (1.16), the fact that the sets e Qk jare disjoint, and H¨older’s inequality yield I1≤CX k,j khk¯ C,2Qk jkv fkq ¯ B,2Qk j |e Qk j| ≤CX k,j Ze Qk j M¯ Ch(x)M¯ B(v f)(x)qdx ≤CZRn M¯ Ch(x)r0dx1/r0ZRn M¯ B(v f)(x)pdxq/p
18 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´ EREZ ≤CZRn h(x)r0dx1/r0ZRn |v(x)f(x)|pdxq/p =CZRn |v(x)f(x)|pdxq/p. The last inequality holds since ¯ C∈Br0and ¯ B∈Bp, and so by Lemma 2.8 M¯ Cis bounded on Lr0and M¯ Bis bounded on Lp. This completes the estimate of I1. To estimate I2we choose the value of the constant ck jto be ck j=1 |Qk j|ZQk j T(fχRn\2Qk j)(y)dy. Let C(t) be as in the estimate of I1. Then, by a standard estimate for Calder´onZygmund singular integrals (see [13, 18]), since q < 1 and by Lemmas 2.3 and 2.7, we obtain I2≤CX k,j 1 |Qk j|ZQk j u(x)qh(x)dx∞ X i=1 2−i1 |2iQk j|Z2iQk j |f(x)|dxq|Qk j| ≤CX k,j 1 |Qk j|ZQk j u(x)qh(x)dx ∞ X i=1 2−iq 1 |2iQk j|Z2iQk j v(x)f(x)v(x)−1dxq|Qk j| ≤CX k,j kuqkC,Qk jkhk¯ C,Qk j|e Qk j| ∞ X i=1 2−iq kv fkq ¯ B,2iQk j kv−1kq B,2iQk j ≤CX k,j kukq A,Qk j khk¯ C,Qk j|e Qk j| ∞ X i=1 2−iq kv fkq ¯ B,2iQk j kv−1kq B,2iQk j . For 0 < β < 1, A(βt)≤βpA(t), so by the definition of the Luxemburg norm, we have that kukA,Qk j≤C2i n/p kukA,2iQk j. Thus, by (1.16) and since p>nit follows I2≤CX k,j khk¯ C,Qk j|e Qk j| ∞ X i=1 2−iq 2inq/p kukq A,2iQk j kv fkq ¯ B,2iQk j kv−1kq B,2iQk j ≤CX k,j khk¯ C,Qk j|e Qk j|inf x∈Qk j M¯ B(v f)(x)q ≤CX k,j Ze Qk j M¯ Ch(x)M¯ B(v f)(x)qdx. We can now argue as we did above for I1to obtain the desired estimate for I2. 3.2. Proof of Theorem 1.7. The proof is almost identical to the one just given and we only indicate the minor changes. We proceed in the same manner with Sd in place of T. We observe that Sdis bounded on L2(Rn) and so it suffices to get the appropriate estimates for I1and I2, where now in I1we write fχQk jin place of fχ2Qk jand in I2we put fχRn\Qk jin place of fχRn\2Qk j. The estimate for I1adapts
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 19 immediately to the dyadic square function since Sdis of weak-type (1,1) and thus satisfies Kolmogorov’s inequality. Since the dyadic square function is more localized than a singular integral, the estimate for I2is much easier. Fix a cube Qk jand set ck j=X Q∈∆ Q⊇Qk j(fχRn\Qk j)Q−(fχRn\Qk j)b Q21/2. Then for any x∈Qk jwe have that Sd(fχRn\Qk j)(x)≡ck j; thus I2= 0 and we are done. 4. Proof of Theorem 1.4 At the heart of the proof of Theorem 1.4 is the following lemma, whose proof we defer for the moment. Lemma 4.1. Given pand (u, v)as in the hypotheses of Theorem 1.4, there exists q, 0< q < 1, such that for all f, h ∈L∞ c(Rn), (4.1) ZRn Mf(x)qM(uqh)(x)dx ≤CZRn |v(x)f(x)|pdxq/p ZRn |h(x)|(p/q)0dx1/(p/q)0 . Proof of Theorem 1.4.Fix qas in Lemma 4.1 and let r=p q>1. Then by duality, ZRn |u(x)Tf(x)|pdxq/p = sup ZRn |Tf(x)|qu(x)qh(x)dx, where the supremum is taken over all non-negative functions h∈L∞ c(Rn) such that khkLr0(Rn)= 1. Fix such a function h. Then by Lemmas 2.6 and 4.1, ZRn |Tf(x)|qu(x)ph(x)dx ≤CZRn Mf(x)qM(uqh)(x)dx ≤CZRn |v(x)f(x)|pdxq/p ZRn |h(x)|r0dx1/r0 =CZRn |v(x)f(x)|pdxq/p. This completes the proof of Theorem 1.4. 4.1. Proof of Lemma 4.1. Fix f; by a standard argument we may assume without loss of generality that f≥0. Further, as we noted above, we may assume without loss of generality that f∈L∞ cand u, v ∈L∞. Fix q, 0 < q < 1, sufficiently close to 1 that there exists > 0 such that 2p−1 + δ= 2(p/q)−1 + . Let r=p/q,w=uqh and a= 4n>2n. For each j, k let Ωkj ={ak−j−1< Mw(x)≤ak−j+1}∩{aj< Mf(x)q≤aj+1}; then ZRn Mf(x)qMw(x)dx =X k,j Z{ak<(Mf)qMw≤ak+1}∩{aj<(Mf)q≤aj+1} Mf(x)qMw(x)dx
20 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´ EREZ ≤X k,j ZΩkj Mf(x)qMw(x)dx. For each integer l, m, let {Rr l}rbe the CZ cubes of wat height al, and let {Ss m}s be the CZ cubes of fat height am/q. Then by Lemma 2.4, for each pair (k, j), {x:Mw(x)> ak−j−1} ⊂ [ r 3Rr k−j−2,{x:Mf(x)q> aj} ⊂ [ s 3Ss j−1. If x∈Ωkj, there exists at least one pair (r, s) such that x∈3Rr k−j−2∩3Ss j−1. Let Ers kj ={x∈Ωkj :x∈3Rr k−j−2∩3Ss j−1}. If the set Ers kj is not empty, then 3Rr k−j−2∩3Ss j−16=∅. Therefore, depending on their relative sizes, we either have 3Rr k−j−2⊂9Ss j−1, or 3Ss j−1⊂9Rr k−j−2. If the first inclusion holds we say that (k, j, r, s)⊂Γ1; if the second holds we say that (k, j, r, s)∈Γ2. Hence, ZRn Mf(x)qMw(x)dx ≤X k,j X r,s ZErs kj Mf(x)qMw(x)dx ≤X k,j X r,s ak−j+1aj+1|Ers kj| ≤X (k,j,r,s)∈Γ1 ak−j+1aj+1|Ers kj|+X (k,j,r,s)∈Γ2 ak−j+1aj+1|Ers kj|=I1+II2. To complete the proof we will estimate each term separately. We consider first I1. Since Ers kj ⊂3Rr k−j−2, by Lemma 2.3, |Ers kj| ≤ 3n|Rr k−j−2| ≤ C|e Rr k−j−2|. On the other hand 3Rr k−j−2⊂9Ss j−1. Thus by Lemma 2.2, I1≤a5X (k,j,r,s)∈Γ11 |Rr k−j−2|ZRr k−j−2 w(x)dx 1 |Ss j−1|ZSs j−1 f(x)dxq |Ers kj| ≤CX j,s X k,r: (k,j,r,s)∈Γ1 1 |Rr k−j−2|ZRr k−j−2 w(x)dx · | e Rr k−j−2| 1 |9Ss j−1|Z9Ss j−1 f(x)dxq ≤CX j,s X k,r: (k,j,r,s)∈Γ1 Ze Rr k−j−2 M(wχ9Ss j−1)(x)dx 1 |9Ss j−1|Z9Ss j−1 f(x)dxq . Since the sets e Rr k−j−2are disjoint and contained in 9Ss j−1, we can apply Yano’s theorem (see Zygmund [56]) to get I1≤CX j,s 1 |9Ss j−1|Z9Ss j−1 M(wχ9Ss j−1)(x)dx 1 |9Ss j−1|Z9Ss j−1 f(x)dx!q |e Ss j−1| ≤CX j,s kwkΦ,9Ss j−1 1 |9Ss j−1|Z9Ss j−1 f(x)dx!q |e Ss j−1|, where Φ(t) = tlog(e+t) and the constant depends only on nand not on the cube Ss j−1. Recall that 2p−1+δ= 2r−1+; hence, if we define D(t) = trlog(e+t)2r−1+,
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 21 then D(tq)≈A(t). Now define e D(t) = tr0log(e+t)−1−(r0−1) ∈Br0. Then we have that Φ−1(t)≈t log(e+t)=t1 r log(e+t)2r−1+ r ×t1 r0log(e+t)−1+ 2r−1+ r≈D−1(t)·e D−1(t). Therefore, recalling that w=uqh, we can apply Lemma 2.7 and (1.18) to get I1≤CX j,s kuqkD,9Ss j−1khke D,9Ss j−1kv fkq ¯ B,9Ss j−1 kv−1kq B,9Ss j−1|e Ss j−1| ≤CX j,s kukq A,9Ss j−1khke D,9Ss j−1kv fkq ¯ B,9Ss j−1 kv−1kq B,9Ss j−1|e Ss j−1| ≤CX j,s Ze Ss j−1 Me Dh(x)M¯ B(v f)(x)qdx ≤CZRn Me Dh(x)M¯ B(v f)(x)qdx ≤CZRn Me Dh(x)r0dx1/r0ZRn M¯ B(v f)(x)pdxq/p ≤CZRn |h(x)|r0dx1/r0ZRn (v(x)f(x))pdxq/p , where the third inequality holds because the sets e Ss j−1are disjoint, and the last inequality holds since by Lemma 2.8, ¯ B∈Bpso M¯ Bis bounded on Lp, and, as we noted above, e D∈Br0, so Me Dis bounded in Lr0. Thus we get the desired bound for I1. We will now estimate I2. The ideas are the same, except that at the key step we will use Kolmogorov’s inequality instead of Yano’s theorem. Since Ers kj ⊂3Ss j−1, by Lemma 2.3, |Ers kj| ≤ C|e Ss j−1|. Further, Mdf(x)q> aj−1on Ss j−1. Thus I2≤a3X (k,j,r,s)∈Γ2 aj+1|Ers kj|1 |Rr k−j−2|ZRr k−j−2 w(x)dx ≤CX (k,j,r,s)∈Γ2 aj+1|e Ss j−1|1 |9Rr k−j−2|Z9Rr k−j−2 w(x)dx ≤CX (k,j,r,s)∈Γ2Ze Ss j−1 Mdf(x)qdx1 |9Rr k−j−2|Z9Rr k−j−2 w(x)dx =CX l,r X (k,j,r,s)∈Γ2 k−j−2=lZe Ss j−1 Mdf(x)qdx1 |9Rr l|Z9Rr l w(x)dx. For fixed land r,e Ss j−1⊂9Rr l. Thus, by Lemma 2.2, for all x∈e Ss j−1, (4.2) Mdf(x) = Md(fχSs j−1)(x)≤Md(fχ9Rr l)(x).
22 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´ EREZ Since t≤Φ(t), kwkL1,9Rr l≤ kwkΦ,9Rr l(see [41]). Therefore, using this, (4.2), Lemma 2.3, and Kolmogorov’s inequality, we get that I2≤CX l,r X (k,j,r,s)∈Γ2 k−j−2=lZe Ss j−1 Md(fχ9Rr l)(x)qdxkwkΦ,9Rr l ≤CX l,r 1 |9Rr l|Z9Rr l Md(fχ9Rr l)(x)qdx · kwkΦ,9Rr l· | e Rr l| ≤CX l,r 1 |9Rr l|Z9Rr l f(x)dxq kwkΦ,9Rr l· | e Rr l|. We can now argue exactly as we did in the estimate of I1to get the desired bound for I2. This completes the proof. Remark 4.2.The term I2is less singular than the term I1: if we did not replace kwkL1,9Rr lby kwkΦ,9Rr l, then a slight modification of our argument would show that we get the desired bound for I2assuming only the weaker condition (1.16). 5. Proof of Theorem 1.8 The proof of Theorem 1.8 is identical in basic idea and organization to the proof of Theorem 1.4, differing only in details. Therefore, rather than give the complete argument, we will outline the changes necessary in the proof of Theorem 1.4. The key changes are in the statement and proof of Lemma 4.1. The new lemma is the following. Lemma 5.1. Given pand (u, v)as in the hypotheses of Theorem 1.8, there exists q, 0< q < 1, such that for all f, h ∈L∞ c(Rn), ZRn M2f(x)qM2(uqh)(x)dx ≤CZRn |v(x)f(x)|pdxq/pZRn |h(x)|(p/q)0dx1/(p/q)0 . Given this inequality, the proof of Theorem 1.8 begins with the same duality argument as the proof of Theorem 1.4. But, instead of Lemma 2.5 we use the following pointwise estimate from [37]: given 0 < q < < 1, (5.1) M# q([b, T]f)(x)≤C M(Tf)(x) + C M2f(x). Thus by Lemma 2.6, we have for every weight wthat ZRn |[b, T]f(x)|qw(x)dx ≤CZRn M# q([b, T]f)(x)qMw(x)dx ≤CZRn M(Tf)qMw(x)dx +CZRn (M2f)qMw(x)dx. The second integral in the last term is dominant. To see this we use the fact that q/ < 1, and Lemmas 2.6 and 2.5 to get ZRn M(Tf)(x)qMw(x)dx =ZRn M(|Tf|)(x)q Mw(x)dx
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 23 ≤CZRn M#(|Tf|)(x)q M2w(x)dx =CZRn M# (Tf)(x)qM2w(x)dx ≤CZRn Mf(x)qM2w(x)dx ≤CZRn M2f(x)qM2w(x)dx. Hence, we have shown that (5.2) ZRn M(Tf)(x)qMw(x)dx ≤CZRn M2f(x)qM2w(x)dx. Fix qas in Lemma 5.1 and let r=p q>1. Then by duality, ZRn |u(x) [b, T]f(x)|pdxq/p = sup ZRn |[b, T]f(x)|qu(x)qh(x)dx, where the supremum is taken over all non-negative functions h∈L∞ c(Rn) such that khkLr0(Rn)= 1. Fix such a function h. By (5.2) and Lemma 5.1 it follows that ZRn |[b, T]f(x)|qu(x)qh(x)dx ≤CZRn M2f(x)qM2(uqh)(x)dx ≤CZRn |v(x)f(x)|pdxq pZRn |h(x)|r0dx1 r0=CZRn |v(x)f(x)|pdxq/p . This completes the proof of Theorem 1.8. 5.1. Proof of Lemma 5.1. Define, as before, Φ(t) = tlog(e+t). It is well known (see, for instance, [45, 12]) that M2f≈MΦf, so in the desired estimate we can replace M2by the maximal function MΦ. We follow the same steps as in the proof of Lemma 4.1, but we replace the Calder´on-Zygmund decomposition by a more general decomposition based on the Orlicz maximal operator MΦ. Its essential properties are exactly the same and are captured in the following definition and lemmas. Definition 5.2. Given a Young function Φ, a non-negative function f∈L1(Rn) (e.g., f∈L∞ c(Rn)), and λ > 0, we define the CZ cubes of fat height λwith respect to Φto be the maximal disjoint dyadic subcubes of the set {x∈Rn:Md,Φf(x)> λ}. Lemma 5.3. Given λ > 0and f∈L1(Rn)non-negative, let {Qj}be the set of CZ cubes of fwith respect to Φ. Then for all j, we have λ < kfkΦ,Qj≤2nλ. Further, Md,Φf(x) = Md,Φ(fχQj)(x)for all x∈Qj. Lemma 5.4. Let f∈L1(Rn). Fix a > 2n, and for k∈Z, let {Qk j}be the CZ cubes of fat height akwith respect to Φ. Then there exist sets {e Qk j},e Qk j⊂Qk j, which are pairwise disjoint for all jand k, and such that there exists α > 1depending only on aand nsuch that |Qk j| ≤ α|e Qk j|. Lemma 5.5. Let f∈L1(Rn)and fix λ > 0. Let {Qj}be the set of CZ cubes of fat height λ/4nwith respect to Φ. Then {x∈Rn:MΦf(x)> λ} ⊂ ∪j3Qj. The proof of each of these lemmas is given in [9] except for the identity in Lemma 5.3, whose proof is identical to the proof of (2.2).
24 D. CRUZ-URIBE, SFO, J. M. MARTELL, AND C. P´ EREZ We now obtain the desired estimate with MΦin place of M2. Proceed as in Lemma 4.1 but fand w=uqhare decomposed with respect to MΦ(in place of M). We estimate I1and I2using the previous lemmas and repeating the computations in Lemma 4.1. We get that I1≤CX j,s 1 |9Ss j−1|Z9Ss j−1 MΦ(wχ9Ss j−1)(x)dx · kfkq Φ,9Ss j−1|e Ss j−1|, I2≤CX l,r 1 |9Rr l|Z9Rr l Md,Φ(fχ9Rr l)(x)qdx · kwkΦ,9Rr l|e Rr l|. For k≥0 define Φk(t) = tlog(e+t)k; then Φ = Φ1. We have the following auxiliary result: the first inequality generalizes Yano’s theorem and is well known (see for instance [12]), and the proof of the second is given below. Lemma 5.6. Let k≥0and 0< q < 1. Then there exists a constant Csuch that for any cube Q 1 |Q|ZQ MΦk(gχQ)(x)dx ≤CkgkΦk+1,Q,1 |Q|ZQ MΦk(gχQ)(x)qdx ≤Ckgkq Φk,Q. If we apply Lemma 5.6 to the estimates for I1and I2we get I1≤CX j,s kwkΦ2,9Ss j−1kfkq Φ,9Ss j−1|e Ss j−1| I2≤CX l,r kfkq Φ,9Rr lkwkΦ,9Rr l|e Rr l| ≤ CX l,r kfkq Φ,9Rr lkwkΦ2,9Rr l|e Rr l|. Hence, both these estimates can be handled in the same way. Let D(t) = tp0log(e+t)2p0−1+δ,e D(t) = tplog(e+t)−1−δ(p−1) ∈Bp, E(t) = trlog(e+t)3r−1+,e E(t) = tr0log(e+t)−1−(r0−1) ∈Br0. Then Φ−1(t)≈D−1(t)·e D−1(t), Φ−1 2(t)≈E−1(t)·e E−1(t), D(t) = B(t) and E(tq)≈ A(t). Therefore, by Lemma 2.7 we have for every cube Qthat kfkΦ,Q ≤Ckf vke D,Q kv−1kB,Q,kwkΦ2,Q ≤CkuqkE,Q khke E,Q ≤Ckukq A,Q khke E,Q. Substitute these values into the above estimates for I1,I2; since e D∈Bp,e E∈Br0, the proof can now be completed exactly as in the proof of Lemma 4.1. Remark 5.7.As we noted in Remark 1.9, the proof of Theorem 1.8 can be adapted to treat the higher order commutators Tk b,k≥2. The ideas are essentially the same. Beginning with the duality argument and applying the analog of (5.1) for higher order commutators (also found in [37]), it is not difficult to see that the proof reduces to obtaining a version of Lemma 5.1 with Mk+1 in place of M2. As Mk+1 ≈MΦk, the decompositions of fand ware made with respect to this Orlicz maximal function. If we let D(t) = tp0log(e+t)(k+1) p0−1+δ,E(t) = trlog(e+t)(k+2) r−1+(e D,e Eremain the same), then by means of Lemma 5.6 we get that the bumps for uand vare, respectively, A(t) = tplog(e+t)(k+2) p−1+δand B(t) = tp0log(e+t)(k+1) p0−1+δ.
SHARP TWO-WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS 25 Proof of Lemma 5.6. We only need to prove show the second inequality. By homogeneity it suffices to assume that kgkΦk,Q = 1. By the properties of Orlicz norms (see [41]), this implies that (5.3) 1 |Q|ZQ Φk(|g(x)|)dx ≤1. The maximal operator MΦksatisfies the modular inequality (5.4) {x∈Rn:MΦkh(x)> λ}≤CZRn Φk(|h(x)|/λ)dx. The proof is standard; see, for instance, [36]. Finally, we note that Φkis submultiplicative: Φk(st)≤Φk(s)Φk(t). Therefore, since 0 < q < 1, if we write the Lq-norm in terms of the level sets, then by (5.3) and (5.4) we have that 1 |Q|ZQ MΦk(g χQ)qdx ≤Z1 0 q λqdλ λ+CZ∞ 1 λq1 |Q|ZRn Φk(|g(x)χQ(x)|/λ)dx dλ λ ≤C+CZ∞ 1 λqΦk(1/λ)dλ λ 1 |Q|ZQ Φk(|g(x)|)dx ≤C. References [1] J. ´ Alvarez and C. P´erez, Estimates with A∞weights for various singular integral operators, Boll. Un. Mat. Ital. A (7) 8 (1994), 123–133. [2] S. Y. A. Chang, J. M. Wilson, and T. H. Wolff, Some weighted norm inequalities concerning the Schr¨odinger operators, Comment. Math. Helvetici 60 (1985), 217–286. [3] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. [4] M. Cotlar and C. Sadosky, On the Helson-Szeg¨o theorem and a related class of modified Toeplitz kernels, Harmonic Analysis in Euclidean Spaces, G. Weiss and S. Wainger eds., vol. 1, 383-407, Proc. Symp. Pure Math. 35, AMS, Providence, 1979. [5] M. Cotlar and C. Sadosky, On some Lpversions of the Helson-Szeg¨o theorem, Conference on Harmonic Analysis in Honor of Antoni Zygmund, W. Beckner et. al. eds., vol. 1, 306-317, Wadsworth International Mathematics Series, Wadsworth, Belmont, 1983. [6] D. Cruz-Uribe, SFO, The product of unbounded Toeplitz operators, Integral Equations Operator Theory 20 (1994), 231-237. [7] D. Cruz-Uribe, SFO, and A. Fiorenza, The A∞property for Young functions and weighted norm inequalities, Houston J. Math., 28 (2002), 169-182. [8] D. Cruz-Uribe, SFO, J.M. Martell, and C. P´erez, Extrapolation from A∞weights and applications, J. Funct. Anal. 213 (2004), 412-439. [9] D. Cruz-Uribe, SFO, and C. P´erez, Sharp two-weight, weak-type norm inequalities for singular integral operators, Math. Res. Let. 6 (1999), 417-428. [10] D. Cruz-Uribe, SFO, and C. P´erez, Two weight extrapolation via the maximal operator, J. Funct. Anal. 174 (2000), 1-17. [11] D. Cruz-Uribe, SFO, and C. P´erez, On the two-weight problem for singular integral operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. I (2002), 821-849. [12] G. Curbera, J. Garc´ıa-Cuerva, J.M. Martell, and C. P´erez, Extrapolation with Weights, Rearrangement Invariant Function Spaces, Modular inequalities and applications to Singular Integrals, to appear in Adv. Math.