Generalized Hörmander conditions and weighted endpoint estimates

GENERALIZED H ¨ ORMANDER’S CONDITIONS AND WEIGHTED ENDPOINT ESTIMATES MAR´ IA LORENTE, JOS´ E MAR´ IA MARTELL, CARLOS P´ EREZ, AND MAR´ IA SILVINA RIVEROS Abstract. We consider two-weight estimates for singular integral operators and their commutators with bounded mean oscillation functions. H¨ormander type conditions in the scale of Orlicz spaces are assumed on the kernels. We prove weighted weak-type estimates for pairs of weights (u, Su) where uis an arbitrary nonnegative function and Sis a maximal operator depending on the smoothness of the kernel. We also obtain sufficient conditions on a pair of weights (u, v) for the operators to be bounded from Lp(v) to Lp,∞(u). One-sided singular integrals, as the differential transform operator, are under study. We also provide applications to Fourier multipliers and homogeneous singular integrals. Contents 1. Introduction 2 2. Preliminaries 5 2.1. Young functions and Orlicz spaces 5 2.2. Muckenhoupt weights 6 2.3. Singular Integral operators and H¨ormader’s type conditions 7 3. Statements of the main results 8 3.1. Singular integral operators 8 3.2. Commutators with BMO functions 10 3.3. One-sided operators 12 4. Applications 13 4.1. The differential transform operator 14 4.2. An example of a one-sided operator with K∈H∞∩Het1/k , k 15 4.3. Multipliers 16 4.4. Kernels related to Hrand MLr17 4.5. Homogeneous Singular Integrals 17 5. Proofs of the main results 18 6. Proofs in the one-sided case 25 References 29 2000 Mathematics Subject Classification. 42B20, 42B25. Key words and phrases. Calder´on-Zygmund operators, homogeneous singular integrals, multipliers, one-sided operators, commutators, BMO, H¨ormander’s condition of Young type, Muckenhoupt weights, two-weight estimates. The first and last authors are partially supported by MCYT Grant MTM2005-08350-C03-02. The first author is also supported by Junta de Andaluc´ıa Grant FQM354. The second author is partially supported by MEC “Programa Ram´on y Cajal, 2005”, by MEC Grant MTM2007-60952, and by UAM-CM Grant CCG07-UAM/ESP-1664. The third author is partially supported by MTM200605622. The last author is partially supported by CONICET, Agencia C´ordoba Ciencias and SECYTUNC. 1 2 M. LORENTE, J. M. MARTELL, C. P´ EREZ, AND M. S. RIVEROS 1. Introduction The Calder´on-Zygmund decomposition is a very powerful tool in Harmonic Analysis. Since its discovery in [6], many results have used it to derive boundedness properties of singular integral operators. For instance, using that the Hilbert or the Riesz transforms are bounded on L2, and by means of the Calder´on-Zygmund decomposition, one proves that these classical operators are of weak-type (1,1). From this starting point, in the literature one can find many boundedness results for the Hilbert and the Riesz transforms: estimates on Lp, one-weight and two-weight norm inequalities, . . . . The Calder´on-Zygmund theory generalizes these ideas to provide a general framework allowing us to deal with singular integral operators. A typical Calder´on-Zygmund convolution operator Tis bounded on L2(Rn) and has a kernel Kon which different conditions are assumed. In the easiest case, Kbehaves as the kernel of the Hilbert or Riesz transforms. That is, Kdecays as |x|−nand its gradient as |x|−n−1. It was already proved in [19] that these assumptions can be relaxed in order to show that T is of weak-type (1,1): it suffices to impose that Ksatisfies the so-called H¨ormander condition (we write K∈H1), Z|x|>c |y| |K(x−y)−K(x)|dx ≤C, y ∈Rn, c > 1. ¿From here, and by interpolation, Tis bounded on Lp(Rn) with 1 < p < ∞. The underlying measure dx can be replaced by w(x)dx where wis a Muckenhoupt Apweight: The Hilbert and the Riesz transforms are bounded on Lp(w) = Lp(w(x)dx) if and only if w∈Apfor 1 < p < ∞. For p= 1, the weak-type (1,1) with respect to wholds if and only if w∈A1. The decay assumed before on the kernel and its gradient guarantees the same weighted estimates for the operator T. However, the H¨ormander condition does not suffice to derive such estimates as it is proved in [18] (see also [26]). One can relax the decay conditions assumed on the kernel and still prove the previous weighted norm inequalities. Namely, it is enough to impose that Ksatisfies the following Lipschitz condition (we write K∈H∗ ∞): |K(x−y)−K(x)| ≤ C|y|α |x|α+n,|x|> c |y|. With this condition in hand, one can show Coifman’s estimate (see [7]): for any 0< p < ∞and any w∈A∞ ZRn |Tf(x)|pw(x)dx ≤CZRn Mf(x)pw(x)dx. (1.1) These estimates can be seen as a control of the operator Tby the Hardy-Littlewood maximal function Mand this allows one to show that Tsatisfies the most of the weighted estimates that Mdoes (see [14] for more details). When relaxing the H∗ ∞condition, the operators become more singular and less smoother. Thus, the Coifman estimates to be expected will have a worse maximal operator on the righthand side. For instance, one has a scale of H¨ormander’s conditions based on the Lebesgue spaces Lrfor 1 ≤r≤ ∞ (see [22], [39] and [43]). A singular integral operator, with kernel satisfying the Lr-H¨ormander condition, 1 < r ≤ ∞, satisfies a Coifman estimate with the maximal operator Mr0in the righthand side (here Mr0f(x) = M(|f|r0)(x)1/r0). These estimates are shown to be sharp in [26]. GENERALIZED H ¨ ORMANDER’S CONDITIONS AND WEIGHTED ENDPOINT ESTIMATES 3 Let us notice that as rgoes to 1 then r0goes to ∞and the corresponding Coifman estimates get worse. In particular, when K∈H1one lacks of Coifman estimates (see [26]). Sometimes, this scale of H¨ormander conditions based on the Lebesgue spaces is too coarse and gives estimates that are not accurate enough. For instance, let us consider the differential transform operator studied in [20] and [4]: T+f(x) = X j∈Z νjDjf(x)−Dj−1f(x),(1.2) where {νj}j∞<∞and Djf(x) = 1 2jZx+2j x f(t)dt . We have that T+is a singular integral operator with kernel Ksupported in (−∞,0), and therefore T+is a one-sided singular integral operator (that is the reason why we write T+). In [4] it was shown that K∈ ∩r≥1Hr(here Hris the H¨ormander condition associated with Lr, see the precise definition below). Thus one can show that T+satisfies a Coifman estimate with Mqin the righthand side for any 1 < q < ∞. Indeed, exploiting the fact that T+is a one-sided operator one can do better: Mqf can be replaced by the pointwise smaller operator M+ qf(the corresponding one-sided maximal function) and A∞by the bigger class A+ ∞(see the precise definitions and more details below). Notice that one can take any 1 < q < ∞, with the case q= 1 remaining open (in general, K /∈H∞). Nevertheless, there are other maximal operators that one can write between M(or M+) and Mq(or M+ q): any iteration of the Hardy-Littlewood maximal function, or maximal operators associated with Orlicz spaces lying between L1and Lqas L(log L)α,α > 0. These ideas motivated [24] on which new classes of H¨ormander conditions based on Orlicz spaces were introduce. Roughly, given a Young function A, associated with the Orlicz space LAone can define a H¨ormander class HA(see Definition 2.3). Thus, a singular integral operator with kernel in HA, is controlled in the sense of Coifman by the maximal operator MA(which is the maximal function associated with the space LA) where Ais the conjugate function of A. This was obtained in [24] as well as the one-sided case (see Theorems 2.4 and 3.11 below). For the differential transform introduced above one can show that K∈Het1/(1+ε)for any ε > 0. Thus T+satisfies a Coifman type estimate with M+ L(log L)1+εon the right hand side —in terms of iterations one can write (M+)3— and this maximal operator is pointwise smaller than M+ qfor any 1 < q < ∞. Coifman’s estimates are important from the point of view of weighted norm inequalities since they encode a lot of information about the singularity of the operator T (see [10] and [14]). In some sense, Tbehaves as the maximal operator that controls it. For instance, one shows that Tis bounded on Lp(w) for 1 < p < ∞,w∈Ap. Also one can see that Tis of weak-type (1,1) for weights in A1,Tis bounded on weighted rearrangement invariant function spaces, Tsatisfies weighted modular inequalities (see [14]), etc. All these one-weight estimates are based on the fact that (1.1) is valid for any weight in A∞and the weight always move within this class. 4 M. LORENTE, J. M. MARTELL, C. P´ EREZ, AND M. S. RIVEROS The situation changes when one works with two-weight inequalities. Let us focus on the end-point estimates for p= 1. In the one weight case, Mis bounded from L1(w) to L1,∞(w) for every w∈A1. Also, there is a version of (1.1) in the sense of L1,∞(w), that is, kTfkL1,∞(w).kMfkL1,∞(w)for every w∈A∞(see [10]). These two facts imply at once that Tis of weak-type (1,1) for weights in A1. In the two-weight case, Vitali’s covering lemma easily gives that for every weight u(a weight is a non-negative locally integrable function) u{x∈Rn:Mf(x)> λ}.1 λZRn |f(x)|Mu(x)dx. However, this estimate is not known for the singular integral operators with smooth kernel. Even for the Hilbert or the Riesz transforms the validity of this estimate is an open question. Reasoning as above, one seeks for pairs of weight (u, Su) for which these operators are of weak-type (1,1), where Swill be a maximal operator worse, in principle, than M. For instance, one can put S=Mqfor every 1 < q < ∞: using that Mqu∈A1and Coifman’s estimate (in L1,∞) one easily obtains the estimate proved in [9] kTfkL1,∞(u)≤ kTfkL1,∞(Mqu).kMfkL1,∞(Mqu).kfkL1(Mqu). As observed before, there are some other maximal operators that lie between Mand Mqas the iterations of Mor ML(log L)α,α > 0. In [32], by means of the Calder´onZygmund decomposition, it was proved that if Tis a singular integral operator with smooth kernel (say K∈H∗ ∞), as the Hilbert or Riesz transform, then for any ε > 0 and any weight u u{x∈Rn:|Tf(x)|> λ}.1 λZRn |f(x)|ML(log L)εu(x)dx. (1.3) Note that in terms of iterations one can write M2. The goal of this paper is to study estimates like (1.3) when the operator Thas a less smoother kernel. That is, if we impose that the kernel of Tsatisfies a H¨ormander condition in the scale of Orlicz spaces, we seek for a maximal operator Sso that T is of weak-type (1,1) with respect to the pair of weights (u, Su). The main technique to be used is the Calder´on-Zygmund decomposition. The bad part is handled by using the smoothness of the kernel. For the good part, one needs a strong two-weight estimate that usually follows from a Coifman estimate (see Theorem 2.6). We also obtain weighted end-point estimates for the commutators of such operators with BMO functions. The corresponding Coifman estimates have been studied in [23]. As one of our main examples is the differential transform operator presented above, we also pay attention to the one-sided operators in which case one can obtain better estimates by replacing a maximal operator by its corresponding one-sided analog. The paper is organized as follows. The following section contains some of the preliminaries and definitions that are needed to state our results. In Section 3 we state our main results on singular integral operators, their commutators with BMO functions and also we consider the one-sided case. Some applications, including the differential transform operator and multipliers, are given in Section 4. Finally Sections 5 and 6 contain the proof of our main results. GENERALIZED H ¨ ORMANDER’S CONDITIONS AND WEIGHTED ENDPOINT ESTIMATES 5 2. Preliminaries 2.1. Young functions and Orlicz spaces. We recall some of the needed background for Orlicz spaces we refer the reader to [37] and [3] for a complete account of this topic. A function A: [0,∞)−→ [0,∞) is a Young function if it is continuous, convex, increasing and satisfies A(0) = 0, A(∞) = ∞. We will assume the Young functions are normalized so that A(1) = 1. We introduce the following localized and averaged Luxemburg norm associated with the Orlicz space LA: given a cube Q kfkA,Q = inf λ > 0 : 1 |Q|ZQ A|f(x)| λdx ≤1. For instance, when A(t) = trwith r≥1 then we have kfkLr,Q =1 |Q|ZQ |f(x)|rdx1 r . It is well known that if A(t)≤CB(t) for t≥t0then kfkA,Q ≤CkfkB,Q. Thus the behavior of A(t) for t≤t0does not matter: if A(t)≈ B(t) for t≥t0the latter estimate implies that kfkA,Q ≈ kfkB,Q. This means that in most of the cases we will not be concerned about the value of the Young functions for tsmall. We can now define the Hardy-Littlewood maximal function associated with Aas MAf(x) = sup Q3x kfkA,Q. When A(t) = tthen MA=Mis the Hardy-Littlewood maximal function. For A(t) = trwith r > 1 we have MAf(x) = M(|f|r)(x)1/r. Given a Young function A, we say that Ais doubling, we write A ∈ ∆2, if A(2 t)≤ CA(t) for every t≥t0>0. For 1 <p<∞,Abelongs to Bpif there exists c > 0 such that Z∞ c A(t) tp dt t<∞. This condition appears first in [34] and it was shown that A ∈ Bpif and only if MA is bounded on Lp(Rn). Abusing on the notation if A(t) = tr,A(t) = etα−1 or A(t) = tr(1 + log+t)α, the Orlicz norms are respectively written as k · kr=k · kLr,k · kexpLα,k · kLr(log L)αand the corresponding maximal operators as Mr=MLr,MexpLαand MLr(log L)α. For k≥0, it is known that ML(log L)kf(x)≈Mk+1f(x) where Mkis the k-times iterated of M(see [33], [38] and [14]). In R, we can also define the one-sided maximal functions associated with a given Young function A: M+ Af(x) = sup b>x kfkA,(x,b)and M− Af(x) = sup a<x kfkA,(a,x). The one-sided Hardy-Littlewood maximal functions M+,M−correspond to the case A(t) = t. Given a Young function A, let Adenote its associate function: the Young function with the property that t≤ A−1(t)A−1(t)≤2t,t > 0. If A(t) = trwith 1 < r < ∞, then A(t) = tr0; if A(t) = trlog(e+t)α, then A(t)≈tr0log(e+t)−α(r0−1). 6 M. LORENTE, J. M. MARTELL, C. P´ EREZ, AND M. S. RIVEROS One has the generalized H¨older’s inequality 1 |Q|ZQ |f g| ≤ 2kfkA,QkgkA,Q. There is a further generalization that turns out to be useful for our purposes, see [31]: If A,B,Care Young functions such that A−1(t)B−1(t)C−1(t)≤t, for all t≥t0>0 (in what follows we assume that t0= 1 for simplicity and clearness in the computations), —sometimes, we will equivalently write A−1(t)B−1(t)≤ C−1(t)— then kf g hkL1,Q ≤CkfkA,B kgkB,Q khkC,Q,kf gkC,Q ≤CkfkA,Q kgkB,Q.(2.1) Remark 2.1. Let us observe that when D(t) = t, which gives L1, then D(t) = 0 if s≤1 and D(t) = ∞otherwise. Although Dis not a Young function one can see that the space LDcoincides with L∞. On the other hand, as the (generalized) inverse is D−1(t)≡1, the previous H¨older’s inequalities make sense with the appropriate changes if one of the three functions is Dor D. We will use this throughout the paper. Remark 2.2. The convexity of Aimplies that A(t)/t is increasing and so t≤CA(t) for all t≥1. This yields that kfkL1,B ≤CkfkA,B for all Young functions A. 2.2. Muckenhoupt weights. We recall the definition of the Muckenhoupt classes Ap, 1 ≤p≤ ∞. Let wbe a non-negative locally integrable function and 1 ≤p < ∞. We say that w∈Apif there exists Cp<∞such that for every ball B⊂Rn 1 |B|ZB w(x)dx1 |B|ZB w(x)1−p0dxp−1 ≤Cp, when 1 < p < ∞, and for p= 1, 1 |B|ZB w(y)dy ≤C1w(x),for a.e. x∈B, which can be equivalently written as Mw(x)≤C1w(x) for a.e. x∈Rn. Finally we set A∞=∪p≥1Ap. It is well known that the Muckenhoupt classes characterize the boundedness of the Hardy-Littlewood maximal function on weighted Lebesgue spaces. Namely, w∈Ap, 1 < p < ∞, if and only if Mis bounded on Lp(w); and w∈A1if and only if Mmaps L1(w) into L1,∞(w). In R, the weighted estimates for the one-sided Hardy-Littlewood maximal function M+(and analogously for M−) are modeled for the classes A+ pwhich are defined as follows. Given 1 < p < ∞,w∈A+ p, if there exists a constant Cp<∞such that for all a<b<c 1 (c−a)pZb a w(x)dxZc b w(x)1−p0dxp−1 ≤Cp. We say that w∈A+ 1if M−w(x)≤C1w(x) for a.e. x∈R. The class A+ ∞is defined as the union of all the A+ pclasses, A+ ∞=∪p≥1A+ p.The classes A− pare defined in a similar way. It is interesting to note that Ap=A+ p∩A− p,Ap(A+ pand Ap(A− p. See [41], [27], [28], [29] for more definitions and results. GENERALIZED H ¨ ORMANDER’S CONDITIONS AND WEIGHTED ENDPOINT ESTIMATES 7 2.3. Singular Integral operators and H¨ormader’s type conditions. Let Tbe a singular integral operator of convolution type, that is, Tis bounded on L2(Rn) and Tf(x) = p.v.ZRn K(x−y)f(y)dy where Kis a measurable function defined away from 0. Convolution operators are considered for simplicity, but the results presented here can be stated for variable kernels with the appropriate changes. The precise statements and the details are left the reader. When n= 1 and we further assumed that the kernel Kis supported on (−∞,0) we say that Tis a one-sided singular integral and we write T+to emphasize it. The results that we present below for (regular) singular integrals apply to T+. However, taking advantage of the extra assumption on the kernel, one can be more precise and get better estimates (see Section 3.3). We introduce the different H¨ormander type conditions assumed on the kernel K. The weakest one is the so-called H¨ormander condition H1(we simply say K∈H1or Ksatisfies the L1-H¨ormander condition): there are constants c > 1 and C > 0 such that Z|x|>c |y| |K(x−y)−K(x)|dx ≤C, y ∈Rn. The strongest one is the classical Lipschitz condition called H∗ ∞(this notation is not standard but we keep H∞for a weaker L∞-condition, see the definition below). We say that K∈H∗ ∞if there are α, C > 0 and c > 1 such that |K(x−y)−K(x)| ≤ C|y|α |x|α+n,|x|> c |y|. Between H1and H∗ ∞one finds the Lr-H¨ormander conditions (which are called HLr= Hrin the definition below). These classes appeared implicitly in the work [22] where it is shown that classical Lr-Dini condition for Kimplies K∈Hr(see also [39] and [43]). However, there are examples of singular integrals like the differential transform operator from Ergodic Theory defined in (1.2), whose kernel K∈Hrfor all 1 ≤r < ∞ but K /∈H∞. As it was obtained in [23], Ksatisfies a H¨ormander condition in the scale of the Orlicz spaces that lies between the intersection of the classes Hrfor 1 ≤r < ∞ and H∞. The same happens with the one-sided discrete square function considered in [42] and [24]. All this has motivated the definition of the LA-H¨ormander conditions in [24]: Definition 2.3. The kernel Kis said to satisfy the LA-H¨ormander condition, we write K∈HA, if there exist c≥1,C > 0such that for any y∈Rnand R > c |y|, ∞ X m=1 (2mR)nkK(· − y)−K(·)kA,|x|∼2mR≤C. We say that K∈H∞if Ksatisfies the previous condition with k·kL∞,|x|∼2mRin place of k·kA,|x|∼2mR. We have used the notation: |x| ∼ sfor s < |x| ≤ 2sand kfkA,|x|∼s=kfχ{|x|∼s}kA,B(0,2s). 8 M. LORENTE, J. M. MARTELL, C. P´ EREZ, AND M. S. RIVEROS Note that if A(t) = tthen HA=H1. On the other hand, since t≤CA(t) for t≥1 we have that HA⊂H1which implies that the classical unweighted Calder´on-Zygmund theory can be applied to T. Also, it is easy to see that H∗ ∞⊂H∞⊂HA. For convenience thorough this paper we write | · | =| · |∞so that everything is adapted to cubes in place of balls (with the appropriate changes everything can be written in terms of balls). For simplicity we also assume that c= 1. Coifman type estimates were proved for kernels in these classes in [24]: Theorem 2.4 ([24]).Let Abe a Young function and let Tbe a singular integral operator with kernel K∈HA. Then for any 0< p < ∞and w∈A∞, ZRn |Tf(x)|pw(x)dx ≤CZRn MAf(x)pw(x)dx, f ∈L∞ c,(2.2) whenever the left-hand side is finite. Note that this improves the previous results in [22], [39] and [43] (for sharpness issues see also [26]). Similar results are also proved for vector-valued and one-sided operators (see [24]). Remark 2.5. Abusing on the notation, as in Remark 2.1, if K∈H∞, then (2.2) holds with MA=M, where A(t) = t. This was obtained in [26] improving the corresponding result for the smaller class H∗ ∞. The previous estimates are useful in applications as one has that Tand MAhave a similar behavior (see [14]). For instance, two-weight estimates can be proved in the following way: Theorem 2.6 ([23]).Let Abe a Young function and 1<p<∞. Suppose that there exist Young functions D,Esuch that E ∈ Bp0and D−1(t)E−1(t)≤ A−1(t)for t≥t0>0. Set Dp(t) = D(t1/p). Let Tbe a linear operator such that its adjoint T∗ satisfies (2.2). Then for any weight u, ZRn |Tf(x)|pu(x)dx ≤CZRn |f(x)|pMDpu(x)dx. (2.3) Remark 2.7. Abusing in the notation, the previous result contains the case A(t) = t on which in (2.2) one has MA=M. Then, Dand Eare conjugate functions and so (2.3) holds for any Dpsuch that D ∈ Bp0. In particular, in (2.3) we can take the pair of weights (u, ML(log L)p−1+δu) for any δ > 0: pick D(t) = tp(1 + log+t)p−1+δwhose conjugate function is D(t)≈tp0/(1 + log+t)1+δ(p0−1) ∈Bp0. 3. Statements of the main results 3.1. Singular integral operators. We are going to obtain endpoint two-weight norm inequalities for singular integral operators where different H¨ormander’s conditions are assumed on the kernel. Namely, we seek for the following weak-type (1,1) estimates with pairs of weights (u, Su) where Swill be a certain maximal function depending on the smoothness of the kernel: u{x∈Rn:|Tf(x)|> λ} ≤ C λZRn |f(x)|Su(x)dx. (3.1) GENERALIZED H ¨ ORMANDER’S CONDITIONS AND WEIGHTED ENDPOINT ESTIMATES 9 Theorem 3.1. Let Tbe a singular integral operator with kernel K. (a)Let Abe a Young function such that its complementary function A ∈ ∆2and assume that there exists r > 1so that lim inft→∞ A(t)/tr>0. If K∈HAthen (3.1) holds for the pairs of weights (u, MAu). (b)Let Abe a Young function and assume that there exist 1< p < ∞, and Young functions Dand Esuch that D−1(t)E−1(t)≤ A−1(t)for t≥t0>0with E ∈ Bp0. If K∈HAthen, (3.1) holds for the pairs of weights (u, MDpu)with Dp(t) = D(t1/p). (c)If K∈H∞, then (3.1) holds for the pairs of weights (u, ML(log L)εu)for any ε > 0. Remark 3.2. In part (c) we improve [32], as we consider a wider class of kernels H∗ ∞(H∞. Remark 3.3. Let us notice that when lim inft→∞ A(t)/tr>0, the pair of weights in (a) is better than the one in (b): one can see that A(t).Dp(t) for t≥1. Take an arbitrary t≥1. The fact that E ∈ Bp0implies E(t).tp0. Also, A(t)≥tas Ais a Young function. Then, from the condition assumed on A,Dand Eit follows that D−1(t).t1/p and therefore A−1(t)≥ D−1(t)E−1(t)&D−1(t)pt1/p0/D−1(t)p−1&D−1(t)p=D−1 p(t). Remark 3.4. We would like to emphasize that in part (a) the associated Coifman estimate in Theorem 2.4 tells us that Tis controlled by MA. Here we show that the pair of weights of the form (u, MAu) is valid. In the previous remark, we have observed that in (b) one gets a bigger maximal operator MDp. In many applications, although we take pvery close to 1, we always obtain a maximal operator pointwise greater than MA. This is the case in (c) which covers the classical Hilbert and Riesz transforms. Here as these operators are controlled by M(in the sense of Coifman) one would wish to show that the pair of weights (u, Mu) is valid. However this remains as an open question and the best known result is (u, ML(log L)εu). There is a general extrapolation principle that allows one to pass from pairs of weights (u, Su), with Sbeing a maximal operator, to general pairs of weights (u, v). The main ideas are implicit in [12], [13] and are further exploited in [11]. Below we present a proof in the one-sided case (see Theorem 3.14), that can be easily adapted to the present situation. Theorem 3.5. Let Fbe a Young function and assume that a given operator Tsatisfies u{x∈Rn:|Tf(x)|> λ} ≤ C λZRn |f(x)|MFu(x)dx (3.2) for every weight uand λ > 0. Given 1< p < ∞, let G,Hbe Young functions such that G−1(t)H−1(t)≤ F−1(t)for all t≥t0>0and H ∈ Bp0. Then for any pair of weights (u, v)satisfying ku1/pkG,Q kv−1/pkLp0,Q ≤C(3.3) 16 M. LORENTE, J. M. MARTELL, C. P´ EREZ, AND M. S. RIVEROS version of Theorem 3.1 part (c), and when k≥1 we employ the one-sided version of Theorem 3.8 part (b). Thus, we conclude the following end-point estimates: given b∈BMO, for every k≥0 and for any ε > 0 u{x∈R:|T+,k bf(x)|> λ} ≤ CZR Ck|f(x)| λM− L(log L)k+εu(x)dx. (4.3) Note that taking ε > 0 small enough, M− L(log L)k+εu(x)≤C(M−)k+2u(x). Remark 4.4. In terms of iterations of the one-sided Hardy-Littlewood maximal function, notice that in (4.2) we have k+ 1 iterations and in (4.3) we have k+ 2, so we obtain an extra iteration. This is because in (c) of Theorem 3.1, in (b) of Theorem 3.8 and in their corresponding versions for one-sided operators we loose a small power of the logarithm. This happens also with Calder´on-Zygmund operators with smooth kernel as the Hilbert and Riesz transforms: they are controlled, in the sense of Coifman, by M, but the end-point estimate holds for the pair of weights (u, M2u) —indeed one can write (u, ML(log L)εu) for any ε > 0—. It is not known, even for the Hilbert and Riesz transforms, whether the pair of weights (u, Mu) is valid for the corresponding weak-type estimate. Notice that in the case of the differential transform operator in both the Coifman inequality and the end-point estimate the number of iterations for the k-th order commutator is k+3. This happens as we already have a small power of the logarithm floating around. ¿From Theorem 3.14 proceeding as in Corollary 4.3 we obtain the following twoweight weak-type estimates: given b∈BMO, for every k≥0 and for any ε > 0 if (u, v) is a pair of weights such that, for all a<b<cwith b−a < c −b, ku1/pkLp(log L)(k+1) p−1+ε,(a,b)kv−1/pkLp0,(b,c)≤C, then T+,k bfmaps Lp(v) into Lp,∞(u). This extends the sharp results obtained in [12] for Calder´on-Zygmund operators with smooth kernels to the setting of one-sided operators. 4.3. Multipliers. Let m∈L∞(Rn) and consider the multiplier operator Tdefined a priori for fin the Schwartz class by c Tf(ξ) = m(ξ)b f(ξ). Given 1 < s ≤2 and 0≤l∈Nwe say that m∈M(s, l) if sup R>0 R|α|kDαmkLs,|ξ|∼R<+∞,for all |α| ≤ l. In [23] it was proved the following: let m∈M(s, l), with 1 < s ≤2, 0 ≤l≤n and l > n/s. Then for all k≥0 and any ε > 0 we have that for all 0 < p < ∞and w∈A∞,ZRn |Tk bf(x)|pw(x)dx ≤CZRn Mn/l+εf(x)pw(x)dx. (4.4) The proof of such estimates consists in obtaining that a family of truncations of the kernel {KN}Nare uniformly in HLr(log L)k r,k with r0=n/l +εThus, taking A(t) = tr,B(t) = tr(1 + log+t)k r we have KN∈HB∩HA,k (this follows easily from KN∈HLr(log L)k r,k). Notice that A−1(t)B−1(t)C−1 k(t).tfor t≥1 and therefore (4.4) follows from Theorem 3.7 for the k-th order commutators of TN(which is the GENERALIZED H ¨ ORMANDER’S CONDITIONS AND WEIGHTED ENDPOINT ESTIMATES 17 operator whose kernel is KN) with constants that are independent of N. A standard approximation argument leads to the desired estimate for Tk b. We refer the reader to [23] for more details. The same argument allows us to apply Theorem 3.1 part (a) and Theorem 3.8 part (a.1) to TN. Observe that A(t) = tr0, then choosing 1 < s < r0we obtain lim inft→∞ A(t)/ts= +∞. Therefore, taking limits we have the following result: Theorem 4.5. Let m∈M(s, l)with 1< s ≤2,0≤l≤nand l > n/s. Then for all k≥0and any ε > 0we have u{x∈Rn:|Tk bf(x)|> λ} ≤ CZRn Ck|f(x)| λMn/l+εu(x)) dx. ¿From this estimate one can obtain weak-type estimates for general pairs of weights by using Theorem 3.5. The precise statements are left to the reader. 4.4. Kernels related to Hrand MLr.Implicit in [39] (see also [22], [43]) and as it was observed in [26] when K∈HLr, that is, when the kernel satisfies the LrH¨ormander condition, then one obtains that Tis controlled by MLr0. In [23] different extensions of that inequality for the higher order commutators where considered. Following the notation of Theorem 3.7 these are the different conditions and maximal operators obtained: for every 1 < r < ∞and k≥0, we have HB,k HB∩HA,k MAf HLr,k HLr∩HLr(log L)−k r,k MLr0(log L)k r0f HLr(log L)k r,k HLr(log L)k r ∩HLr,k MLr0f HLr(log L)k,k HLr(log L)k∩HLr(log L)−k(r−1),k MLr0(log L)kf≈(MLr0)k+1 Table 1. Examples of different Hr-conditions Thus, applying Theorem 3.1 part (a) and Theorem 3.8 part (a.1) we obtain that Tk bsatisfies (3.6) with the different pairs of weights (u, MAu) in the previous table. 4.5. Homogeneous Singular Integrals. Denote by Σ = Σn−1the unit sphere on Rn. For x6= 0, we write x0=x/|x|. Let us consider Ω ∈L1(Σ). This function can be extended to Rn\ {0}as Ω(x) = Ω(x0) (abusing on the notation we call both functions Ω). Thus Ω is a function homogeneous of degree 0. We assume that RΣΩ(x0)dσ(x0) = 0. Set K(x) = Ω(x)/|x|nand let Tbe the operator associated with the kernel K. Given a Young function Awe define the LA-modulus of continuity of Ω as $A(t) = sup |y|≤t kΩ(·+y)−Ω(·)kA,Σ. Given Ω ∈LB(Σ) and Tbe as above. Let k≥0 and A,Bbe Young functions such that A−1(t)B−1(t)C−1 k(t)≤tfor all t≥1. If Z1 0 $B(t)dt t+Z1 01 + log 1 tk$A(t)dt t<∞, 18 M. LORENTE, J. M. MARTELL, C. P´ EREZ, AND M. S. RIVEROS then it was proved in [23] that K∈HB∩HA,k and therefore ZRn |Tk bf(x)|pw(x)dx ≤CZRn MAf(x)pw(x)dx, for every 0 < p < ∞and w∈A∞. Once it is known that K∈HB∩HA,k one can apply Theorems 3.1 and 3.8 to derive the corresponding two-weight end-point estimates. The precise statements and further details are left to the interested reader. 5. Proofs of the main results Proof of Theorem 3.1. Without loss of generality we can assume that uis bounded and has compact support (otherwise we prove the corresponding estimate for uN= min{u, N}χB(0,N)with bounds independent of Nand apply the monotone convergence theorem). We assume that 0 ≤f∈L∞ c(Rn) and consider the standard Calder´on-Zygmund decomposition of fal level λ: there exists a collection of maximal (and so disjoint) dyadic cubes {Qj}j(with center xjand sidelength 2 rj) such that λ < 1 |Qj|ZQj f≤2nλ. (5.1) We write f=g+hwhere g=fχRn\∪jQj+X j fQjχQj, h =X j hj=X j (f−fQj)χQj where fQjdenotes the average of fover Qj. Let us recall that 0 ≤g(x)≤2nλa.e. and also that each hjhas vanishing integral. We set ˜ Qj= 2 Qj,˜ Ω = ∪j˜ Qj, and ˜u=uχRn\˜ Ω. Then, u{x∈Rn:|Tf(x)|> λ} ≤ u(˜ Ω) + u{x∈Rn\˜ Ω : |Th(x)|> λ/2} +u{x∈Rn\˜ Ω : |Tg(x)|> λ/2} =I+II +III. We estimate each term separately. The estimates for Iand II are obtained in the same way in the three cases (a), (b) and (c). We show that I.1 λZRn f(x)Mu(x)dx, II .1 λZRn f(x)MAu(x)dx, (5.2) where, in case (c), as K∈H∞=HL∞it is understood that A(t) = tso MA=ML1= M. Let us observe that both estimates lead us to the desired conclusions in the three cases (a), (b) and (c). Regarding I,Mu is controlled by MAuin (a) —as Ais a Young function—, by MDpuin (b) —as we pointed out in Remark 3.3 that D−1(t).t1/p for t≥1 which yields Dp(t)≥tfor t≥1— and by ML(log L)εuin (c). For II,MAuis the desired weight in (a); in (b) we observed in Remark 3.3 that MAu.MDpu; and in (c) we have MAu=Mu ≤ML(log L)εu. Let us show the first estimate in (5.2). By (5.1) we have I=u(∪j˜ Qj)≤X j u(˜ Qj) = 2nX j u(˜ Qj) |˜ Qj||Qj| ≤ 2n λX j u(˜ Qj) |˜ Qj|ZQj f(x)dx GENERALIZED H ¨ ORMANDER’S CONDITIONS AND WEIGHTED ENDPOINT ESTIMATES 19 ≤2n λX jZQj f(x)Mu(x)dx =2n λZRn f(x)Mu(x)dx. Next, we estimate II: as the functions hjhas vanishing integral II =unx∈Rn\˜ Ω : X j Thj(x)> λ/2o≤2 λX jZRn\˜ Ω |Thj(x)|u(x)dx ≤2 λX jZRn\˜ ΩZQj (K(x−y)−K(x−xQj)) hj(y)dyu(x)dx ≤2 λX jZQj |hj(y)|ZRn\˜ Qj |K(x−y)−K(x−xQj)|u(x)dx dy. We claim that for every y∈Qjwe have ZRn\˜ Qj |K(x−y)−K(x−xQj)|u(x)dx .ess inf Qj MAu. (5.3) This estimate drives us to II .1 λX j ess inf Qj MAuZQj |hj(y)|dy .1 λX j ess inf Qj MAuZQj f(y)dy ≤1 λX jZQj f(y)MAu(y)dy ≤1 λZRn f(y)MAu(y)dy. We obtain (5.3): using the generalized H¨older’s inequality for Aand A(when K∈H∞ we understand that A(t) = tand so we have the corresponding L1−L∞H¨older’s estimate) ZRn\˜ Qj |K(x−y)−K(x−xQj)|u(x)dx ≤ ∞ X k=1 Z|x−xQj|∼2krj |K(x−y)−K(x−xQj)|u(x)dx . ∞ X k=1 (2krj)nkK(· − y)−K(· − xQj)kA,|x−xQj|∼2krjkukA,|x−xQj|≤2k+1 rj ≤Cess inf Qj MAu, where in the last estimate we have used that K∈HA. To complete the proof, it remains to estimate III. Here, the proof changes in each of the cases. We start with (a). As lim inft→∞ A(t)/tr>0 then there exists c=cr such that A(t)≥c trfor every t≥1. On the other hand, using that A ∈ ∆2there exist 1 <s<∞(indeed we can take s>r) such that A(t)≤C tsfor every t≥1 (this follows by iterating the ∆2-condition). Then, taking p>swe have III =u{x∈Rn\˜ Ω : |Tg(x)|> λ/2} ≤ 2p λpZRn |Tg(x)|p˜u(x)dx ≤2p λpZRn |Tg(x)|pMr˜u(x)dx .1 λpZRn MAg(x)pMr˜u(x)dx, (5.4) 20 M. LORENTE, J. M. MARTELL, C. P´ EREZ, AND M. S. RIVEROS where in the last inequality we have used Theorem 2.4 and the fact that Mr˜u∈A1⊂ A∞as r > 1. Notice that one has to check that the left-hand side of (2.2) is finite. Indeed, as we have assumed that u∈L∞we have ZRn |Tg(x)|pMr˜u(x)dx ≤ kukL∞ZRn |Tg(x)|pdx .kukL∞ZRn g(x)pdx .kukL∞λp−1ZRn g(x)dx =kukL∞λp−1ZRn f(x)dx < ∞, where we have used that Tis bounded on Lp(Rn) as K∈HA⊂H1; and also that f and uare bounded with compact support. We can continue with the estimate of III: as A(t)≤C tsfor every t≥1 it follows that III .1 λpZRn Msg(x)pMr˜u(x)dx =1 λpZRn M(gs)(x)p/s Mr˜u(x)dx .1 λpZRn g(x)pMr˜u(x)dx .1 λpZRn g(x)pMA˜u(x)dx, (5.5) where we have used that Mr˜u∈A1, therefore Mis bounded on Lp/s(Mr˜u) and also that A(t)≥c trfor every t≥1. We claim that Z∪jQj g(x)MA˜u(x)dx .Z∪jQj f(x)MA˜u(x)dx. (5.6) ¿From (5.5), this estimate and the fact that 0 ≤g(x)≤2nλa.e. yield III .1 λZRn g(x)MA˜u(x)dx =1 λZRn\∪jQj f(x)MA˜u(x)dx +1 λZ∪jQj g(x)MA˜u(x)dx .1 λZRn f(x)MA˜u(x)dx ≤1 λZRn f(x)MAu(x)dx, which is the desired estimate for III. To complete the proof of (a) we need to show (5.6). We first obtain that for any Young function C, any weight vwith MCv < ∞a.e, and any cube Qwe have MC(vχRn\2Q)(y)≈ess inf z∈QMC(vχRn\2Q)(z),a.ey∈Q. (5.7) Let y∈Qand Rbe any cube such that y∈R. If R\2Q= Ø then kvχRn\2QkC,R = 0. Otherwise, we have that `(R)> `(Q)/2 which implies that Q⊂5R. Then, kvχRn\2QkC,R .kvχRn\2QkC,5R≤ess inf z∈QMC(vχRn\2Q)(z), and taking the supremum over all the cubes R3ywe conclude the desired estimate. Next, we use (5.7) to obtain (5.6): Z∪jQj g(x)MA˜u(x)dx =X jZQj g(x)MA˜u(x)dx =X j fQjZQj MA(˜uχRn\2Qj)(x)dx .X jZQj f(x)dx ess inf z∈Qj MA(˜uχRn\2Qj)(z)≤X jZQj f(x)MA(˜uχRn\2Qj)(x)dx =Z∪jQj f(x)MA˜u(x)dx. GENERALIZED H ¨ ORMANDER’S CONDITIONS AND WEIGHTED ENDPOINT ESTIMATES 21 This completes the proof of (a). To show (b), we only have to estimate III. The argument is very similar, the main change consists of proving (5.5) with Dpin place of A. Once we have that, the argument presented above adapts trivially to the present situation and so the desired estimate for III follows. Note that our hypotheses guarantee that we can apply Theorem 2.4 to the adjoint of T—let us observe that T∗=˜ Twhere ˜ Tis the singular operator with kernel ˜ K(x) = K(−x)∈HA— and then Theorem 2.6 yields III =u{x∈Rn\˜ Ω : |Tg(x)|> λ/2} ≤ 2p λpZRn |Tg(x)|p˜u(x)dx .1 λpZRn g(x)pMDp˜u(x)dx. (5.8) As just mentioned, the ideas used before applied straightforward and the desired estimate follows at once. Finally, we show (c). Given ε > 0 we pick p > 1 and δ > 0 so that p−1+δ=ε(note that pis taken very close to 1 and δvery small). Then, by Remark 2.7, (2.3) holds for the pair of weights (u, ML(log L)εu). Then, the previous case mutatis mutandis leads us to the desired estimate. Remark 5.1. There is another argument to derive (c): Given ε > 0 we pick p > 1 and δ > 0, so that p−1 + 2 δ=ε. Let A(t) = t(1 + log+)δ/p. Note that H∞⊂HAand so K∈HA. We take D(t) = tp(1+log+t)p−1+2 δand E(t)≈tp0/(1+log+t)1+δ(p0−1) ∈Bp0. Then, we can apply (b) to obtain the desired estimate for the pair of weights (u, MDpu). To conclude we observe that Dp(t) = D(t1/p) = t(1 + log+t)ε.  Proof of Theorem 3.8. The argument follows the scheme of the proof Theorem 3.1, which corresponds to the case k= 0, and we only give the main changes. We proceed by induction to obtain (a). The proof of (b) follows as in Theorem 3.1 from (a.2) by a suitable choice of Aand B(see Remark 5.1). We assume that the cases m= 0,1, . . . , k −1 are proved and we show the desired estimate for Tk b. Thus, we fix a weight u∈L∞ cand 0 ≤f∈L∞ c. By homogeneity we can also assume that kbkBMO = 1. We recall some properties of BMO to be used later. Given b∈BMO, a cube Q, j≥0 and q > 0, by John-Nirenberg’s theorem we have k(b−bQ)jkLq,Q ≤ k(b−bQ)jkCk,Q =kb−bQkj exp L,Q ≤Ckbkj BMO.(5.9) On the other hand, for every l≥1 and b∈BMO, we have |bQ−b2lQ| ≤ l X m=1 |b2m−1Q−b2mQ| ≤ 2n l X m=1 kb−b2mQkL1,2mQ≤2nlkbkBMO.(5.10) We perform the Calder´on-Zygmund decomposition of fat level λ. Let g,h=Pjhj, Qj,˜ Qj,˜ Ω and ˜ube as in the proof of Theorem 3.1. Then, u{x∈Rn:|Tk bf(x)|> λ} ≤ u(˜ Ω) + u{x∈Rn\˜ Ω : |Tk bh(x)|> λ/2} +u{x∈Rn\˜ Ω : |Tk bg(x)|> λ/2} 22 M. LORENTE, J. M. MARTELL, C. P´ EREZ, AND M. S. RIVEROS =I+II +III, and we estimate each term separately. For Iwe obtain the first estimate in (5.2) exactly as before. Then, I.1 λZRn f(x)Mu(x)dx ≤ZRn Ck|f(x)| λMu(x)dx and we observe that Mu is pointwise controlled by either MAu,MDpuor ML(log L)k+εu. So the desired estimate follows in each of the cases. Next, we estimate II by using the induction hypothesis and the conditions assumed on the kernel. As in [33] we can write Tk bh(x) = X j Tk bhj(x) = k−1 X m=0 Ck,mTm bX j (b−bQj)k−mhj(x) +X j (b(x)−bQj)kThj(x) = F1(x) + F2(x),(5.11) and we estimate each function in turn. For F1we would like to use the induction hypothesis. We start with (a.1). If 0≤m≤k−1 then HA,k ⊂HA,m and so K∈HB∩HA,m. Also, as Ck(t)≤ Cm(t) we have A−1(t)B−1(t)C−1 m(t)≤ A−1(t)B−1(t)C−1 k(t)≤t. Thus the hypotheses on (a) are satisfied for every 0 ≤m≤k−1 and therefore u{x∈Rn\˜ Ω : |F1(x)|> λ/4} ≤ k−1 X m=0 ˜unx:Tm bX j (b−bQj)k−mhj(x)> λ/Co . k−1 X m=0 ZRn Cm Pj(b−bQj)k−mhj λ MA˜u dx . k−1 X m=0 X jZQj Cm|b−bQj|k−m|hj| λMA˜u dx . k−1 X m=0 X j ess inf Qj MA˜uZQj Cm|b−bQj|k−m|hj| λdx, where in the last estimate we have used (5.7). Let us observe that C−1 k(t)C−1 k−m(t). C−1 m(t). Then, Young’s inequality implies ZQj Cm|b−bQj|k−m|hj| λdx .ZQj Ck|hj| c λ dx +ZQj Ck−m(c|b−bQj|k−m)dx =ZQj Ck|hj| λdx +ZQj ec|b−bQj|dx .ZQj Ck|hj| λdx +|Qj|(5.12) as kbkBMO = 1 implies, by John-Nirenberg’s theorem, that kb−bQjkexp L,Qj≤c. Besides, using that Ck(t)δ, 0 < δ < 1, is concave and so subadditive it follows that Ck GENERALIZED H ¨ ORMANDER’S CONDITIONS AND WEIGHTED ENDPOINT ESTIMATES 23 is quasi-subadditive —that is, Ck(t1+t2).Ck(t1) + Ck(t2)—. Therefore, by Jensen’s inequality for Ck ZQj Ck|hj| λdx ≤ZQj Ckf λdx +|Qj| CkfQj λ≤2ZQj Ckf λdx. Also, (5.1) implies |Qj| ≤ 1 λZQj f dx ≤ZQj Ckf λdx. Plugging these estimates into (5.12) we obtain u{x∈Rn\˜ Ω : |F1(x)|> λ/4}. k−1 X m=0 X j ess inf Qj MA˜uZQj Ckf λdx .X jZQj Ckf λMA˜u dx ≤ZRn Ckf λMAu dx. This gives the desired estimate for F1in case (a.1). Notice that the same computations hold in case (a.2) replacing everywhere MAby MDp. Next, we estimate F2: u{x∈Rn\˜ Ω : |F2(x)|> λ/4} ≤ 4 λX jZRn\˜ Ω |b(x)−bQj|k|Thj(x)|u(x)dx ≤4 λX jZRn\˜ Ω |b(x)−bQj|kZQj |K(x−y)−K(x−xQj)|| hj(y)|dy u(x)dx ≤4 λX jZQj |hj(y)|ZRn\˜ Qj |K(x−y)−K(x−xQj)| |b(x)−bQj|ku(x)dx dy. We claim that for every cube Q(whose center is xQ) and for every y∈Qwe have ZRn\2Q |K(x−y)−K(x−xQ)| |b(x)−bQ|ku(x)dx .ess inf QMAu. (5.13) This estimate applied to each Qjimplies u{x∈Rn\˜ Ω : |F2(x)|> λ/4}.1 λX j ess inf Qj MAuZQj |hj(y)|dy .1 λX j ess inf Qj MAuZQj f(y)dy ≤1 λX jZQj f(y)MAu(y)dy ≤1 λZRn f(y)MAu(y)dy ≤ZRn Ck|f(x)| λMAu(y)dy. Note that this leads to the desired estimate in (a.1) and also in (a.2) (we observed in Remark 3.3 that MAu.MDpu). Collecting the obtained inequalities for F1and F2 we complete the estimate of II. 24 M. LORENTE, J. M. MARTELL, C. P´ EREZ, AND M. S. RIVEROS We show (5.13). Let Qbe a cube with center xQand sidelength 2 r. Using (5.10); the generalized H¨older’s inequality for Aand A, and also for A,Band Ck; and (5.9) ZRn\2Q |K(x−y)−K(x−xQ)| |b(x)−bQ|ku(x)dx . ∞ X l=1 Z|x−xQ|∼2lr |K(x−y)−K(x−xQ)| |b(x)−b2l+1 Q|ku(x)dx + ∞ X l=1 lkZ|x−xQ|∼2lr |K(x−y)−K(x−xQ)|u(x)dx . ∞ X l=1 (2lr)nkK(· − y)−K(· − xQ)kB,|x−xQj|∼2lrk(b−b2l+1 Q)kkCk,2l+1 QkukA,2l+1 Q + ∞ X l=1 (2lr)nlkkK(· − y)−K(· − xQ)kA,|x−xQ|∼2lrkukA,2l+1 Q .ess inf QMAu, where we have used that K∈HB∩HA,k. To complete the proof we need to estimate III. The proof is almost identical to that of Theorem 3.1. For the case (a.1), in (5.4) we apply Theorem 3.7 in place of Theorem 2.4. Once we have that estimate, the proof follows the same computations once we check that |Tk bg|pMr˜u∈L1(Rn) (we show this below). For the case (a.2) we need to show that Tk bsatisfies the corresponding estimate in (5.8). But this follows from Theorem 2.6 as we can apply Theorem 3.7 to the adjoint of Tk b—note that (Tk b)∗= (T∗)k −band T∗is a singular integral operator with kernel ˜ K(x) = K(−x)∈ HB∩HA,k—. As just mentioned we only need to check that |Tk bg|pMr˜u∈L1(Rn). As u∈L∞ it suffices to see that Tk bg∈Lp(Rn) for plarge enough. This is trivial if one assumes that b∈L∞as our assumption on Kimplies that K∈H1and thus Tis bounded on Lp(Rn) for every 1 < p < ∞: kTk bgkLp(Rn)= k X m=0 Cm,k bk−mT(bmg)Lp(Rn).kbkk L∞kgkLp(Rn) ≤ kbkk L∞λ(p−1)/p kfk1/p L1(Rn)<∞. Thus, we obtain (3.6) with Su =MAuunder the additional assumption that b∈L∞. We pass to an arbitrary b∈BMO: for any N > 0 we define bN(x) = b(x) if −N≤ b(x)≤N,bN(x) = Nif b(x)> N and bN(x) = −Nif b(x)<−N. It is not hard to prove that |bN(x)−bN(y)| ≤ |b(x)−b(y)|and hence kbNkBMO ≤2kbkBMO. Therefore, as bN∈L∞we can use (3.6) with bNin place of band so u{x∈Rn:|Tk bNf(x)|> λ} ≤ CZRn CkkbNkk BMO |f(x)| λMAu(x)dx ≤CZRn Ckkbkk BMO |f(x)| λMAu(x)dx (5.14) GENERALIZED H ¨ ORMANDER’S CONDITIONS AND WEIGHTED ENDPOINT ESTIMATES 25 where Cdoes not depend on N. Since f∈L∞ cit follows that for 0 ≤m≤k, (bN)mf−→ bmfas N→ ∞ in Lqfor q > 1. The fact that Tis bounded on Lq implies T((bN)mf)−→ T(bmf) as N→ ∞ in Lq. Passing to a subsequence the convergence is almost everywhere and so using that Tk bNf(x) = k X m=0 Cm,k bN(x)k−mT(bm Nf)(x) it follows that Tk bNjf(x)−→ Tk bf(x) for a.e. x∈Rnas j→ ∞. Then, we clearly have that χ{Tk bf>λ}(x)≤lim infj→∞ χ{Tk bNj f>λ}(x) a.e. Thus, Fatou’s lemma and (5.14) drive us to the desired estimate for Tk b. This completes the proof of (a). To obtain (b), we proceed as in Remark 5.1. Given ε > 0 we pick p > 1 and δ > 0 so that (k+1) p−1+2 δ=k+ε. Let A(t) = exp(t1 k+δ/p )−1 and B(t) = exp(tp/δ)−1. Note that we have A−1(t)B−1(t)C−1 k(t).t. Also, H∞⊂HBand Het1/k ,k ⊂HA,k (as A(t).et1/k −1 for t≥1). Then K∈HB∩HA,k. We apply (a.2) with D(t) = tp(1 + log+t)(k+1) p−1+2 δand E(t)≈tp0/(1 + log+t)1+δ(p0−1) ∈Bp0(note that we have D−1(t)E−1(t)≤ A−1(t)). Then we obtain the desired estimate for the pair of weights (u, MDpu). To conclude we observe that Dp(t) = D(t1/p) = t(1 + log+t)k+ε. Remark 5.2. The proof for the multilinear commutators follows the same scheme, we just give some of the changes, leaving details to the reader. To estimate II we use ideas from [36] and replace (5.11) by |T~ bh(x)|.X σ1,σ2T~ bσ2X j πσ1(~ b−~ λj)hj(x)+X j |π{1,...,k}(~ b−~ λj)| |Thj(x)| =F1(x) + F2(x), where the first sum runs over all partitions σ1,σ2of {1, . . . , k}with σ16= Ø; T~ bσ2is the multilinear commutator associated with the vector ~ bσ2= (bσ2(l))l;πσ1(~v) = Qlvσ1(l) and π{1,...,k}(~v) = Qk l=1 vl; and ~ λj=(b1)Qj, . . . (bk)Qj. With this in hand we estimate F1using the induction hypothesis as #σ2≤k−1, and we estimate F2using that K∈HB,k (see [23]). The estimate for III is obtained by using [23, Theorem 7.1] and observing that (T~ b)∗= (T∗)−~ band T∗is a singular integral operator with kernel ˜ K(x) = K(−x)∈ HB,k. 6. Proofs in the one-sided case Proof of Theorem 3.12, Part (i).The proof follows the same pattern as the proof in Theorem 3.1. We will only highlight some of the details. Again, we can assume that uis bounded and has compact support, also 0 ≤f∈L∞ c(R). Let Ω = {x∈R:M+f(x)> λ}=[ j Ij=[ j (aj, bj)