Sharp weighted bounds for multilinear maximal functions and Calderón-Zygmund operators

arXiv:1211.5115v1 [math.CA] 21 Nov 2012 SHARP WEIGHTED BOUNDS FOR MULTILINEAR MAXIMAL FUNCTIONS AND CALDER´ ON-ZYGMUND OPERATORS WENDOL´ IN DAMI´ AN, ANDREI K. LERNER, AND CARLOS P´ EREZ Abstract. In this paper we prove some sharp weighted norm inequalities for the multi(sub)linear maximal function Mintroduced in [18] and for multilinear Calder´on-Zygmund operators. In particular we obtain a sharp mixed “Ap−A∞” bound for M, some partial results related to a Buckley-type estimate for M, and a sufficient condition for the boundedness of Mbetween weighted Lpspaces with different weights taking into account the precise bounds. Next we get a bound for multilinear Calder´on-Zygmund operators in terms of dyadic positive multilinear operators in the spirit of the recent work [16]. Then we obtain a multilinear version of the “A2conjecture”. Several open problems are posed. 1. Introduction The modern theory of weighted norm inequalities for some of the main operators in Harmonic Analysis originated in the beginning of the 70’s in the works by R. Hunt, B. Muckenhoupt, R. Wheeden, R. Coifman and C. Fefferman. In particular, it was realized that the key role in this theory is played by the so-called Apcondition. Much later, the question about the sharp dependence of the Lp(w) operator norm in term of the the Apconstant or characteristic of the weight appeared. First, for the Hardy-Littlewood maximal operator this problem was solved by S. Buckley [2]. It turned out that for singular integrals the question is much more complicated. In [21], S. Petermichl solved it for the Hilbert transform. Recently, T. Hyt¨onen [9] gave a complete solution for general Calder´on-Zygmund operators solving the so-called A2conjecture. A bit later this result was improved in [12] in the case p= 2 and for general pin [10]. A further improvement was obtained in [11] where a non-probablistic proof was found together with a q-variation estimate. We refer to these papers for a more detailed history and for some other closely related results like (1.3). The aim of this paper is to give some multilinear analogues of the above mentioned results in the spirit of the theory of multiple weights developed recently in [18]. We introduce some notation. Given ~ f= (f1,...,fm), we define the multi(sub)linear 2010 Mathematics Subject Classification. 42B20, 42B25. Key words and phrases. Multilinear maximal operator, Calder´on-Zygmund theory, sharp weighted bounds. 1 2 W. DAMI ´ AN, A.K. LERNER, AND C. P´ EREZ maximal operator Mby M(~ f)(x) = sup Q∋x m Y i=1 1 |Q|ZQ |fi(yi)|dyi, where the supremum is taken over all cubes Qcontaining x. It is shown in that paper that this operator controls in several ways the class of multilinear Calder´on-Zygmund operators (see section 2.2). A particular instance of this intimate relationship is the class of weights characterizing the weighted Lpspaces for which both operators are bounded. To define this class of weights we let ~w = (w1,...,wm) and ~ P= (p1,...,pm). Set 1 p=1 p1+···+1 pmand ν~w =Qm i=1 wp/pi i. We say that ~w satisfies the A~ Pcondition if [~w]A~ P= sup Q1 |Q|ZQ ν~wm Y i=1 1 |Q|ZQ w1−p′ i ip/p′ i<∞. It is easy to see that in the linear case (that is, if m= 1) [~w]A~ P= [w]Apis the usual Apconstant. In [18] the following multilinear extension of the Muckenhoupt Aptheorem for the maximal function was obtained: the inequality (1.1) kM(~ f)kLp(ν~w)≤C m Y i=1 kfikLpi(wi) holds for every ~ fif and only if ~w satisfies the A~ Pcondition. The first question we are going to study is the question about the sharp dependence of Cin (1.1). In the standard situation, namely when m= 1, two different types of sharp weighted inequalities for the Hardy-Littlewood maximal operator M are known. First, Buckley’s theorem [2] says that for any 1 < p < ∞, (1.2) kMkLp(w)≤Cn,p[w] 1 p−1 Ap. Until very recently this result was thought to be sharp since the exponent of [w]Ap cannot be improved. However, recently T. Hyt¨onen and the third author [12] (see also [13] for another proof) showed that (1.2) can be improved in the following way. Define the A∞constant of wby [w]A∞= sup Q 1 w(Q)ZQ M(wχQ). This definition goes back to the characterization of the A∞class of weights given by N. Fujii [7] (see also M. Wilson [22]). It is not difficult to show that [w]A∞≤cn[w]Ar for any r≥1. It was shown in [12] that (1.3) kMkLp(w)≤Cn,p([w]Ap[σ]A∞)1/p, where σ=w1−p′. This estimate implies (1.2) since [σ]A∞≤c[σ]Ap′=c[w] 1 p−1 Ap, and therefore ([w]Ap[σ]A∞)1/p ≤c[w] 1 p−1 Ap. SHARP WEIGHTED BOUNDS 3 Our multilinear results related to (1.2) and (1.3) are somewhat surprising. They show that contrary to the case m= 1,multilinear versions of (1.2) and (1.3) are independent of each other. We are able to get a full analogue of (1.3). However such an analogue does not yield an expected full analogue of (1.2). Our first main result is the following. Theorem 1.1. Let 1< pi<∞, i = 1,...,m and 1 p=1 p1+...+1 pm. Then the inequality (1.4) kM(~ f)kLp(ν~w)≤Cn,m, ~ P[~w] 1 p A~ P m Y i=1 ([σi]A∞)1 pi m Y i=1 kfikLpi(wi) holds if ~w ∈A~ P, where σi=w1−p′ i i,i= 1,...,m. Furthermore the exponents are sharp in the sense that they cannot be replaced by smaller ones. Even though the result of this theorem is sharp it would be of interest to find an extension of Buckley’s estimate (1.2). However this task seems to be more complicated. Currently we can get only several partial results expressed in the following theorem. Theorem 1.2. Let 1< pi<∞, i = 1,...,m and 1 p=1 p1+...+1 pm. Denote by α=α(p1,...,pm)the best possible power in (1.5) kM(~ f)kLp(ν~w)≤Cn,m,p [~w]α A~ P m Y i=1 kfikLpi(wi). Then we have the following results: (i) for all 1< p1, . . . , pm<∞,m mp−1≤α≤1 p1 + Pm i=1 1 pi−1; (ii) if p1=p2=···=pm=r > 1, then α=m r−1. It is easy to see that the upper and lower bounds for αin (i) coincide if m= 1. The upper bound for αin (i) is a corollary of Theorem 1.1 after an application of Lemma 3.1 (see section 3). However, in the case of (ii) αcoincides with the lower bound in (i). This says that if m≥2, then the upper bound in (i) is not sharp, in general. Hence, contrary to the linear situation, (1.4) cannot be used in order to get a sharp bound in terms of [~w]A~ P. Perhaps, the explanation of this is that the right-hand side of (1.4) involves m+ 1 independent suprema while the definition of [~w]A~ Pinvolves only one supremum or else Lemma 3.1 is not sharp. Resuming, the problem of finding the sharp αin (1.5) remains open except the case considered in (ii). We also give a sufficient condition for the “two-weighted” boundedness of M with precise bounds generalizing the corresponding linear result from [20] and its multilinear counterpart in [19]. Let Xbe a Banach function space. By X′we denote the associate space to X. Given a cube Q, define the X-average of fover Qand 4 W. DAMI ´ AN, A.K. LERNER, AND C. P´ EREZ the maximal operator MXby kfkX,Q =kτℓQ(fχQ)kX, MXf(x) = sup Q∋x kfkX,Q, where ℓQdenotes the side length of Qand where τδf=f(δx), δ > 0, x ∈Rn. Theorem 1.3. Let 1< pi<∞, i = 1,...,m and 1 p=1 p1+...+1 pm. Let Xibe a Banach function space such that MX′ iis bounded on Lpi(Rn). Let uand v1,...,vm be the weights satisfying K= sup Qu(Q) |Q|1 p m Y i=1 kv−1/pi ikXi,Q <∞. Then kM(~ f)kLp(u)≤Cn,mK m Y i=1 kMX′ ikLpi(Rn)kfikLpi(vi). This result can be seen as a two weight version of (1.4) when considering function spaces Xgiven by X=Lrp′for 1 < p, r < ∞so that kMX′kLp(Rn)=kM(rp′)′kLp(Rn)≈(r′)1/p. Another interesting example is given when considering the Orlicz space space X= LBwhere Bis a Young function for which kMX′kLp(Rn)=kM¯ BkLp(Rn)is finite. In particular if B(t) = tp′(log(e+t))p′−1+δ,δ > 0, 1 < p < ∞it follows from [20] that kMX′kLp(Rn)=kM¯ BkLp(Rn)≈(1 δ)1/p. We turn now to some multilinear analogues of the sharp weighted results for singular integrals. First, we recall some linear results. Let Tbe a Calder´on-Zygmund operator. Then it was proved by T. Hyt¨onen [9] in full generality that (1.6) kTkLp(w)≤CT,n,p[w]max(1,1 p−1) Ap(1 < p < ∞). Observe that it suffices to prove (1.6) in the case p= 2; then for any other pthe result follows by the sharp extrapolation theorem. Note also that (1.6) in the case p= 2 was usually referred as the A2conjecture. In two recent papers [16,17] a different proof of (1.6) was found by the second author. This proof shows that there is an intimate relationship between the (continuous) singular integral Tand some very special dyadic type operators. These operators are defined by means of the concept of sparseness. Given a sparse family S={Qk j}of cubes from a dyadic grid D, (these notions will be defined in Section 2.1) we consider the operator AS,D defined by AS,Df(x) = X j,k fQk jχQk j(x). SHARP WEIGHTED BOUNDS 5 It was proved in [16,17] that for any Banach function space X, (1.7) kTfkX≤cT,n sup S,D kAS,D|f|kX. A rather simple argument found in [3] shows that kAS,DkL2(w)≤c[w]A2. Therefore if X=L2(w), then (1.7) implies (1.6) for p= 2. The second main result of this paper is an extension of (1.7) to the multilinear setting. Here, the dyadic operator that appears naturally is a multilinear version of the operator AD,Sgiven by AD,S(~ f)(x) = X j,k m Y i=1 (fi)Qk jχQk j(x). Given ~ f= (f1,...,fm), denote ~ |f|= (|f1|,...,|fm|). Theorem 1.4. Let T(~ f)be a multilinear Calder´on-Zygmund operator and let Xbe a Banach function space over Rnequipped with Lebesgue measure. Then, for any appropriate ~ f, kT(~ f)kX≤cT,m,n sup D,S kAD,S(~ |f|)kX, where the supremum is taken over arbitrary dyadic grids Dand sparse families S ∈ D. Similarly to the linear case, we would like to apply this result when X=Lp(ν~w). However, the exponent pis allowed to be smaller than one, to be more precise 1/m < p < ∞(see section 2.2). Therefore, if 1/m < p < 1, Theorem 1.4 cannot be applied since in this case the space Lp(ν~w) is not a Banach space. This raises an interesting question whether the condition in Theorem 1.4 that Xis a Banach space can be relaxed until that Xis a quasi-Banach space. It is natural to consider first this question in the linear situation. Observe that in the current proof of (1.7) the fact that Xis a Banach space was essential. The third main result of this paper can be seen as a multilinear version of the A2conjecture (1.6). Indeed, one of the main results obtained in [18] is that if ~w ∈A~ P, then an analogue of (1.1) holds with T(~ f) instead of M(~ f). Hence, the question about the sharp dependence on [~w]A~ Pin the corresponding inequality is quite natural. It is interesting that contrary to the linear situation where (1.7) implies the A2conjecture, we are currently able to apply Theorem 1.4 only in one particular case being the content of the following theorem. Theorem 1.5. Let T(~ f)be a multilinear Calder´on-Zygmund operator. Assume that p1=p2=···=pm=m+ 1. Then (1.8) kT(~ f)kLp(ν~w)≤CT,m,n[~w]A~ P m Y i=1 kfikLpi(wi). 6 W. DAMI ´ AN, A.K. LERNER, AND C. P´ EREZ Several remarks about this result are in order. As we already mentioned, Theorem 1.5 can be regarded as a multilinear “A2conjecture”. However, it is natural to ask how to extend it to all 1 < pi<∞. This leads to several interesting problems. As we explained above, (1.6) can be obtained from the case p= 2 by the sharp version of the extrapolation theorem of Rubio de Francia obtained in [5] (see also [6] or [4] for different and simpler proofs). It would be very desirable to get a multilinear analogue of this result. Having such an analogue, inequality (1.8) probably would be a starting point to extrapolate from. Observe, however, that (1.6) can be proved also without the use of extrapolation. Indeed, one can easily prove (1.6) with AD,Sinstead of Tfor all 1 < p < ∞(as it was done in [3] for p= 2), and then apply (1.7) with X=Lp(w). The proof of (1.6) for AD,Sis very close in spirit to the proof of Buckley’s inequality (1.2) found in [14]. Hence, it is natural to ask whether it is possible to find a similar proof for a multilinear version of AD,S. But this leads to a problem of finding αin Theorem 1.2 by the method in [14]. Part (ii) of Theorem 1.2 indeed obtained by an adaptation of this method. However how to do that in the case of different piis not clear. The paper is organized as follows. Some preliminaries are contained in Section 2. In Section 3 we prove all theorems related to M. Section 4 is devoted to the proof of Theorem 1.4. Finally, Theorem 1.5 is proved in Section 5. 2. Preliminaries 2.1. Dyadic grids. Recall that the standard dyadic grid in Rnconsists of the cubes 2−k([0,1)n+j), k ∈Z, j ∈Zn. Denote the standard grid by D. By a general dyadic grid Dwe mean a collection of cubes with the following properties: (i) for any Q∈Dits sidelength ℓQis of the form 2k, k ∈Z; (ii) Q∩R∈ {Q, R, ∅} for any Q, R ∈D; (iii) the cubes of a fixed sidelength 2kform a partition of Rn. We say that {Qk j}is a sparse family of cubes if: (i) the cubes Qk jare disjoint in j, with kfixed; (ii) if Ωk=∪jQk j, then Ωk+1 ⊂Ωk; (iii) |Ωk+1 ∩Qk j| ≤ 1 2|Qk j|. With each sparse family {Qk j}we associate the sets Ek j=Qk j\Ωk+1. Observe that the sets Ek jare pairwise disjoint and |Qk j| ≤ 2|Ek j|. Given a cube Q0, denote by D(Q0) the set of all dyadic cubes with respect to Q0, that is, the cubes from D(Q0) are formed by repeated subdivision of Q0and each of its descendants into 2ncongruent subcubes. Observe that if Q0∈D, then each cube from D(Q0) will also belong to D. We will use the following proposition from [12]. Proposition 2.1. There are 2ndyadic grids Dαsuch that for any cube Q⊂Rn there exists a cube Qα∈Dαsuch that Q⊂Qαand ℓQα≤6ℓQ. SHARP WEIGHTED BOUNDS 7 Lemma 2.2. For any non-negative integrable fi, i = 1,...,m, there exist sparse families Sα∈Dαsuch that for all x∈Rn, M(~ f)(x)≤(2 ·12n)m 2n X α=1 ADα,Sα(~ f)(x). Proof. First, by Proposition 2.1, (2.1) M(~ f)(x)≤6mn 2n X α=1 MDα(~ f)(x). Consider Md(~ f) taken with respect to the standard dyadic grid. We will use exactly the same argument as in the Calder´on-Zygmund decomposition. For cn which will be specified below and for k∈Zconsider the sets Ωk={x∈Rn:Md(~ f)(x)> ck n}. Then we have that Ωk=∪jQk j, where the cubes Qk jare pairwise disjoint with k fixed, and ck n< m Y i=1 (fi)Qk j≤2mnck n. From this and from H¨older’s inequality, |Qk j∩Ωk+1|=X Qk+1 l⊂Qk j |Qk+1 l| < c−k+1 m nX Qk+1 l⊂Qk j m Y i=1 ZQk+1 l fi1/m ≤c−k+1 m n m Y i=1 ZQk j fi1/m ≤2nc−1/m n|Qk j|. Hence, taking cn= 2m(n+1), we obtain that the family {Qk j}is sparse, and Md(~ f)(x)≤2m(n+1)AD,S(~ f)(x). Applying the same argument to each MDα(~ f) and using (2.1), we get the statement of the lemma.  2.2. Multilinear Calder´on-Zygmund operators. Let Tbe a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values into the space of tempered distributions, T:S(Rn)× · · · × S(Rn)→S′(Rn). We say that Tis an m-linear Calder´on-Zygmund operator if, for some 1 ≤qj<∞, it extends to a bounded multilinear operator from Lq1× · · · × Lqmto Lq, where 8 W. DAMI ´ AN, A.K. LERNER, AND C. P´ EREZ 1 q=1 q1+···+1 qm, and if there exists a function K, defined off the diagonal x=y1= ···=ymin (Rn)m+1, satisfying T(f1,...,fm)(x) = Z (Rn)m K(x, y1,...,ym)f1(y1). . . fm(ym)dy1. . . dym for all x6∈ ∩m j=1 supp fj, |K(y0, y1,...,ym)| ≤ A m P k,l=0 |yk−yl|mn , and |K(y0,...,yj, . . . , ym)−K(y0,...,y′ j,...,ym)| ≤ A|yj−y′ j|ǫ m P k,l=0 |yk−yl|mn+ǫ, for some ǫ > 0 and all 0 ≤j≤m, whenever |yj−y′ j| ≤ 1 2max 0≤k≤m|yj−yk|. It was shown in [8] that if 1 r1+···+1 rm=1 r, then an m-linear Calder´on-Zygmund operator satisfies T:Lr1(Rn)× · · · × Lrm(Rn)→Lr(Rn) when 1 < rj<∞for all j= 1,···, m. Similarly if 1 ≤rj≤ ∞ for all j= 1,···, m, we have (2.2) T:L1(Rn)× · · · × L1(Rn)→L1/m,∞(Rn) 2.3. A “local mean oscillation decomposition”. The non-increasing rearrangement of a measurable function fon Rnis defined by f∗(t) = inf{α > 0 : |{x∈Rn:|f(x)|< α}| < t}(0 < t < ∞). Given a measurable function fon Rnand a cube Q, the local mean oscillation of fon Qis defined by ωλ(f;Q) = inf c∈R(f−c)χQ∗λ|Q|(0 < λ < 1). By a median value of fover Qwe mean a possibly nonunique, real number mf(Q) such that max |{x∈Q:f(x)> mf(Q)}|,|{x∈Q:f(x)< mf(Q)}|≤ |Q|/2. It is easy to see that the set of all median values of fis either one point or the closed interval. In the latter case we will assume for the definiteness that mf(Q) is the maximal median value. Observe that it follows from the definitions that (2.3) |mf(Q)| ≤ (fχQ)∗(|Q|/2). SHARP WEIGHTED BOUNDS 9 Given a cube Q0, the dyadic local sharp maximal function m#,d λ;Q0fis defined by m#,d λ;Q0f(x) = sup x∈Q′∈D(Q0) ωλ(f;Q′). The following theorem was proved in [16] (its very similar version can be found in [15]). Theorem 2.3. Let fbe a measurable function on Rnand let Q0be a fixed cube. Then there exists a (possibly empty) sparse family of cubes Qk j∈ D(Q0)such that for a.e. x∈Q0, |f(x)−mf(Q0)| ≤ 4m#,d 1 2n+2 ;Q0f(x) + 2 X k,j ω1 2n+2 (f;Qk j)χQk j(x). 2.4. Banach function spaces. For a general account of Banach function spaces we refer to [1, Ch. 1]. We mention only several notions which will be used below. The associate space X′consists of measurable functions ffor which kfkX′= sup kgkX≤1ZRn |f(x)g(x)|dx < ∞. This definition implies the following H¨older inequality: (2.4) ZRn |f(x)g(x)|dx ≤ kfkXkgkX′. Further [1, p. 13], (2.5) kfkX= sup kgkX′=1 ZRn |f(x)g(x)|dx. 3. Proof of Theorems 1.1,1.2 and 1.3 In the proof of Theorem 1.1 we shall use the following reverse H¨older property of A∞weights proved in [12]: if w∈A∞, then (3.1) 1 |Q|ZQ wr(w)1/r(w) ≤21 |Q|ZQ w, where r(w) = 1 + 1 τn[w]A∞ and τn= 211+n. Observe that r′(w)≈[w]A∞. Proof of Theorem 1.1.By (2.1), it suffices to prove the theorem for the dyadic maximal operators MDα. Since the proof is independent of the particular dyadic grid, without loss of generality we consider Mdtaken with respect to the standard dyadic grid D. Let a= 2m(n+1). and Ωk={x∈Rn:Md(~ f)(x)> ak}. We have seen in the proof of Lemma 2.2 that Ωk=∪jQk j, where the family {Qk j}is sparse and 16 W. DAMI ´ AN, A.K. LERNER, AND C. P´ EREZ We shall also need the following auxiliary operator Ml,α(~ f1,...,m−1, g)(x) = X j,k m−1 Y i=1 (fi)Pl,α j,k  1 |Pl,α j,k |ZQk j gχPl,α j,k (x). This object appears naturally in the following duality relation: (4.5) ZRn Tl,α(~ f)g dx =ZRn Ml,α(~ f1,...,m−1, g)fmdx. Lemma 4.2. For any cube Q∈Dα, ωλ(Ml,α(~ f1,...,m−1, g); Q)≤c(λ, m, n)lgQ m−1 Y i=1 (fi)Q. Proof. Let Q∈Dαand let x∈Q. We have Ml,α(~ f1,...,m−1, g)(x) = X k,j:Pm,α j,k ⊂Q +X k,j:Q⊆Pm,α j,k . The second sum is a constant (denote it by c) for x∈Q, while the first sum involves only the functions fiwhich are supported in Q. We get the following simple estimate: (4.6) |Ml,α(~ f1,...,m−1, g)−c|χQ(x)≤ m−1 Y i=1 M(fiχQ)(x)Tl(gχQ)(x), where Tlg(x) = X j,k 1 |Pm,α j,k |ZQk j gχPm,α j,k (x). It was proved in [17, Lemma 3.2] that kTlgkL1,∞≤c(n)lkgkL1. Using this estimate, the weak type (1,1) of M, and reiterating the well known property of rearrangements, (fg)∗(t)≤f∗(t/2)g∗(t/2), t > 0, we get using (4.6) ωλ(Ml,α(~ f1,...,m−1, g); Q)≤ m−1 Y i=1 M(fiχQ)Tl(gχQ)∗λ|Q| ≤ m−1 Y i=1 M(fiχQ)∗λ|Q|/2iTl(gχQ)∗λ|Q|/2m−1 ≤c(λ, m, n)lgQ m−1 Y i=1 (fi)Q, which completes the proof.  SHARP WEIGHTED BOUNDS 17 4.2. Proof of (4.3). By (4.4) it is enough to prove (4.3) with Tl,α(~ f), for each α= 1,...,2n, instead of TS,l(~ f) on the left-hand side. By the standard limiting argument one can assume that the sum defining Tl,α(~ f) is finite. Then the sum defining the corresponding operator Ml,α(~ f1,...,m−1, g) in (4.5) will be finite too. This means that the support of Ml,α(~ f1,...,m−1, g) is compact. One can cover it by at most 2ncubes Qν∈Dαsuch that mMl,α(~ f1,...,m−1,g)(Qν) = 0, ν = 1,···,2n. Applying Theorem 2.3 along with Lemma 4.2, we get that there exists a sparse family Sα∈Dα(Qν) such that for a.e. x∈Qν, Ml,α(~ f1,...,m−1, g)(x) ≤c(m, n)lMDα(~ f1,...,m−1, g)(x) + X Qk j∈Sαm−1 Y i=1 (fi)Qk jgQk jχQk j(x), where MDα(~ f1,...,m−1, g)(x) = sup x∈Q∈Dα m−1 Y i=1 (fi)QgQ. Applying to this maximal operator the same argument as in the proof of Lemma 2.2 and combining with the previous estimate, we get that there exists two sparse families Sα,1and Sα,2from Dαsuch that for a.e. x∈Qν, Ml,α(~ f1,...,m−1, g)(x)≤c(m, n)l 2 X κ=1 X Qk j∈Sα,κ m−1 Y i=1 (fi)Qk jgQk jχQk j(x). Hence, by H¨older’s inequality (2.4), ZQν Ml,α(~ f1,...,m−1, g)fmdx ≤c(m, n)l 2 X κ=1 ZRn ADα,Sα,κ (~ f)g dx ≤2c(m, n)lsup D,S kAD,S(~ f)kXkgkX′. Summing up over Qνand using (4.5), we get ZRn Tl,α(~ f)g dx ≤2n+1c(m, n)lsup D,S kAD,S(~ f)kXkgkX′. By (2.5), taking here the supremum over g≥0 with kgkX′= 1 gives (4.3) for Tl,α(~ f), and therefore the proof is complete. 18 W. DAMI ´ AN, A.K. LERNER, AND C. P´ EREZ 5. Proof of Theorem 1.5 First, we apply Theorem 1.4 with X=Lp(ν~w) (observe that p=m+1 m). Fix S ∈ D. Assume that fi≥0. By duality kAD,S(~ f)kLp(ν~w)= sup kgk Lp′ν−1/(p−1) ~w =1 X j,k m Y i=1 (fi)Qk jZQk j g. Observe that by our choice of piwe have p/p′ i= 1. Denote A~ P(~w;Q) = 1 |Q|ZQ ν~wm Y i=1 1 |Q|ZQ σi. We have X j,k m Y i=1 (fi)Qk jZQk j g =X j,k A~ P(~w;Qk j)m Y i=1 1 σi(Qk j)ZQk j fi 1 ν~w(Qk j)ZQk j g|Qk j| ≤2[~w]A~ PX j,k ZEk j m Y i=1 MD σi(fiσ−1 i)MD ν~w (gν−1 ~w )dx ≤2[~w]A~ PZRn m Y i=1 MD σi(fiσ−1 i)MD ν~w (gν−1 ~w )dx. Now we apply H¨older’s inequality: ZRn m Y i=1 MD σi(fiσ−1 i)MD ν~w (gν−1 ~w )dx ≤ k m Y i=1 MD σi(fiσ−1 i)kLp(ν−(p−1) ~w )kMD ν~w (gν−1 ~w )kLp′(ν~w). First, kMD ν~w (gν−1 ~w )kLp′(ν~w)≤ckgkLp′ν−1/(p−1) ~w =c. 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Lerner, Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel E-mail address:[email protected] Carlos P´ erez, Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Sevilla, 41080 Sevilla, Spain E-mail address:[email protected]