New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory

Advances in Mathematics, V. 220, (2009) 1222-1264. NEW MAXIMAL FUNCTIONS AND MULTIPLE WEIGHTS FOR THE MULTILINEAR CALDER ´ ON-ZYGMUND THEORY ANDREI K. LERNER, SHELDY OMBROSI, CARLOS P´ EREZ, RODOLFO H. TORRES, AND RODRIGO TRUJILLO-GONZ´ ALEZ Abstract. A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller that the m-fold product of the Hardy-Littlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of Calder´on-Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calder´on-Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators. 1. Introduction The groundbreaking work of Calder´on and Zygmund in the 50’s [3] is the basis for what is today named after them Calder´on-Zygmund theory. Their initial work on operators given by convolution with singular kernels was motivated by connections with potential theory and elliptic partial differential equations, and by the need to study operators which are higher-dimension analogs of the classical Hilbert transform. The tools developed over the years to deal with these and related problem in Rnform the core of what are nowadays called real-variable techniques. The theory has had quite a success in the solution of many problems in both real and complex analysis, operator theory, approximation theory, and partial differential equations. This success is, in part, a consequence of the broad extension of the methods employed to different geometrical and multivariable contexts, which include homogeneous and non-homogeneous spaces, and multiparameter, non-linear, and multiliear settings. We refer to Coifman-Meyer [12], Christ [6], Fefferman [20], Stein [48], 2000 Mathematics Subject Classification. 42B20, 42B25. Key words and phrases. Multilinear singular integrals, Calder´on-Zygmund theory, maximal operators, weighted norm inequalities, commutators. A.K. Lerner’s research supported by the Spanish Ministry of Education under the programm “Programa Ram´on y Cajal, 2006”. S, Ombrosi’s research supported by a fellowship from the same institution. These two authors and C. P´erez are also supported by the same institution under research grant MTM2006-05622. R. Trujillo-Gonzlez is also supported by the the same institution under grant MTM2005-07347. R.H. Torres’ research supported in part by the National Science Foundation under grants OISE 0126272 and DMS 0400423. 1 2 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ Grafakos-Torres [27] and Volberg [49] for surveys and historical details about these different aspects of the subject. Adapting the methods of the Calder´on-Zygmund theory to each different context is, however, not always immediate. The theory provides a blueprint for the kind of results to be expected but, typically, the general approach needs to be complemented with the development of tools intrinsic to each particular new situation being faced. In particular, it is of relevance in each application to identify appropriate maximal functions that control in various ways many operators and functionals quantities that need to be estimated. As we will describe in this article, this is also the case for the multilinear Calder´on-Zygmund theory. A collection of maximal functions that we will introduce will give us a way to obtain several sharp bounds for multilinear Calder´onZygmund operators and their commutators. The multilinear version of the Calder´on-Zygmund theory originated in the works of Coifman and Meyer in the 70’s, see e.g. [10], [11], and it was oriented towards the study of the Calder´on commutator. Later on the topic was retaken by several authors; including Christ and Journ´e [8], Kenig and Stein [33], and Grafakos and Torres [25]. This last work provides a comprehensive approach to general multilinear Calder´onZygmund operators that we will follow in this article. As it is well-known, linear Calder´on-Zygmund operators map Lpinto itself for 1 < p < ∞, with an L1→L1,∞estimate as one end-point and an L∞→BMO estimate as the other. It is then natural that the first results obtained for multilinear Calder´onZygmund operators (see Section 2 below for technical definitions) were of the form Lp×Lq→Lr, with 1 < p, q, r < ∞satisfying the H¨older relation 1/p + 1/q = 1/r. The fact that positive results also hold for r > 1/2 was somehow overlooked until Lacey and Thiele obtained their boundedness results for the bilinear Hilbert transform [35], [36]. The bilinear Hilbert transform is an operator far more singular than the bilinear Calder´on-Zygmund operators and yet it satisfies bounds for r > 2/3 (it is not known yet whether the bounds are also true for r > 1/2). It was then shown in [33] and [25] that the full range r > 1/2 is achieved for bilinear Calder´on-Zygmund operators, with an end-point estimate of the form L1×L1→L1/2,∞. (An m-linear version also holds; see (2.5) below.) Weighted estimates and commutators in this multilinear setting were then studied in [26] and [45]. These works opened up some new problems that we resolve in this article. The first set of problems that we consider relates directly to multilinear singular integrals. It was shown in [26] that if Tis an m-linear Calder´on-Zygmund operator, then T(f1,· · · , fm) is controlled in terms of Lp-norms by Qm j=1 Mfj, where Mis the usual Hardy-Littlewood maximal operator. As a consequence, it was deduced that if 1 p1+· · ·+1 pm=1 pand p0= min{pj}>1, then Tis a bounded from Lp1(w)×· · ·×Lpm(w) into Lp(w), provided that the weight wis in the class Ap0. It is a simple observation that the same approach shows that (1.1) T:Lp1(w1)× · · · × Lpm(wm)→Lp(ν), MAXIMAL FUNCTIONS AND MULTILINEAR CALDER ´ ON-ZYGMUND OPERATORS 3 where ν=Qm j=1 wp/pj jand wjis in Apj. Such weights νwere used in [24] to obtain multilinear extrapolation results. Nevertheless, the question of the existence of a multiple weight theory was posed in [27], and it has been since then an open problem whether the control of Tby Qm j=1 Mfjis optimal and whether the conditions on wjfor which (1.1) holds cannot be improved. In this article we answer these questions by studying a multi(sub)linear maximal function Mdefined by M(~ f)(x) = sup x∈Q m Y i=1 1 |Q|ZQ |fi(yi)|dyi. This operator is strictly smaller than the m-fold product of M. We develop the corresponding theory of weights for this new maximal function which, in turn, gives the right class of multiple weights for m-linear Calder´on-Zygmund operators. We use some analogous tools to study a second set of problems related now to multilinear versions of the commutators of Coifman, Rochberg and Weiss [13]. We recall that the operators introduced in [13] are definined by [b, T]f=b T(f)−T(bf), where bis a locally integrable function in Rn, usually called the symbol, and Tis a Calder´onZygmund singular integral. The original interest in the study of such operators was related to generalizations of the classical factorization theorem for Hardy spaces. Further applications have then been found in partial differential equations [4] [5] [17] [28]. Recently multiparameter versions have also received renewed attention; see e.g. [21] and [34]. The main result from [13] states that if bis in BMO, then [b, T] is a bounded operator on Lp(Rn), 1 <p<∞. In fact, the BMO membership of bis also a necessary condition for the Lp-boundedness of the commutator when, for example, T=H, the Hilbert transform. An interesting fact is that, unlike what it is done with singular integral operators, the proof of the Lp-boundedness of the commutator does not rely on a weak type (1,1) inequality. In fact, simple examples show that in general [b, T] fails to be of weak type (1,1) when b∈BMO. Instead, it was proved by P´erez [41] that a weak-L(log L) type estimate holds (see (3.15) below). Given a collection of locally integrable functions ~ b= (b1, . . . , bm), we define the m-linear commutator of ~ band the m-linear Calder´on-Zygmund operator Tto be (1.2) T~ b(f1,· · · , fm) = m X j=1 Tj ~ b(~ f), where each term is the commutator of bjand Tin the j-th entry of T, that is, Tj ~ b(~ f) = bjT(f1,· · · , fj,· · · , fm)−T(f1,· · · , bjfj,· · · , fm).1 1We chose this definition to follow [45] and for symmetry and simplicity in some statements, but for most estimates it will be enough to consider only one term Tj ~ b(~ f). On the other hand, we shall see that in the end-point result the presence of just one symbol produces, surprisingly, the appearance of non-linear functional estimates in all the entries of T. 4 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ This definition coincides with the linear commutator [b, T] when m= 1. The mlinear commutators were considered by P´erez and Torres in [45]. They proved that if ~ b∈(BMO)m, 1 <p<∞, and p1, p2,· · · , pmare such that 1 p1+· · · +1 pm=1 p, then T~ b:Lp1× · · · × Lpm−→ Lp. Observe that a crucial condition p > 1 was assumed in this result. The restriction arose in [45] because of the method used which as in the linear case [13], rely on strong Apestimates (hence limiting the approach to p > 1). The experience with the linear commutators and multilinear Calder´on-Zygmund operators, however, suggests that the optimal range should be 1/m<p<∞. We will see in this article that this is in fact the case. Moreover, the question of the existence of an end-point result along the lines of the work [41] was also stated in [45], and we find an answer involving an appropriate weak-L(log L) estimate when p= 1/m. The bounds that we obtain hold also for the new multiple weights and we achieve them by using yet other new maximal functions, Mi L(log L),i= 1, . . . , m, and ML(log L), defined by the expressions Mi L(log L)(~ f)(x) = sup Q3x kfikL(log L),Q Y j6=i 1 |Q|ZQ fjdx and ML(log L)(~ f)(x) = sup Q3x m Y j=1 kfjkL(log L),Q. (See Section 2 below for more details about the norm k·kL(log L),Q.) Observe that ML(log L)is bigger than Mreflecting the presence of the BMO symbols. One can see that ML(log L)(~ f) is pointwise controlled by a multiple of Qm j=1 M2fj(x), but this product is too big to derive the sharp weighted estimate for commutators that we are interested in. That is why we use instead ML(log L). The article is organized as follows. Section 2 contains some basic definitions and facts concerning multilinear singular integrals, weights, sharp maximal functions, and Orlicz spaces needed throughout the rest of this work. The reader familiar with the subject, however, may skip directly to Section 3 where all the theorems are stated. The proofs of the results involving Mand multilinear Calder´on-Zygmund operators are presented in Section 4, while Section 5 contains the pointwise and strong-type estimates for Mi L(log L)and the commutators. The proof of the end-point estimate for Mi L(log L)and the commutators are postponed until Section 6. Various examples (and counterexamples) are collected in Section 7. 2. Preliminaries 2.1. 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)→S0(Rn). MAXIMAL FUNCTIONS AND MULTILINEAR CALDER ´ ON-ZYGMUND OPERATORS 5 Following [25], 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 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 x /∈ ∩m j=1supp fj; (2.1) |K(y0, y1, . . . , ym)| ≤ A m P k,l=0 |yk−yl|mn ; and (2.2) |K(y0, . . . , yj, . . . , ym)−K(y0, . . . , y0 j, . . . , ym)| ≤ A|yj−y0 j|ε m P k,l=0 |yk−yl|mn+ε, for some ε > 0 and all 0 ≤j≤m, whenever |yj−y0 j| ≤ 1 2max0≤k≤m|yj−yk|. It was shown in [25] that if 1 r1+· · · +1 rm=1 r, then an m-linear Calder´on-Zygmund operator satisfies (2.3) T:Lr1× · · · × Lrm→Lr when 1 < rj<∞for all j= 1,· · · , m; and (2.4) T:Lr1× · · · × Lrm→Lr,∞, when 1 ≤rj<∞for all j= 1,· · · , m, and at least one rj= 1. In particular, (2.5) T:L1× · · · × L1→L1/m,∞ 2.2. Weights. By a weight we mean a non-negative measurable function. We recall that a weight wbelongs to the class Ap, 1 <p<∞, if sup Q1 |Q|ZQ w(y)dy 1 |Q|ZQ w(y)1−p0dyp−1 <∞. This number is called the Apconstant of w. A weight wbelongs to the class A1if there is a constant Csuch that 1 |Q|ZQ w(y)dy ≤Cinf Qw, and the infimum of these constants Cis called the A1constant of w. Since the Ap classes are increasing with respect to p, the A∞class of weights is defined in a natural way by A∞=∪p>1Apand the A∞constant of w∈A∞is the smallest of the infimum of the Apconstant such that w∈Ap. 6 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ A well-known result obtained by Muckenhoupt [40] is that the Hardy-Littlewood maximal function, Mf(x) = sup Q3x 1 QZQ |f(y)|dy, satisfies M:Lp(w)→Lp(w) if and only if wis in Ap(see [38] for a new simple proof which yields the sharp Apconstant). He also obtained a characterization of the weak-type inequalities for M. Namely, M:Lp(w)→Lp,∞(ν) if and only if (2.6) sup Q1 |Q|ZQ ν(y)dy 1 |Q|ZQ w(y)1−p0dyp−1 <∞. We will also need some results about one-sided weights. Given an interval I= [a, b], we denote I+= [b, 2b−a]. A weight wis said to belong to the A+ pcondition if sup I1 |I|ZI w(x)dx1 |I|ZI+ w(x)−1/(p−1)dxp−1<∞. It is a known fact in the theory of one-sided weights (see, e.g., [39]) that if wsatisfies the A+ pcondition, then there exists a constant csuch that for any interval I, (2.7) w(I)≤cw(I+). 2.3. Sharp maximal operators. For δ > 0, let Mδbe the maximal function Mδf(x) = M(|f|δ)1/δ(x) = sup Q3x 1 |Q|ZQ |f(y)|δdy1/δ . Also, let M#be the usual sharp maximal function of Fefferman and Stein [19], M#(f)(x) = sup Q3x inf c 1 |Q|ZQ |f(y)−c|dy ≈sup Q3x 1 |Q|ZQ |f(y)−fQ|dy, where as usual fQ=1 |Q|RQf(y)dy denotes the average of fover Q. We will use the following form of the classical result of Fefferman and Stein [19]. See also [32]. Let 0 < p, δ < ∞and let wbe a weight in A∞. Then, there exists C > 0 (depending on the A∞constant of w), such that (2.8) ZRn (Mδf(x))pw(x)dx ≤CZRn (M# δf(x))pw(x)dx, for all function ffor which the left hand side is finite. Similarly, if ϕ: (0,∞)→(0,∞) is doubling, then there exists a constant c(depending on the A∞constant of wand the doubling condition of ϕ) such that (2.9) sup λ>0 ϕ(λ)w({y∈Rn:Mδf(y)> λ})≤csup λ>0 ϕ(λ)w({y∈Rn:M# δf(y)> λ}) for every function fsuch that the left hand side is finite. Extension of these estimates for a large class of spaces can be found in [16]. MAXIMAL FUNCTIONS AND MULTILINEAR CALDER ´ ON-ZYGMUND OPERATORS 7 2.4. Orlicz spaces and normalized measures. We need some basic facts from the theory of Orlicz spaces that we will state without proof. For more information and a lively exposition about these spaces the reader may consult the book by Wilson [50] or [46]. Let Φ : [0,∞)→[0,∞) be a Young function. That is, a continuous, convex, increasing function with Φ(0) = 0 and such that Φ(t)→ ∞ as t→ ∞. The Orlicz space with respect to the measure µ,LΦ(µ), is defined to be the set of measurable functions fsuch that for some λ > 0, ZRn Φ|f(x)| λdµ < ∞. The space LΦis a Banach space when endowed with the Luxemburg norm kfkΦ=kfkLΦ= inf nλ > 0 : ZRn Φ|f(x)| λdµ ≤1o. The Φ-average of a function fover a cube Qis defined to be LΦ(µ) with µthe normalized measure of the cube Qand it is denoted by kfkΦ,Q. That is, kfkΦ,Q = inf{λ > 0 : 1 |Q|ZQ Φ|f(x)| λdx ≤1}. It is a simple but important observation that kfkΦ,Q >1 if and only if 1 |Q|ZQ Φ (|f(x)|)dx > 1. Another useful observation is that if Φ1and Φ2are two Young functions with Φ1(t)≤ Φ2(t), for t≥t0>0, then (2.10) kfkΦ1,Q ≤CkfkΦ2,Q, which can be seen as a generalized Jensen’s inequality. Associated to each Young function Φ, one can defined a complementary function (2.11) ¯ Φ(s) = sup t>0 {st −Φ(t)}. Such ¯ Φ is also a Young function and the ¯ Φ-averages it defines are related to the LΦaverages via a the generalized H¨older’s inequality. Namely, (2.12) 1 |Q|ZQ |f(x)g(x)|dx ≤2kfkΦ,Q kgk¯ Φ,Q. A particular case of interest, an especially in this paper, are the Young functions Φ(t) = t(1 + log+t) and Ψ(t) = et−1, defining the classical Zygmund spaces L(log L), and exp Lrespectively. The corresponding averages will be denoted by k·kΦ,Q =k·kL(log L),Q and k·kΨ,Q =k·kexp L,Q. 8 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ Observe that the above function Φ is submultiplicative. That is, for s, t > 0 Φ(st)≤Φ(s) Φ(t). This submultiplicative property will be used several times in this article. A computation shows that the complementary function of Ψ defined by (2.11) satisfies ¯ Ψ(t)≤Φ(t), and so from the generalized H¨older inequality (2.12) and (2.10) we also get (2.13) 1 |Q|ZQ |f(x)g(x)|dx ≤Ckfkexp L,Q kgkL(log L),Q. This inequality allows to write the following formula that will be used in this article: (2.14) 1 |Q|ZQ |b(y)−bQ|f(y)dy ≤CkbkBMOkfkL(log L),Q. for any function b∈BMO and any non negative function f. This inequality follows from (2.13) and the John-Nirenberg inequality [31] for BMO functions: there are dimensional positive constants c1<1 and c2>2 such that 1 |Q|ZQ exp(c1|b(y)−bQ| kbkBMO )dy ≤c2 which easily implies that for appropriate constant c > 0 kb−bQkexp L,Q ≤ckbkBMO. In view of this result and its applications it is natural to define as in [42] a maximal operator ML(log L)f(x) = sup Q3x kfkL(log L),Q, where the supremum is taken over all the cubes containing x. (Other equivalent definitions can be found in the literature.) We will also use the pointwise equivalence (2.15) ML(log L)f(x)≈M2f(x). This equivalence was obtained in [43] using Stein’s lemma [47] (see [16] for a different argument) and it is shown in [41] the relationship with linear commutators. Finally, we will employ several times the following simple Kolmogorov inequality. Let 0 < p < q < ∞, then there is a constant C=Cp,q such that for any measurable function f (2.16) kfkLp(Q, dx |Q|)≤CkfkLq,∞(Q, dx |Q|). MAXIMAL FUNCTIONS AND MULTILINEAR CALDER ´ ON-ZYGMUND OPERATORS 9 3. Main results 3.1. The key pointwise estimate. Definition 3.1. Given ~ f= (f1, . . . , fm), we define the maximal operator Mby M(~ f)(x) = sup Q3x m Y i=1 1 |Q|ZQ |fi(yi)|dyi, where the supremum is taken over all cubes Qcontaining x. With some abuse, will refer to Mas a multilinear maximal function, even though it is obviously only sublinear in each entry. The main result connecting multilinear Calder´on-Zygmund operators and this multilinear maximal function is the following. Theorem 3.2. Let Tbe an m-linear Calder´on-Zygmund operator and let δ > 0such that δ < 1/m. Then for all ~ fin any product of Lqj(Rn)spaces, with 1≤qj<∞, (3.1) M# δ(T(~ f))(x)≤CM(~ f)(x). The linear version of this estimate can be found in [1] (see also [30] for a earlier result related to (3.1)). We note that (3.1) improves the inequality (3.2) M# δ(T(~ f))(x)≤C m Y j=1 Mfj(x) obtained in [45]. Since Mis trivially controlled by the m-fold product of M, (3.1) can be used to recover all the weighted estimates results of [26] and [45] that follow from (3.2). The point here, however, is that (3.1) opens up the possibility of considering more general weights. We exploit this possibility in our next result. 3.2. Weighted estimates for the multilinear maximal function. We investigate the boundedness properties of Mon various weighted spaces. Theorem 3.3. Let 1≤pj<∞, j = 1, . . . , m and 1 p=1 p1+· · · +1 pm. Let νand wjbe weights. Then the inequality (3.3) kM(~ f)kLp,∞(ν)≤c m Y j=1 kfjkLpj(wj) holds for any ~ fif and only if (3.4) sup Q1 |Q|ZQ ν1/p m Y j=1 1 |Q|ZQ w1−p0 j j1/p0 j<∞, where 1 |Q|RQw1−p0 j j1/p0 jin the case pj= 1 is understood as (inf Qwj)−1. 16 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ for all ~ f= (f1, .., fm)bounded with compact support. 4. Proofs of the results for Mand Calder´ on-Zygmund operators Proof of Theorem 3.2. We will use ideas from [26], [37], and [45], although with some modifications. Fix a point xand a cube Qcontaining x. As is well-known, to obtain (3.1) it suffices to prove for 0 < δ < 1 m (4.1) 1 |Q|ZQ|T(~ f)(z)|δ− |cQ|δdz1/δ ≤CM(~ f)(x), for some constant cQto be determined. In fact we will show, using ||α|r−|β|r| ≤ |α−β|r, 0< r < 1, that (4.2) 1 |Q|ZQ |T(~ f)(z)−cQ|δdz1/δ ≤CM(~ f)(x). Let fj=f0 j+f∞ j, where f0 j=fjχQ∗, j = 1, . . . , m, where Q∗= 3Q. Then m Y j=1 fj(yj) = m Y j=1f0 j(yj) + f∞ j(yj) =X α1,...,αm∈{0,∞} fα1 1(y1). . . fαm m(ym) = m Y j=1 f0 j+X0fα1 1(y1). . . fαm m(ym), where each term of P0contains at least one αj6= 0. Write then (4.3) T(~ f)(z) = T(~ f0)(z) + X0T(fα1 1,· · · , fαm m)(z). Applying Kolmogorov’s inequality (2.16) to the term T(~ f0(z)) = T(f0 1, . . . , f0 m)(z) with p=δand q= 1/m, we derive 1 |Q|ZQ |T(~ f0(z)|δdz1/δ ≤Cm,δkT(~ f0(z)kL1/m,∞(Q, dx |Q|) ≤C m Y j=1 1 |3Q|Z3Q |fj(z)|dz ≤CM(~ f)(x), since T:L1× · · · × L1→L1/m. In order to study the other terms in (4.3), we set now c=X0T(fα1 1, . . . , fαm m)(x), MAXIMAL FUNCTIONS AND MULTILINEAR CALDER ´ ON-ZYGMUND OPERATORS 17 and we will show that, for any z∈Q, we also get an estimate of the form (4.4) X0|T(fα1 1,· · · , fαm m)(z)−T(fα1 1,· · · , fαm m)(x)| ≤ CM(~ f)(x). Consider first the case when α1=· · · =αm=∞and define T(~ f∞) = T(f∞ 1,· · · , f∞ m). By the regularity condition (2.2), for any z∈Qwe obtain |T(~ f∞)(z)−T(~ f∞)(x)| ≤CZ (Rn\3Q)m |x−z|ε (|z−y1|+· · · +|z−ym)nm+ε m Y i=1 |fi(yi)|d~y ≤C ∞ X k=1 Z (3k+1Q)m\(3kQ)m |x−z|ε (|z−y1|+· · · +|z−ym)nm+ε m Y i=1 |fi(yi)|d~y ≤C ∞ X k=1 |Q|ε/n (3k|Q|1/n)nm+εZ(3k+1Q)m m Y i=1 |fi(yi)|d~y ≤C ∞ X k=1 1 3kε m Y i=1 |fi|3k+1Q≤CM(~ f)(x) (here we have used the notation Em=E× · · · × Eand d~y =dy1. . . dym). What remains to be considered are the terms in (4.4) such that αj1=· · · =αjl= 0 for some {j1, . . . , jl}⊂{1, . . . , m}and 1 ≤l < m. By (2.2), |T(fα1 1, . . . , fαm m)(z)−T(fα1 1, . . . , fαm m)(x)| ≤Y j∈{j1,...,jl}Z3Q |fj|dyjZ (Rn\3Q)m−l |x−z|εQj6∈{j1,...,jl}|fj|dyj (|z−y1|+· · · +|z−ym)nm+ε ≤Y j∈{j1,...,jl}Z3Q |fj|dyj ∞ X k=1 |Q|ε/n (3k|Q|1/n)nm+εZ (3k+1Q)m−lY j6∈{j1,...,jl} |fj|dyj ≤C ∞ X k=1 |Q|ε/n (3k|Q|1/n)nm+εZ(3k+1Q)m m Y i=1 |fi(yi)|d~y, and we arrived at the expression considered in the previous case. This gives (4.4) and concludes the proof of the theorem.  Proof of Theorem 3.3. The proof is very similar to the one in the linear situation (see [40]). We consider only the case when pj>1 for all j= 1, . . . , m. Minor modifications for the case of some pj= 1 can be done exactly as in the linear situation. Suppose that (3.3) holds. 18 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ Then for any ~ fwe clearly get (4.5) ZQ ν1/p m Y j=1 |fj|Q≤C m Y j=1 kfjχQkLpj(wj). Setting here fj=w−1/(pj−1) j, we obtain (3.4). Assume now that (3.4) holds. Then, by H¨older’s inequality we obtain (4.5). It follows easily from (4.5) that M(~ f)(x)≤C m Y j=1 Mc ν(|fj|pjwj/ν)(x)1/pj, where Mc νdenotes the weighted centered maximal function. From this, using the wellknown fact (based on the Besicovitch covering theorem) that Mc νis of weak type (1,1) with respect to ν, and the H¨older inequality for weak spaces (see, [23, p.15]), we obtain kM(~ f)kLp,∞(ν)≤Ck m Y j=1 Mc ν(|fj|pjwj/ν)1/pjkLp,∞(ν) ≤C m Y j=1 kMc ν(|fj|pjwj/ν)1/pjkLpj,∞(ν) =C m Y j=1 kMc ν(|fj|pjwj/ν)k1/pj L1,∞(ν) ≤C m Y j=1 kfjkLpj(wj). The theorem is proved.  Proof of Theorem 3.6. Consider first the case when there exists at least one pj>1. Without loss of generality we can assume that p1, . . . , pl= 1,0≤l < m, and pj>1 for j=l+ 1, . . . , m. Suppose that ~w satisfies the multilinear A~ Pcondition. Fix j≥l+ 1 and define the numbers qj=pm−1 + 1 pjand qi=pi pi−1 qj p, i 6=j, i ≥l+ 1. We first prove that w1−p0 j j∈Amp0 jfor j≥l+ 1, i.e., (4.6) ZQ w−1/(pj−1) jZQ w p pjqj j qjpj p(pj−1) ≤c|Q| mpj pj−1. MAXIMAL FUNCTIONS AND MULTILINEAR CALDER ´ ON-ZYGMUND OPERATORS 19 Since m X i=l+1 1 qi =1 m−1+1/pj1 p+ m X i=l+1,i6=j (1 −1/pi)= 1, applying the H¨older inequality, we obtain ZQ w p pjqj j=ZQm Y i=l+1 w p piqj i m Y i=l+1,i6=j w−p piqj i ≤ZQ m Y i=l+1 wp/pi i1/qjm Y i=l+1,i6=jZQ w−1/(pi−1) i1/qi. From this inequality and the A~ Pcondition we easily get (4.6). Next we show that ν~w ∈Amp. Setting sj= (m−1/p)p0 j,j≥l+ 1, we have Pm j=l+1 1 sj= 1 and, therefore, by H¨older’s inequality, (4.7) ZQ m Y j=l+1 w−p pj(pm−1) j≤ m Y j=l+1 ZQ w−1/(pj−1) j1/sj. Hence, ZQ (ν~w)−1 pm−1≤ l Y j=1 (inf Qwj)−p pm−1 m Y j=l+1 ZQ w−1/(pj−1) j1/sj. Combining this inequality with the A~ Pcondition gives ν~w ∈Amp. Suppose now that l > 0, and let us show that w1/m j∈A1, j = 1, . . . , l. Fix 1 ≤i0≤l. By H¨older’s inequality and (4.7), ZQ w1/m i0≤ZQ wp i0 m Y j=l+1 wp/pj j1/pmZQ m Y j=l+1 w−p pj(pm−1) j1−1/pm ≤ZQ wp i0 m Y j=l+1 wp/pj j1/pm m Y j=l+1 ZQ w1−p0 j j1 mp0 j This inequality combined with the A~ Pcondition proves w1/m i0∈A1. Thus we have proved that ~w ∈A~ P⇒(3.6). To prove that (3.6) is sufficient for ~w ∈A~ P, we first observe that for any weight wj, (4.8) 1 ≤1 |Q|ZQ ν−1 pm−1 ~w m−1/p m Y j=1 1 |Q|ZQ w 1 pj(m−1)+1 jm−1+1/pj. Indeed, let α=1 1+pm(m−1) and αj=1/p+m(m−1) 1/pj+m−1. Then Pm j=1 1/αj= 1,and by H¨older’s inequality, ZQ να ~w ≤ m Y j=1 ZQ w αpαj pj j1/αj= m Y j=1 ZQ w 1 pj(m−1)+1 jαp(m−1+1/pj). 20 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ Using again the H¨older inequality, we have 1≤1 |Q|ZQ να ~w 1 |Q|ZQ ν−1 pm−1 ~w α(pm−1). This inequality along with the previous one yields (4.8). Finally, (4.8) combined with (3.6) easily gives that ~w ∈A~ P. It remains to consider the case when pj= 1 for all j= 1,· · · , m. Assume that ~w ∈A(1,··· ,1), i.e., (4.9) 1 |Q|ZQm Y j=1 wj1/m!m ≤c m Y j=1 inf Qwj. It is clear that (4.9) implies that w1/m j∈A1,j= 1, . . . , m and ν~w ∈A1. Conversely, combining these last conditions with H¨older’s inequality we obtain 1 |Q|ZQm Y j=1 wj1/m!m ≤cinf Qm Y j=1 wj≤c 1 |Q|ZQm Y j=1 wj1/m2!m2 . ≤c m Y j=1 1 |Q|ZQ w1/m jm ≤c m Y j=1 inf Qwj. This proves that ~w ∈A(1,··· ,1) is equivalent to w1/m j∈A1,j= 1, . . . , m and ν~w ∈A1. The theorem is proved.  Proof of Theorem 3.7. The necessity follows immediately from Theorem 3.3, so we only have to prove the sufficiency. We give two proofs based on different ideas. They parallel to some extend the different proofs given in the linear situation in [9], and [7]. 1st Proof. Assume that ~w ∈A~p. Then by Theorem 3.6 each w−1 pj−1 jsatisfies the reverse H¨older inequality, i.e., there exist rj>1 and c > 0 such that for all 1 ≤r≤rjand for any cube Q, (4.10) 1 |Q|ZQ w−r pj−1 j1/r ≤c1 |Q|ZQ w−1 pj−1 j. Let ξ= min 1≤j≤mrjand q= max 1≤j≤m pm pm + (1 −1/ξ)(pj−1), and observe that qpj>1 for any j. We claim that the following pointwise inequality holds: (4.11) M(~ f)(x)≤c m Y j=1 Mc ν~w (|fj|pjwj/ν~w)qx1/qpj. MAXIMAL FUNCTIONS AND MULTILINEAR CALDER ´ ON-ZYGMUND OPERATORS 21 Then the proof of the theorem follows from H¨older’s inequality and the boundedness of the centered maximal operator. To verify the claim we first observe that, by H¨older’s inequality, (4.12) ZQ |fj| ≤ ZQ |fj|pjqwq jν1−q ~w 1 qpjZQwq jν1−q ~w −1 qpj−11−1 qpj. Set γj=qpj−1 (1−q)(pm−1). By the definition of q,γj>1 for any j. Applying again H¨older’s inequality, we get (4.13) ZQwq jν1−q ~w −1 qpj−1≤ZQ w−qγ0 j qpj−1 j1/γ0 jZQ ν−1 pm−1 ~w 1/γj. Note now that for any j, q(pj−1)γ0 j qpj−1=q(pj−1) q(pj−1) −(1 −q)pm ≤ξ. Therefore, by (4.10), ZQ w−qγ0 j qpj−1 j=ZQ w−1 pj−1 q(pj−1)γ0 j qpj−1 j (4.14) ≤c|Q|1−q(pj−1)γ0 j qpj−1ZQ w−1 pj−1 j q(pj−1)γ0 j qpj−1. Applying (4.13), (4.14) and the fact that ν~w ∈Apm (see Theorem 3.6), we obtain ZQwq jν1−q ~w −1 qpj−11−1 qpj ≤c|Q|−pm(1−q) qpjZQ w−1 pj−1 j1−1/pjZQ ν−1 pm−1 ~w (1−q)(pm−1) qpj ≤c ν~w(Q) 1−q qpjZQ w−1 pj−1 j1−1/pj. Finally, combining this inequality with (4.12) and the A~ Pcondition we can estimate m Y j=1 |fj|Q≤c m Y j=1 1 ν~w(Q)ZQ (|fj|pjwj/ν~w)qν~w1/qpj. This yields (4.11) and hence the theorem is proved.  2nd Proof. We first give the proof for the dyadic version of Mdefined by Md(~ f)(x) = sup x∈Q∈D m Y i=1 1 |Q|ZQ |fi(yi)|dyi, 22 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ where Dis the family of all dyadic cubes in Rn. Observe that kMd(~ f)kLp(ν~w)≤c m Y j=1 kfjkLpj(wj) is equivalent to (4.15) kMd(~ fσ)kLp(ν~w)≤c m Y j=1 kfjkLpj(σj), where σj=w−1 pj−1 jand ~ fσ= (f1σ1, . . . , fmσm). Fix a > 2mn. For each integer klet Ωk={x∈Rn:Md(~ f)(x)> ak}. It is easy to see that a full analogue of the classical Calder´on–Zygmund decomposition holds for Md(~ f) and, therefore, there is a family of maximal non-overlapping dyadic cubes {Qk,j}for which Ωk=∪jQk,j and (4.16) ak< m Y i=1 1 |Qk,j|ZQk,j |fiσi(yi)|dyi≤2nmak. It follows that ZRn Md(~ fσ)pν~wdx =X kZΩk\Ωk+1 Md(~ fσ)pν~wdx ≤apX k akpν~w(Ωk) = apX k,j akpν~w(Qk,j) ≤apX k,j m Y i=1 1 |Qk,j|ZQk,j |fiσi(yi)|dyi!p ν~w(Qk,j) =apX k,j m Y i=1 1 σi(Qk,j)ZQk,j |fiσi(yi)|dyi!p m Y i=1 σi(Qk,j) |Qk,j|!p ν~w(Qk,j) ≤cX k,j m Y i=1 1 σi(Qk,j)ZQk,j |fiσi(yi)|dyi!pm Y i=1 σi(Qk,j)p/pi (where in the last estimate we have used the A~ Pcondition). Set now Ek,j =Qk,j \Qk,j ∩Ωk+1. We claim that there exists a constant β > 0 such that (4.17) |Qk,j|< β |Ek,j| MAXIMAL FUNCTIONS AND MULTILINEAR CALDER ´ ON-ZYGMUND OPERATORS 23 for each k, j. Indeed, by (4.16) and H¨older’s inequality, |Qk,j ∩Ωk+1|=X Qk+1,l⊂Qk,j |Qk+1,l| <1 a(k+1)/m X Qk+1,l⊂Qk,j m Y i=1 ZQk+1,l |fiσi|!1/m ≤ 1 ak+1 m Y i=1 ZQk,j |fiσi|!1/m ≤2n a1/m |Qk,j|, which proves (4.17) with 1/β = 1 −2n/a1/m. By Theorem 3.6, each σisatisfies the Aqicondition for appropriate qi>1. It follows from the definition of Apand H¨older’s inequality (see, e.g., [23, p. 693]) that there exists a constant csuch that, for any cube Qand any measurable subset E⊂Q, |E| |Q|qi ≤cσi(E) σi(Q). Combining this inequality with (4.17), we get σi(Qk,j)≤γiσi(Ek,j) for each i= 1, . . . , m and each k, j. Hence, using H¨older’s inequality and the fact that the sets Ek,j are pairwise disjoint, we obtain (with γ= maxiγi) ZRn Md(~ fσ)pν~wdx ≤c γ X k,j m Y i=1 1 σi(Qk,j)ZQk,j |fiσi(yi)|dyi!pm Y i=1 σi(Ek,j)p/pi. ≤c γ m Y i=1 X k,j 1 σi(Qk,j)ZQk,j |fiσi(yi)|dyipiσi(Ek,j)!p/pi . ≤c γ m Y i=1 X k,j ZEk,j Mσi(fi)piσi!p/pi ≤cγ m Y i=1 ZRn Mσi(fi)piσip/pi ≤c m Y i=1 ZRn |fi|piσip/pi , where in the last inequality we have used the boundedness of Mσion Lpi(σi). The proof is complete in the dyadic case. Observe that it is enough to assume that the weight ~w satisfies the A~ Pcondition only for dyadic cubes. 24 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ To pass from the dyadic version to the general situation we use established tools to handle such passage. We need the following easy variant of a result of Fefferman-Stein [18] which can be also found in [22, p. 431]. Lemma 4.1. For each integer k, each ~ f, all xin Rnand p > 0there exists a constant c, depending only on n,mand p, so that Mk(~ f)(x)p≤c |Qk|ZQkτ−t◦ Md◦τt(~ f)(x)pdt. Here τtg(x) = g(x−t),Qkis the cube centered at the origin with side length 2k+2, and Mkis the operator defined as Mbut with cubes having sides of length smaller than 2k. Clearly, to estimate the left-hand side of (3.7) it suffices to estimate kMk(~ f)kLp(ν~w). It follows from the above pointwise inequality and Fubini’s theorem that kMk(~ f)kLp(ν~w)≤csup t kτ−t◦ Md◦τtkLp(ν~w). We have now to estimate kτ−t◦ Md◦τtkLp(ν~w)with constant independent of t. The bound τ−t◦ Md◦τt:Lp1(w1)× · · · × Lpm(wm)→Lp(~w) with constant independent of tis equivalent to the bound Md:Lp1(τtw1)× · · · × Lpm(τtwm)→Lp(τt(~w)) with constant independent of t. But τt(~w) satisfies the A~ Pwith constant independent of tbecause A~p is invariant under translation and, hence, we can apply the first part of the proof (i.e, the one considered for the dyadic maximal case).  Proof of Corollary 3.8. It is enough to prove (3.8) when the right-hand side is finite (or there is nothing to prove). We have using (2.8) and (3.1) kT(~ f)kLp(w)≤ kMδ(T(~ f))kLp(w)≤CkM# δ(T(~ f))kLp(w)≤CkM(~ f)kLp(w), which gives the desired result provided we can show that kMδ(T(~ f))kLp(w)is finite. Note that since wis in A∞,wis also in Ap0with 0 <max(1, pm)< p0<∞. So with δ < p/p0<1/m we have, in addition to the above inequalities, kMδ(T(~ f))kLp(w)≤ kMp/p0(T(~ f))kLp(w)=CkM(T(~ f)p/p0)kp0/p Lp0(w) ≤Ck(T(~ f)p/p0)kp0/p Lp0(w)≤CkT(~ f)kLp(w). It is enough then to prove that kT(~ f)kLp(w)is finite for each family ~ fof bounded functions with compact support for which kM(~ f)kLp(w)is finite. We will see that this is always the case. The standard arguments are as follows. The weight wis also in Lq loc for qsufficiently close to 1 so that its dual exponent q0 satisfies pq0>1/m. Then, for any ball Bcenter at the origin kT(~ f)kLp(B,w)is finite MAXIMAL FUNCTIONS AND MULTILINEAR CALDER ´ ON-ZYGMUND OPERATORS 25 by H¨older’s inequality and the unweighted theory for T. On the other hand, outside a sufficiently large ball B, (4.18) M(~ f)(x)≥C1|x|−mn ≥C2|Tf(x)| (with constants depending on ~ fof course). From the assumption kM(~ f)kLp(w)finite and (4.18), we conclude kT(~ f)kLp(Rn\B,w)≤CkM(~ f)kLp(Rn\B,w)<∞. Similar arguments give the weak-type estimate (3.9).  Proof of Corollary 3.9. Since ν~w is in A∞and the intersection of the space of simple functions with Lp(w) is dense in Lp(w) for any weight w([2, p. 211], the corollary immediately follows from the previous one and the boundedness properties of Mon weighted spaces.  Proof of Theorem 3.11. For simplicity we consider only the one-dimensional case. The higher dimensional one is only notationally more complicated. Recall that the sum of the m-linear Riesz transforms is T(~ f) (x) = p.v. ZRmPm j=1(x−yj) (Pm j=1 |x−yj|2)m+1 2 f1(y1). . . fm(ym)dy1. . . dym. Clearly, it suffices to show that if (3.11) holds, then ~w ∈A~ P. We follow similar a argument to the one used in the linear case (see [22, p. 417]) but with some modifications. First, we suppose that all functions fi≥0 and that supp(fj)⊂I. Then, if x∈I+ and yj∈Ifor all j, we get that Pm j=1(x−yj) (Pm j=1 |x−yj|)m+1 =1 (Pm j=1 |x−yj|)m≥cm |I|m Therefore, if x∈I+we have T(~ f) (x)≥cm m Y j=1 |fj|I, and hence I+⊂ {x:|T(~ f)(x)|> λ}, whenever 0 <λ<cmQm j=1 |fj|I. Arguing exactly as in Theorem 3.3, we obtain from this that for any interval I, (4.19) 1 |I|ZI+ ν~w1/p m Y j=1 1 |I|ZI w−1/(pj−1) j1−1/pj≤c. In a similar way we can prove that (3.11) implies that for any interval I, (4.20) 1 |I|ZI ν~w1/p m Y j=1 1 |I|ZI+ w−1/(pj−1) j1−1/pj≤c. 32 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ + sup t>0 C Φ(1 t)w({y∈Rn:M(~ f)(y)> tm}) ≤sup t>0 C Φ(1 t)w({y∈Rn:M1 L(logL)(~ f)(y)> tm}). We need to verify now that (5.3) sup t>0 1 Φ(1 t)w({y∈Rn:Mδ(Tb(~ f))(y)> tm})<∞. and (5.4) sup t>0 1 Φ(1 t)w({y∈Rn:Mε(T(~ f))(y)> tm})<∞. We will only show (5.3) because the proof of (5.4) is very similar but easier. Recall that we are assuming that wis bounded, so sup t>0 1 Φ(1 t)w({y∈Rn:Mδ(Tb(~ f))(y)> tm}) ≤ kwkL∞sup t>0 1 Φ(1 t)|{y∈Rn:Mmδ |T~ b~ f|1/m(y)> t}| Now, using Φ(t)≥t,mδ < 1, and the fact η < 1 =⇒Mη:L1,∞(Rn)→L1,∞(Rn) (which is a consequence of M:Lr,∞(Rn)→Lr,∞(Rn), r > 1 ), we obtain sup t>0 1 Φ(1 t)|{y∈Rn:Mmδ |Tb~ f|1/m(y)> t}| ≤sup t>0 t|{y∈Rn:Mmδ |Tb~ f|1/m(y)> t}| ≤Csup t>0 t|{y∈Rn:|Tb~ f(y)|1/m > t}|. Recalling that ~ fhas compact support, we may assume that supp ~ f⊂B(0, R) for some R > 0. Write then sup t>0 t|{y∈Rn:|Tb~ f(y)|1/m > t}| ≤ sup t>0 t|{y∈B2R:|Tb~ f(y)|1/m > t}| + sup t>0 t|{y /∈B2R:|Tb~ f(y)|1/m > t}| =I+II. For Iwe estimate the L1norm instead and then use H¨older’s inequality to compute I≤ZB2R |Tbf(y)|1/mdy ≤CR(1−1/p)nZRn |Tb~ f|p/mdy1/p . This last term is finite by the strong case if we choose psufficiently large. MAXIMAL FUNCTIONS AND MULTILINEAR CALDER ´ ON-ZYGMUND OPERATORS 33 For II, we can control as before Tb(~ f)(x) by M(~ f)(x) is we assume that bis bounded. Then we have IIm≤Csup t>0 tm|{y∈Rn:M(~ f)(y)1/m > t}|m =CkM(~ f)kL1/m,∞≤C m Y i=1 ZRn |fi|dx < ∞. Summarizing we have shown that (5.5) sup t>0 t|{y∈Rn:|Tb~ f(y)|1/m > t}| <∞, which gives in turn (5.3), provided wis bounded and bis bounded. As already explained, we can pass to a general win A∞using monotone convergence and in this way we obtain the result for arbitrary win A∞and bin L∞, and with the constant in (3.20) depending on the BMO norm of b. We now eliminate the assumption bbounded. Observe first that it is enough to prove (3.20) with the level set {y∈Rn:|T~ b(~ f)(y)|> tm}replaced by {y∈B(0, N) : |T~ b(~ f)(y)|> tm}for arbitrary N > 0 and with a constant on the right-hand side independent of N. Then, we can approximate bby {bj}as before and use now that, for each compact set, an appropriate subsequence {|Tbj~ f|} also converges to |Tb~ f|in measure. Taking limit in jgives then the required estimate for arbitrary bin BMO with a constant independent of N. Finally taking the sup in Ncompletes the proof of the theorem.  6. Proof of the end-point estimate for the multilinear commutator Proof of Theorem 3.18. We need the following preliminary lemma. Lemma 6.1. Assume that ~w = (w1, . . . , wm)satisfies the A~ Pcondition. Then there exists a finite constant r > 1such that ~w ∈A~ P/r. Proof. By Theorem 3.6, each σj=w−1 pj−1 jbelongs to A∞and, hence, there are constants cj, tj>1, depending on the A∞constant of σj, such that for any cube Q 1 |Q|ZQ w−tj pj−1 j1 tj≤cj |Q|ZQ w−1 pj−1 j. Let rj>1 be selected so that tj pj−1=1 pj rj−1. Then, if r= min{r1,· · · , rm}and c= max{c1,· · · , cm}, we have 1 |Q|ZQ ν~w1/p/r m Y j=1 1 |Q|ZQ w−1/(pj r−1) j1−1 pj r, 34 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ =1 |Q|ZQ ν~wr/p m Y j=1 1 |Q|ZQ w−1/(pj r−1) j(pj r−1) r pj ≤1 |Q|ZQ ν~wr/p m Y j=1 1 |Q|ZQ w−1/(pj rj−1) j(pj rj−1) r pj =1 |Q|ZQ ν~wr/p m Y j=1 1 |Q|ZQ w−tj pj−1 j pj−1 tj r pj, ≤crm 1 |Q|ZQ ν~wr/p m Y j=1 1 |Q|ZQ w−1 pj−1 j(pj−1) r pj≤crm[w]r A~ P. Since, ~w = (w1, . . . , wm) satisfies the A~ Pcondition the proof of the lemma is finished.  Now, by Theorem 3.19 and since ν~w is also in A∞, ZRn |T→ b(~ f)(x)|pν~w(x)dx ≤CZRn ML(log L)(~ f)(x)pν~w(x)dx. To finish the proof we use a bigger operator than ML(log L)that is enough for our purposes. Indeed, if r > 1 and since Φ(t) = t(1 + log+(t)) ≤tr,t > 1 we have by the generalized Jensen’s inequality (2.10). kfkL(log L),Q ≤c1 |Q|ZQ |f(y)|rdy1/r, and we can therefore estimate the maximal operator ML(log L)by the larger one Mr(~ f)(x) = sup Q3x m Y j=1 1 |Q|ZQ |fj|r1/r, to obtain kT~ b(~ f)kLp(ν~w)≤ckMr(~ f)kLp(ν~w). Now, to prove kMr(~ f)kLp(ν~w)≤c m Y j=1 kfjkLpj(wj) is equivalent to prove kM(~ f)kLp/r(ν~w)≤c m Y j=1 kfjkLpj/r(wj). By Theorem 3.7, this is equivalent to show that ~w ∈A~ P/r and we already know that this is true for some r > 1 because of Lemma 6.1.  MAXIMAL FUNCTIONS AND MULTILINEAR CALDER ´ ON-ZYGMUND OPERATORS 35 Proof of Theorem 3.17. Without loss of generality we may assume i= 1. Also, by homogeneity, we may assume that t= 1. Finally, we may also assume that ~ f≥0. Define the set Ω = {x∈Rn:M1 L(logL)(~ f)(x)>1} It is easy to see that Ω is open and we may assume that it is not empty (or there is nothing to prove). To estimate the size of Ω, it is enough to estimate the size of every compact set Fcontained in Ω. We can cover any such Fby a finite family of cubes {Qj}for which (6.1) 1 <kf1kΦ,Qj m Y j=2 (fj)Qj Using Vitali’s covering lemma, we can extract a subfamily of disjoint cubes {Qi}such that (6.2) F⊂ ∪i3Qi. By homogeneity, 1<  m Y j=2 (fj)Qi Φ,Qi and by the properties of the norm k·kΦ,Q, this is the same as 1<1 |Qi|ZQi Φ f1(y) m Y j=2 (fj)Qi!dy, Using now that Φ is submultiplicative and Jensen’s inequality 1< m Y j=1 1 |Qi|ZQi Φ(fj(y)). Finally by the condition on the weights and H¨older’s inequality at discrete level, ν~wFm≈ X i ν~w(Qi)!m ≤ X i m Y j=1 inf Qw1/m j|Qi|1/m 1 |Qi|ZQi Φ(fj(y)) dy1/m!m ≤ X i m Y j=1 ZQi Φ(fj(y))wj(y)dy1/m!m ≤ m Y j=1 ZRn Φ(|fj(y)|)wj(y)dy, which concludes the proof.  Proof of Theorem 3.16. We have all the ingredients to prove Theorem 3.16. As in the proof of Theorem 3.15, by linearity it is enough to consider the operator with only one symbol. By homogeneity it is enough to assume t= 1 and hence we 36 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ must prove ν~wx∈Rn:|Tb(~ f)(x)|>1m≤C m Y j=1 ZRn Φ(|fj(x)|)wj(x)dx. Now, since Φ is submultiplicative, we have by Theorem 3.19 and Theorem 3.17 ν~wx∈Rn:|Tb(~ f)(x)|>1m≤Csup t>0 1 Φ(1 t)mν~wx∈Rn:|Tb(~ f)(x)|> tmm ≤Csup t>0 1 Φ(1 t)mν~w({y∈Rn:M1 L(logL)(~ f)(x)> tm})m ≤Csup t>0 1 Φ(1 t)m m Y j=1 ZRn Φ(|fj(x)| t)wj(x)dx ≤Csup t>0 1 Φ(1 t)m m Y j=1 ZRn Φ(|fj(x)|) Φ(1 t)wj(x)dx ≤C m Y j=1 ZRn Φ(|fj(x)|)wj(x)dx as we wanted to prove.  7. Remarks, examples and counterexamples In this section we provide the examples and counterexamples mentioned earlier in the article and which establish that several of the estimates obtained are, in appropriate senses, sharp. Remark 7.1. The two conditions in (3.6) are independent of each other. Set ~w = (w1, w−p2/p1 1). We have then ν~w = 1 which trivially belongs to A2pfor any w1. If we select w1so that w−1 p1−1 16∈ L1 loc, we see that the first condition in (3.6) does not hold. Conversely, let n= 1, m = 2 and p1=p2= 2. Set w1=w2=|x|−2. Then the first condition in (3.6) holds (because w−1 j=|x|2∈A4), while ν~w =|x|−26∈ L1 loc, and hence ν~w 6∈ A2. Remark 7.2. The condition ~w ∈A~ Pdoes not imply in general wj∈L1 loc for any j. Take, for instance, w1=χ[0,2](x) |x−1|+χR/[0,2](x) and wj(x) = 1 |x|for j= 2, ..., m. Then, using the definition, it is not difficult to check that ν~w ∈A1. We also have infQν~w ∼Qm j=1 infQwp/pj j. These last two facts together easily imply that ~w ∈A~ P. MAXIMAL FUNCTIONS AND MULTILINEAR CALDER ´ ON-ZYGMUND OPERATORS 37 Remark 7.3. The classes A~ Pare not increasing Let us consider the partial order relation between vectors ~ P= (p1, . . . , pm) and ~ Q= (q1, . . . , qm) given by ~ P.~ Qif pj≤qjfor all j. Then, for ~ P.~ Qwe have m Y j=1 Apj⊆ m Y j=1 Aqj, but A~ Pis not contained in A~ Q. To see this, consider n= 1, m= 2, ~ P= (p1, p2) = (2,2), and ~w = (w1, w2) = (|x|−5/3,1) Then since w1/2 1∈A1it is easy to see that ~w ∈A~ P. Also, since wraised to an appropriate large power becomes non-locally integrable, it is easy to show that ~w /∈A~ Q if, for instance, ~ Q= (2,6). Remark 7.4. The assumption pj>1for all jis essential in Theorem 3.7 even in the unweighted case. Let n= 1, m= 2, and suppose that (3.7) holds with wj≡1 for p1= 1 and p2>1. Then taking f1to be the Dirac mass at the origin and f2=g, where gis any non-increasing function on (0,∞), we get that g(x) x≤1 x2Zx 0 g(t)dt ≤ M(f1, f2)(x), and hence (3.7) would imply Z∞ 0 (g(x)/x)pdx1/p ≤cZ∞ 0 g(x)p21/p2, where 1/p = 1 + 1/p2. The simple choice of g(x) = x1−1/p(log(1/x))−1/pχ(0,1/2) shows that this inequality is not true. Remark 7.5. The estimate (3.7) does not hold if M(~ f)is replaced by Qm j=1 M(fj), and therefore Corollary 3.9 cannot be obtained from the known estimates in [26]. Let m= 2 and let p1, p2≥1 satisfy 1/p1+ 1/p2= 1/p > 1. Chose ε > 0 such that p2< p2/p −ε. Set now w1= 1 and w2=|x|ε−p2/p. Then it is easy to check that ~w ∈A~ P(this follows from the fact that wp/p2 2∈A1). Nevertheless, the inequality (7.1) kMf1Mf2kLp,∞(ν~w)≤Ckf1kLp1(w1)kf2kLp2(w2) does not hold for all f1, f2. Indeed, set f1=χ[0,1] and f1=Nχ[N,N+1], for Nbig enough. It is clear that [0,1] ⊂ {x:Mf1Mf2>1/2}, 38 A. K. LERNER, S. OMBROSI, C. P´ EREZ, R. H. TORRES, AND R. TRUJILLO-GONZ ´ ALEZ so the left-hand side of (7.1) is bigger than some constant c > 0. Furthermore, kf1kLp1(w1)= 1 and kf2kLp2(w2)∼N1+ ε p2−1 p. We see then that (7.1) would imply c≤N1+ ε p2−1 p, which is a contradiction. Remark 7.6. A weak-type analogue of (3.12) is not true for arbitrary weights wjif at least one pj= 1. Let n= 1 and m= 2. Let 1 ≤p1<∞and p2= 1. For k≥4 set Jk= (k+1 4k, k +1 2k). Let now w1(x) = P∞ k=4 kχJk(x) and w2(x) = P∞ k=4 1 kχJk(x). Suppose that the inequality (7.2) kMf1Mf2kLp,∞(ν~w)≤ckf1kLp1(Mw1)kf2kL1(Mw2) holds with a constant cindependent of f1and f2. Set f1=χ(0,1) and f2=PN k=1 δk, where δkis the Dirac mass at the point k. Simple computations show that N [ k=4 Jk⊂ {x:Mf1(x)Mf2(x)>1}. On the other hand, kf1kLp1 Mw1 ≤cand Mw2(k)≤c/k. Therefore, (7.2) would imply PN k=1 1 k≤c, which is a obviously a contradiction. Remark 7.7. An estimate of the form (7.3) |{x:|Tb(~ f)|> λm}| ≤ C(kb||BMO) kfi λkY j6=i kΦ(|fj| λ)kL1!1/m cannot hold for characteristics functions of intervals if k · k is finite on characteristic functions and satisfies kλfk=λkfk. In particular (a bounded) mapping property of the form T→ b:L1× · · · × L1→L1/m,∞ does not hold. For m= 1 this was already shown in [41]. 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Lerner, Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Sevilla, 41080 Sevilla, Spain E-mail address:[email protected] Sheldy Ombrosi, Departamento de Matem´ atica, Universidad Nacional del Sur, Bah´ ıa Blanca, 8000, Argentina Current address: Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad de Sevilla, 41080 Sevilla, Spain 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] Rodolfo H. Torres, Department of Mathematics, University of Kansas, 405 Snow Hall 1460 Jayhawk Blvd, Lawrence, Kansas 66045-7523, USA E-mail address:[email protected] Rodrigo Trujillo-Gonz´ alez, Departamento de An´ alisis Matem´ atico, Universidad de La Laguna, 38271 La Laguna, S.C. de Tenerife, Spain E-mail address:[email protected]