Sharp weighted estimates for multilinear commutators

Sharp weighted estimates for multilinear commutators C. P´erez∗and R. Trujillo-Gonz´alez† Journal of the London mathematical society 65 (2002), 672–692. Abstract We consider multilinear commutators with vector symbol ~ b= (b1,··, bm) defined by T~ b(f)(x) = ZRn  m Y j=1 (bj(x)−bj(y)) K(x, y)f(y)dy, where Kis a Calder´on-Zygmund kernel. We prove the following a priori estimates for w∈A∞. For 0 < p < ∞there exists a constant Csuch that kT~ b(f)kLp(w)≤Ck~ bk kML(logL)1/r (f)kLp(w) and sup t>0 1 Φ(1 t)w({y∈Rn:|T~ bf(y)|> t})≤Csup t>0 1 Φ(1 t)w({y∈Rn:ML(logL)1/r (||~ b||f)(y)> t}) where ||~ b|| =Qm j=1 kbjkoscexpLrj, Φ(t) = tlog1/r(e+t), 1 r=1 r1+··+1 rm, and ML(logL)αis an Orlicz type maximal operator. This extends, with a different approach, classical results by Coifman [5] (see also [6] [24]). As a corollary we deduce that the operators T~ bare bounded on Lp(w) when w∈Apand that they satisfy corresponding weighted L(logL)1/r type estimates with w∈A1. ∗Partially supported by DGESIC Grant PB980106. †Partially supported by Gobierno de Canarias PI1999/105. 1991 Mathematics Subject Classification: 42B20, 42B25. Keywords: Calder´on-Zygmund singular integral operators, commutators, Apweights, maximal functions 1 1 Introduction The main purpose of this article is to prove sharp estimates for multilinear commutators involving nonstandard symbols. This will be established by means of appropriate maximal operators which somehow control the commutators. This illustrates the classical Calder´on–Zygmund principle which roughly states that any singular integral operator is controlled by a suitable maximal operator. Background Motivated by the work of Calder´on on commutators, Coifman, Rochberg and Weiss introduced in [7] the operator Tbf(x) = ZRn (b(x)−b(y))K(x, y)f(y)dy, (1.1) where Kis a kernel satisfying the standard Calder´on-Zygmund estimates (see section 3.1) and where b, the “symbol” of the operator, is any locally integrable function. The operator is called commutator since Tb= [b, T] = bT −T(b·) where Tis the Calder´on-Zygmund singular integral operator associated to K. The main result from [7] states that [b, T] is a bounded operator on Lp(Rn), 1 < p < ∞, when the symbol bis a BMO function. These commutators have proved to be of interest in many situations. We shall only mention the recent results in the theory of non divergence elliptic equations with discontinuous coefficients [2] [3] [9]. There is also an interesting connection, as pointed out in [21], with the nonlinear commutator considered by R. Rochberg and G. Weiss in [23] and defined by f→Nf =T(flog |f|)−Tf log |Tf|. This is in turn is related to the Jacobian mapping of vector function and with nonlinear P.D.E. as shown in [13] [14]. A natural generalization of the commutator [b, T] is given by Tm bf(x) = ZRn (b(x)−b(y))mK(x, y)f(y)dy, (1.2) where m∈N. The case m= 0 recaptures the Calder´on-Zygmund singular integral operator. Observing that Tm b= (m times) z }| { [b, ··,[b, T]] we have that for m∈N,Tm bis also a bounded operator on Lp(Rn), 1 < p < ∞, when bis a BMO function. It was shown in [21] that there is an intimate connection between the commutator Tm band iterations of the maximal operators. Indeed, the main theorem from [7] was sharpened in [21] as follows: for any 0 < p < ∞and any w∈A∞there is a constant Csuch that ZRn |Tm bf(x)|pw(x)dx ≤Ckbkmp BMO ZRn (Mm+1f(x))pw(x)dx, (1.3) 2 where Mm+1 denotes the m+ 1 iterations of the Hardy-Littlewood maximal operator, namely Mm= (m times) z }| { M◦··◦M. Furthermore, this inequality is sharp since, by the Lebesgue differentiation theorem, Mm+1 can not be replaced by the smaller operator Mm. Estimate (1.3) can also be seen as a generalization of a by now classical result of Coifman for Calder´on-Zygmund singular integral operators. See [5] and also [6]. There is a weak version of inequality (1.3) obtained in [20]. Indeed, if we let Φm(t) = tlogm(e+t), for any w∈A∞and b∈BMO there is a constant Csuch that sup t>0 1 Φm(1 t)w({y∈Rn:|Tm bf(y)|> t})≤Csup t>0 1 Φm(1 t)w({y∈Rn:Mm+1f(y)> t}).(1.4) A priori inequalities of the form (1.3) (or (1.4)) encode a good amount of information about the behavior of the operator. The first observation is that in any case they reflect the higher degree of singularity of Tm b, as compared with T, since a larger operator than M, namely Mm+1, is needed to balance the inequality. As a second instance, if p > 1 we can apply m+ 1 times Muckenhoupt’s theorem to conclude from (1.3) the well known fact that high order commutators are bounded on Lp(w) whenever w∈Ap. It should be mentioned that this Apestimate follows, as it is well known, from an estimate due to Str¨omberg (see [15, p. 268], [25, p. 417]). This method is powerful and can be applied to different situations. As a sample we remit to [10] for a very nice application to commutators with strongly singular integral operators. However, this estimate of Str¨omberg is not sharp enough neither to derive (1.3) nor (1.4). In fact it can be shown from (1.4) the following (Llog Ltype) endpoint estimate first deduced in [20]: let w∈A1and b∈BMO, then there exists a constant Csuch that for all λ > 0 w({y∈Rn:|Tm bf(y)|> λ})≤CZRn |f(y)| λ(1 + log+(|f(y)| λ))mw(y)dy. (1.5) As a third instance, it was shown in [21] that estimate (1.3) implies a sharp two weighted inequalities for the commutator of the form ZRn |Tm bf(x)|pw(x)dx ≤Ckbkmp BMO ZRn |f(x)|pM[(m+1)p]+1w(x)dx, where no condition on the weight wis assumed. This result is an extension of the case m= 0 proved in [18] generalizing some previous partial results by M. Wilson [26]. The approach considered in [18] is different from that in [26] and it combines (1.3) with certain sharp two weighted estimates for the Hardy–Littlewood maximal function derived in [19]. Finally, there is a relationship between (1.4) and the endpoint behavior of the operator. Indeed, an interesting observation that occurs in the development of the theory of commutators is that the Lptheory, p > 1, was developed without appealing to any endpoint estimate. On the other hand, it is well known that in the classical Calder´on-Zygmund Lptheory, p > 1, a crucial step is to show that the operators are of weak type (1,1). However, this is not the case of the commutator [b, T] when b∈BMO as shown in [20] and estimate (1.5) is the right replacement. 3 This can be seen as another way of expressing the fact that commutators have a higher degree of singularity as compared with the Calder´on-Zygmund singular integral operators. Results of this paper In this paper we obtain similar estimates for a wider class of commutators. Given a Calder´on- Zygmund singular integral operator Twith kernel Kin Rn×Rnand a vector ~ b= (b1, .., bm) of locally integrable functions, we define the multilinear operator T~ bf(x) = ZRn  m Y j=1 (bj(x)−bj(y)) K(x, y)f(y)dy, (1.6) generalizing the commutator (1.2). We remit the reader to [6] for an extensive study of multilinear operators. We will be considering the following class of symbols: for r≥1 and for any locally integrable function f, we define kfkoscexpLr= sup Q kf−fQkexpLr,Q being the supremum taken over all the cubes Qwith sides parallels to the axes. Here kgkexpLr,Q is the mean value of gon the cube Qwith respect to the Young function Φ(t) = etr−1. See Section 2.2 for the precise definition. As usual, fQdenotes the average of fon Q. For r≥1 we define the space OscexpLrby OscexpLr={f∈L1 loc(Rn) : kfkoscexpLr<∞}. In the particular case of r= 1, OscexpL1coincides with the BMO space by John-Niremberg’s theorem. Also we have OscexpLr BMO for any r > 1. Other examples are provided by Trudinger’s inequality for Riesz potentials. To be more precise, for any 0 < α < n and any f∈Ln/α(Rn), the Riesz potential Iαfof order αbelongs to OscexpL(n/α)0(cf. [12, 27]) For the statement of our results we introduce some notation that will simplify the presentation. Along this paper mwill always be the number of symbols of the operator T~ bwhere ~ b= (b1, .., bm) is a family of mlocally integrable functions. If r1, .., rmare mpositive real numbers, we denote 1 r=1 r1 +··+1 rm and ||~ b|| = m Y j=1 kbjkoscexpLrj. Our main results are the following. Theorem 1.1 Let 0< p < ∞,w∈A∞. Suppose that T~ bfis the commutator (1.6) where bis as above such that bi∈OscexpLri,ri≥1,1≤i≤m. Then there exists a constant C > 0such that ZRn |T~ b(f)(x)|pw(x)dx ≤C||~ b||pZRn (ML(logL)1/r (f)(x))pw(x)dx (1.7) 4 for all bounded functions fwith compact support. For the precise definition of the operator ML(logL)αsee Section 2.2. Inequality (1.7) shows that the maximal operator ML(logL)1/r is the one that controls the multilinear commutators Tb. Since ri≥1 for each i, it follows that ML(logL)1/r is pointwise smaller than ML(logL)mand this in turn is known to be equivalent to m+ 1 iterations of the Hardy–Littlewood maximal operator Mm+1. Hence we can use again Muckenhoupt’s theorem and deduce the boundedness of Tbon Lp(w) for p > 1 and any w∈Ap: Corollary 1.2 Let 1<p<∞and w∈Ap. Suppose that T~ bfis the commutator (1.6) where bis as above such that bi∈OscexpLri,ri≥1,1≤i≤m. Then there exists a constant C > 0 such that ZRn |T~ b(f)(x)|pw(x)dx ≤C||~ b||pZRn |f(x)|pw(x)dx (1.8) for all bounded functions fwith compact support. As we mentioned above this estimate is a generalized version of Coifman’s result in [5]. However, our approach is different and will be based on a pointwise inequality that roughly can be expressed as M# δ(T~ bf)(x)≤C||~ b||ML(logL)1/r f(x) + R(f)(x) (1.9) for an appropriate positive but small enough number δ.Rdenotes certain “remainder” operator which is smoother, less singular, than ML(logL)1/r in some suitable sense. See Lemma 3.1 for a complete and precise statement of this estimate. Moreover, reasoning as in [21], Theorem 2, from this theorem we deduce the following estimate for general weights. As usual [α] will denote the integer part of α. Theorem 1.3 Let 1< p < ∞and let T~ bbe as in Theorem 1.1. Then there exists a constant C such that for any weight w ZRn |T~ b(f)(x)|pw(x)dx ≤C||~ b||pZRn |f(x)|pM[( 1 r+1)p]+1(w)(x)dx for all bounded functions fwith compact support. Remark 1.4 We remark that the number of iterations of the maximal function needed in the Theorem is optimal as can be seen in [21], Section 5. In fact it follows from the proof of Theorem 1.3 that there is a sharper estimate: ZRn |T~ b(f)(x)|pw(x)dx ≤C||~ b||pZRn |f(x)|pML(log L)(1 r+1)p−1+(w)(x)dx where  > 0, being the result false for = 0. 5 The key pointwise estimate (1.9) is also used to derive the next endpoint result. Recall that commutators with BMO functions are not of weak type (1,1). Recall that we denote ||~ b|| =Qm j=1 kbjkoscexpLrjand Φ(t) = tlog1/r(e+t), where 1 r= 1 r1+··+1 rm. Theorem 1.5 Let w∈A1. There exists a constant C > 0such that for all λ > 0 w({y∈Rn:|Tbf(y)|> λ})≤CZRn Φ(||~ b|| |f(y)| λ)w(y)dy, (1.10) for all bounded functions fwith compact support and for all ~ b. The proof is based on the following result which generalizes inequality (1.4). Theorem 1.6 Let w∈A∞. Then, there exists a positive constant Csuch that sup t>0 1 Φ(1 t)w({y∈Rn:|Tbf(y)|> t})≤Csup t>0 1 Φ(1 t)w({y∈Rn:MΦ(||~ b||f)(y)> t}) (1.11) for all bounded functions fwith compact support. In any of the above results, if we choose b1=·· =bmand r1=·· =rm= 1, we recover completely the corresponding results from [20] and [21]. As usual Cwill denote a positive constant that can change its value on each statement. If the constant depends on precise parameters it will be pointed out. 2 Preliminaries In this section we introduce the basic tools needed for the proof of the main results. 2.1 The Fefferman-Stein inequality for A∞weights A weight will always mean a positive function which is locally integrable. We say that a weight wbelongs to the class Ap, 1 < p < ∞, if there is a constant Csuch that 1 |Q|ZQ w(y)dy 1 |Q|ZQ w(y)1−p0dyp−1 ≤C for each cube Qand where as usual 1 p+1 p0= 1. A weight wbelongs to the class A1if there is a constant Csuch that 1 |Q|ZQ w(y)dy ≤Cinf Qw. We will denote the infimum of the constants Cby [w]Ap. Observe that [w]Ap≥1 by Jensen’s inequality. 6 Since the Apclasses are increasing with respect to p, the A∞class of weights is defined in a natural way by A∞=∪p>1Ap. However, the following characterization it is more interesting: there are positive constants cand ρsuch that for any cube Qand any measurable set Econtained in Qthen w(E) w(Q)≤c|E| |Q|ρ . A simple fact that will be useful is that, for any w∈Apand any m > 0, wm= min{w, m} ∈ Apwith [wm]Ap≤Cp[w]Ap. For more information on Apweights we remit the reader to the references [11] and [25]. We recall now the definitions of classical maximal operators. If, as usual, Mdenotes the Hardy–Littlewood maximal operator, we consider for δ > 0 Mδf(x) = M(|f|δ)1/δ(x) = sup Q3x 1 |Q|ZQ |f(y)|δdy!1/δ , M#(f)(x) = sup Q3x inf c 1 |Q|ZQ |f(y)−c|dy ≈sup Q3x 1 |Q|ZQ |f(y)−fQ|dy and a variant of this sharp maximal operator that will become the main tool in our scheme M# δf(x) = M#(|f|δ)(x)1/δ. The main inequality between these operators to be used is a version of the classical one due to C. Fefferman and E. Stein (see [24], [16]). Lemma 2.1 Let wbe an A∞weight, then there exists a constant cdepending upon the A∞ condition of wsuch that for all λ,  > 0 w({y∈Rn:Mf(y)> λ, M#f(y)≤λ})≤c ρw({y∈Rn:Mf(y)>λ 2}). As a consequence we have the following estimates for δ > 0. a) Let ϕ: (0,∞)→(0,∞)doubling. Then, there exists a constant cdepending upon the A∞ condition of wand the doubling condition of ϕsuch that sup λ>0 ϕ(λ)w({y∈Rn:Mδf(y)> λ})≤csup λ>0 ϕ(λ)w({y∈Rn:M# δf(y)> λ}) (2.1) for every function such that the left hand side is finite. b) Let 0<p<∞there exists a positive constant Cupon the A∞condition of wand psuch that ZRn (Mδf(x))pw(x)dx ≤CZRn (M# δf(x))pw(x)dx, (2.2) for every function fsuch that the left hand side is finite. 7 2.2 Orlicz maximal functions By a Young function Φ we shall mean a continuous, nonnegative, strictly increasing and convex function on [0,∞) with lim t→0+ Φ(t) t= lim t→∞ t Φ(t)= 0. We define the Φ-averages of a function fover a cube Qby kfkΦ,Q =kfkΦ(L),Q = inf{λ > 0 : 1 |Q|ZQ Φ|f(x)| λdx ≤1}. For Orlicz norms we are usually only concerned about the behavior of Young functions for large values of t’s. Given two functions Band C, we write B(t)≈C(t) if B(t)/C(t) is bounded and bounded below for t≥c > 0. We also recall that if B(t)≤C(t) for t≥c > 0, then kfkB,Q ≤CkfkC,Q with Can absolute constant. For more information on the subject see the reference [22]. Associate to this average we can define a maximal operator MΦgiven by MΦf(x) = MΦ(L)f(x) = sup Q3x kfkΦ,Q, where the supremum is taken over all the cubes containing x. For example, if Φ(x) = exr−1, then k·kexpLr,Q and MexpLr, denote respectively the Φ-average and the maximal operator associated to Φ. Similarly we have for Φ(x) = xlogr(e+x) k · kL(logL)r,Q and ML(logL)r. Observe that by the above remarks, Mf ≤CML(logL)rffor any r > 0. These examples will be relevant in our work. Finally, we will be using the following known pointwise inequality: if m∈Nthen ML(logL)m∼Mm+1 = (m+1 times) z }| { M◦··◦M, the m+ 1 iterations of the Hardy-Littlewood maximal operator. A generalization of this equivalence can be found in [8]. We begin with some technical lemmas on convex functions whose proofs are standard. Lemma 2.2 ([17], Lema 2.1) If Φ0,Φ1, ..., Φmare real-valued, non-negative, nondecreasing, left continuous functions defined on [0,∞)such that, with the definition Φ−1 i(x) = inf{y: Φi(y)> x}, it verify Φ−1 1(x)Φ−1 2(x)· ·Φ−1 m(x)≤Φ−1 0(x),(2.3) 8 then for all 0≤x1, x2, .., xm<∞ Φ0(x1x2· ·xm)≤Φ1(x1)+Φ2(x2) + ··+Φm(xm) Lemma 2.3 Let Φ0be a convex function such that Φ0(0) = 0 and let Φ1, .., Φmas in the previous lemma all verifying (2.3). If f1, f2, .., fmare functions satisfying that kfikΦi,Q <∞ for all 1≤i≤mand for a given cube Q, then kf1···fmkΦ0,Q ≤mkf1kΦ1,Q ···kfmkΦm,Q.(2.4) PROOF: Taking ε > 0 small enough, by Lemma 2.2 and convexity 1 |Q|ZQ Φ0f1(x)f2(x)· ·fm(x) m(kf1kΦ1,Q +ε)(kf2kΦ2,Q +ε)· ·(kfmkΦm,Q +ε)dx ≤ 1 m 1 |Q|ZQ Φ0f1(x)f2(x)· ·fm(x) (kf1kΦ1,Q +ε)(kf2kΦ2,Q +ε)· ·(kfmkΦm,Q +ε)dx ≤ 1 m 1 |Q|ZQΦ1f1(x) kf1kΦ1,Q +ε+··+Φmf1(x) kfmkΦm,Q +εdx ≤1 which implies that kf1f2· ·fmkΦ0,Q ≤m(kf1kΦ1,Q +ε)· ·(kfmkΦm,Q +ε). Now, taking letting ε→0 we get (2.4). The main example that we will be using is the following: 1 |Q|ZQ |f1···fmg| ≤ Ckf1kexpLr1,Q ···kfmkexpLrm,QkgkL(logL)1/r,Q,(2.5) where r1, .., rm≥1 and 1 r=1 r1 +··+1 rm . Indeed, in this case we can write for any x≥0 that log 1 r1(1 + x)···log 1 rm(1 + x)x log 1 r1+··+1 rm(e+x) ≤x. Then (2.3) holds for Φ0(x) = x, Φ−1 i(x) = log 1 ri(1 + x), i= 1, .., m and Φ−1 m+1(x) = x log 1 r1+··+1 rm(e+x) . The inverse functions are given by Φi(x) = exri−1, i= 1, .., m, and Φm+1(x)≈xlog 1 r1+··+1 rm(e+ x). 9 Now, as in (3.4) and making use again of Kolmogorov’s inequality and (2.5), we can estimate IV by IV ≤C |2B|Z2B |b1(y)−λ1|···|bm(y)−λm||f(y)|dy ≤Ckb1−λ1kexpLr1,2B···kbm−λmkexpLrm,2BkfkL(logL)1/r,2B ≤CML(logL)1/r f(x),(3.10) where recall that 1 r=1 r1 +··+1 rm . Finally, for V, choosing c= (T((b1−λ1)··(bm−λm)f2)))Band repeating the argument used to get (3.5), it follows from (2.5) and (3.6) that V≤C ∞ X k=1 2−kγ 1 (2k+1R)nZ2k+1B  m Y j=1 |bj(w)−λj| |f(w)|dw ≤C ∞ X k=1 2−kγ   m Y j=1 kbj−λjkexpLrj,2k+1B kfkL(logL)1/r,2k+1B ≤C"∞ X k=1 2−kγkm#  m Y j=1 kbjkoscexpLrj ML(logL)1/r f(x).(3.11) Finally, from (3.10) and (3.11) we conclude III ≤CML(logL)1/r f(x), which together with (3.8) and (3.9) gives (3.1) and the proof of the lemma is finished. We are now in a position to prove Theorem 1.1 Proof of Theorem 1.1: We may assume that ZRn (ML(logL)1/r f(x))pw(x)dx < ∞,(3.12) since otherwise there is nothing to be proved. To apply the Fefferman-Stein inequality (2.2) we first take for granted that kMδ(T~ bf)kLp(w) is finite. We will check this to the end of the proof. 16 We proceed by induction on m. For m= 1, by (2.2) and Lemma 3.1 we can estimate kTb1fkLp(w)≤ kMδ(Tb1f)kLp(w) ≤CkM# δ(Tb1f)kLp(w) ≤Ckb1koscexpLr1hkMε(Tf)kLp(w)+kML(logL)1/r1fkLp(w))i ≤Ckb1koscexpLr1hkTfkLp(w)+kML(logL)1/r1fkLp(w)i ≤Ckb1koscexpLr1hkMfkLp(w)+kML(logL)1/r1fkLp(w)i ≤Ckb1koscexpLr1kML(logL)1/r1fkLp(w) where the fourth inequality follows since w∈A∞and therefore there exists q > 1 such that w∈Aq, then we can choose εsuch that 0 < ε < p/q (we may take q > p if necessary). The fifth holds by the classical estimate (1.3) for m= 0 (see [6], Chapter 1). Suppose now that for m−1 the theorem is true and let’s prove it for m. So, the same argument used above and by the induction hypothesis gives kT~ bfkLp(w)≤ kMδ(T~ bf)kLp(w) ≤CkM# δ(T~ bf)kLp(w) ≤Chkb1koscexpLr1· ·kbmkoscexpLrmkML(logL)1/r fkLp(w) + m−1 X j=1 X σ∈Cm j kbσkoscexpLσkMε(T~ bσ0)kLp(w)  ≤Chkb1koscexpLr1· ·kbmkoscexpLrmkML(logL)1/r fkLp(w) + m−1 X j=1 X σ∈Cm j kbσkoscexpLσkbσ0koscexpLσ0kML(logL)1/rσ0fkLp(w)  ≤Ckb1koscexpLr1· ·kbmkoscexpLrmkML(logL)1/r fkLp(w), since ML(logL)1/rσ0≤C ML(logL)1/r Let us check now that for appropriate δwe have kMδ(T~ bf)kLp(w)<∞. Indeed, as above since w∈A∞, there exists q > 1 such that w∈Aqand we can choose δsmall enough so that p/δ > q. Then by Muckehoupt’s theorem all is reduced to checking that kTbfkLp(w)<∞. Suppose that the symbols bk’s and the weight ware all bounded functions. Since fhas compact support we may assume that the support of fis contained in the ball BR=B(0, R). Then we can split the integral as ZRn |T~ bf(x)|pw(x)dx =Z|x|≤2R |T~ bf(x)|pw(x)dx +Z|x|>2R |T~ bf(x)|pw(x)dx. 17 The first integral can be easily estimated making use of the L∞–boundedness of the bk’s and wand the Lq–boundedness for q > 1 of the Calder´on–Zygmund operator T. For the second term, by the properties of the kernel Kand the boundedness of the symbols bk’s, since |x|>2R, we have the following pointwise estimate |T~ bf(x)| ≤ CZBR |b1(x)−b1(y)|···|bm(x)−bm(y)||f(y)| |x−y|ndy ≤C |x|nZB(0,|x|) |f(y)|dy ≤CMf(x) ≤CML(logL)1 rf(x).(3.13) Thus, Z|x|>2R |T~ bf(x)|pw(x)dx ≤CZ|x|>2R (ML(logL)1 rf(x))pw(x)dx which is finite by the assumption (3.12). For the general case, we will truncate the symbols bk’s and the weight was follows (cf [6], p. 40). We denote by ~ bNthe vector of truncated elements by N, i.e., ~ bN= (bN 1, .., bN m) where each bN kis the truncation of bkas it is defined in (2.6). Observe that in our case (2.7) becomes kbN kkoscexpLrk≤CkbkkoscexpLrk(3.14) with C > 0 a constant independent of N. Analogously we consider the truncations of the weight wby wN= inf{w, N}that satisfy [wN]A∞≤C[w]A∞.(3.15) Then (1.7) holds for the operator T~ bNand the weight wN. Combining (3.14) and (3.15), this estimate gives ZRn |T~ bNf(x)|pwN(x)dx ≤C  m Y j=1 kbjkp oscexpLrj ZRn (ML(logL)1/r f(x))pw(x)dx. Next, taking into account that fhas compact support, we deduce that any product bN i1··bN ikf converges in any Lqfor q > 1 to bi1· ·bikfas N→ ∞. Hence, the classical Lq–boundedness of the operator Tgives, at least for a subsequence, that |T~ bNf(x)|pwN(x) converges pointwise almost everywhere to |T~ bf(x)|pw(x) and by Fatou’s lemma we conclude the theorem for this general case. The theorem is proved. 3.2 Proof of Theorem 1.5 We adapt here some of the arguments from [20]. Since the proof of Theorem 1.5 is based on Theorem 1.6 we prove this first, namely we must show that sup t>0 1 Φ(1 t)w({y∈Rn:|T~ bf(y)|> t})≤Csup t>0 1 Φ(1 t)w({y∈Rn:MΦ(||~ b||f)(y)> t}) (3.16) 18 for all bounded functions fwith compact support. Recall that Φ(t) = Φ~ b(t) = tlog1/r(e+t). In fact we are going to prove something stronger than (3.16) namely: For every ~ b,ϕ: (0,∞)→(0,∞) doubling with ϕ(t)≤C t,t > 0, and for every 0< δ < 1 there exists a constant Csuch that sup t>0 ϕ(t)w({y∈Rn:Mδ(T~ bf)(y)> t})≤Csup t>0 ϕ(t)w({y∈Rn:MΦ(||~ b||f)(y)> t}) (3.17) for all bounded functions fwith compact support. By the Lebesgue differentiation theorem and taking ϕ(t) = Φ(1 t)−1=t log1/r(e+1 t)it is clear that (3.17) implies (3.16). By making use of the weighted version of the Fefferman-Stein lemma 2.1, more precisely estimate (2.1), we have that sup t>0 ϕ(t)w({y∈Rn:Mδ(T~ bf)(y)> t})≤Csup t>0 ϕ(t)w({y∈Rn:M# δ(T~ bf)(y)> t}) (3.18) whenever the left hand side is finite. Therefore (3.17) will follow from sup t>0 ϕ(t)w({y∈Rn:M# δ(T~ bf)(y)> t})≤Csup t>0 ϕ(t)w({y∈Rn:MΦ(||~ b||f)(y)> t}).(3.19) We first check that the left hand side of (3.18) is finite for all bounded function fwith compact support. By proceeding as in the proof of Theorem 1.1, we may assume that band w are bounded. For the general case of unbounded symbols and unbounded weight we reproduce the argument used in the proof of Theorem 1.1 taking into account this time the weak–(1,1) boundedness of the operator Twhich gives the convergence in measure. Suppose that supp f ⊂BR=B(0, R). Hence, since 0 < δ < 1, it follows ϕ(t)w({y∈Rn:Mδ(T~ bf)(y)> t})≤C ϕ(t)|{y∈Rn:Mδ(χB2RT~ bf)(y)> t/2}| +C ϕ(t)|{y∈Rn:Mδ(χRn\B2RT~ bf)(y)> t/2}| =I+II. For Iwe use that Mis of weak type–(1,1) and the fact that ϕ(t)≤C t, then I≤Ct |{y∈Rn:M(χB2RT~ bf)(y)> t/2}| ≤CZB2R |T~ bf(y)|dy ≤CRn/2ZRn |Tf(y)|2dy1/2 , which is finite since Tis a Calder´on–Zygmund operator and using that the symbols bk’s are bounded. 19 For II we take into account the pointwise estimate (3.13) and the well known fact that (Mf)δ∈A1, then we have II ≤C t |{y∈Rn:Mδ(Mf)(y)> Ct}| ≤Ct |{y∈Rn:Mf(y)> Ct}| ≤CZRn |f(y)|dy < ∞. Combining the homogeneity and the linearity of T~ b, it is easy to see that we may assume that ||~ b|| = 1 in both (1.10) and (3.16). To prove (3.19) we proceed by induction on m. 3.3 The case m= 1 This case is essentially taken from [20] and we repeat it, with minor modifications, for the sake of completeness. In this case the operator Tbis simply defined by one single function b Tbf= [b, T]f=b T(f)−T(bf), where Tis any Calder´on–Zygmund operator. Recall that by homogeneity we may assume that ||b|| =kbkoscexpLr= 1 and therefore what we must prove is sup t>0 ϕ(t)w({y∈Rn:M# δ([b, T]f)(y)> t})≤Csup t>0 ϕ(t)w({y∈Rn:ML(log L)1/r (f)(y)> t}) (3.20) for all bounded functions fwith compact support. Now applying Lemma 3.1 with any αsuch that δ < α < 1 we have that the left hand side of (3.20) is estimated by Csup t>0 ϕ(t)w({y∈Rn:ChML(log L)1/r (f)(y) + Mα(Tf)(y)i> t}) ≤Csup t>0 ϕ(t)w({y∈Rn:ML(log L)1/r (f)(y)> t}) + Csup t>0 ϕ(t)w({y∈Rn:Mα(Tf)(y)> t}) where we also have used the doubling condition of ϕ. Next, considering the estimate M# α(Tf)(y)≤CαMf(y),(3.21) which holds for all 0 < α < 1 (see [1], Theorem 2.1), if we further select αsuch that 0 < δ < α < 1, the Fefferman-Stein’s lemma yields sup t>0 ϕ(t)w({y∈Rn:M# δ([b, T]f)(y)> t})≤ ≤Csup t>0 ϕ(t)w({y∈Rn:ML(log L)1/r (f)(y)> t}) + Csup t>0 ϕ(t)w({y∈Rn:M# α(Tf)(y)> t}) ≤Csup t>0 ϕ(t)w({y∈Rn:ML(log L)1/r (f)(y)> t}) + Csup t>0 ϕ(t)w({y∈Rn:M(f)(y)> t}) ≤Csup t>0 C ϕ(t)w({y∈Rn:ML(log L)1/r (f)(y)> t}), since trivially M(f) = ML(f)≤ML(log L)1/r (f). This finishes the proof of (3.20). 20 3.4 The general case Suppose now that (3.19) holds for m−1 and let’s prove it for m. Recall that Φ(t)=Φb(t) = tlog1/r(e+t) with 1 r=1 r1+··+1 rm. Then, by Lemma 3.1, sup t>0 ϕ(t)w({y∈Rn:M# δ(T~ bf)(y)> t}) ≤sup t>0 ϕ(t)w({y∈Rn:C MΦ(f)(y) + m X j=1 X σ∈Cm j kbσkoscexpLrσMε0(T~ bσ0f)(y) > t}) ≤Cmsup t>0 ϕ(t)w({y∈Rn:MΦ(f)(y)> t}) +Cm m X j=1 X σ∈Cm j sup t>0 ϕ(t)w({y∈Rn:Mε0T~ bσ0kbσkoscexpLrσf(y)> t}) Since ε0<1 and we already checked that the distribution set on the left hand side is finite, combining the Fefferman–Stein’s lemma together with the induction hypothesis on (3.19) to T~ bσ0we can estimate the last expression by Cm m X j=1 X σ∈Cm j sup t>0 ϕ(t)w({y∈Rn:M# ε0T~ bσ0kbσkoscexpLrσf(y)> t}) ≤Cm m X j=1 X σ∈Cm j sup t>0 ϕ(t)w({y∈Rn:MΦ~ bσ0 (kbσ0koscexpLrσ0kbσkoscexpLrσf)(y)> t}) ≤Cm m X j=1 X σ∈Cm j sup t>0 ϕ(t)w({y∈Rn:MΦ~ bσ0 (f)(y)> t}) since k~ bσ0koscexpLrσ0k~ bσkoscexpLrσ=||~ b|| = 1. Finally, using the trivial observation that MΦ~ bσ0 (f)≤ MΦ~ bσ(f) = MΦ(f) we have sup t>0 ϕ(t)w({y∈Rn:M# δ(T~ bf)(y)> t})≤Cmsup t>0 ϕ(t)w({y∈Rn:MΦ(f)(y)> t}) and the claim (3.19) is proved. We need the following lemma concerning estimates of the maximal operator MΦ, which is a more general version than the one given in Lemma 8.3 [20]. The proof is standard and we shall omit it. Lemma 3.2 Let w∈A1, then there exists a positive constant Csuch that for any t > 0and any locally integrable function f w({y∈Rn:MΦf(y)> t})≤CZRn Φ(|f(y)| t)w(y)dy. 21 We are now in position to prove Theorem 1.5 Proof of the Theorem 1.5: By homogeneity it is enough to assume t=||b|| = 1 and hence we must prove w({y∈Rn:|Tbf(y)|>1})≤CZRn Φ(|f(y)|)w(y)dy. Now, since Φ is submultiplicative, namely Φ(ab)≤2Φ(a) Φ(b), a, b ≥0 we have by Theorem 1.6 and Lemma 3.2 w({y∈Rn:|Tbf(y)|>1})≤Csup t>0 1 Φ(1 t)w({y∈Rn:|Tbf(y)|> t}) ≤Csup t>0 1 Φ(1 t)w({y∈Rn:MΦf(y)> t}) ≤Csup t>0 1 Φ(1 t)ZRn Φ(|f(y)| t)w(y)dy ≤Csup t>0 1 Φ(1 t)ZRn Φ(|f(y)|)Φ(1 t)w(y)dy ≤CZRn Φ(|f(y)|)w(y)dy and the proof is concluded. References [1] Alvarez, J., and P´ erez, C.,Estimates with A∞weights for various singular integral operators, Boll. 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P.,Weakly Differentiable Functions, Graduate Texts in Mathematics, Vol. 120, Springer-Verlag, New York/Berlin, 1989. Carlos P´ erez Departmento de An´ alisis Matem´ atico Facultad de Matem´ aticas Universidad de Sevilla 41080 Sevilla Spain E-mail address: [email protected] Rodrigo Trujillo-Gonz´ alez Departmento de An´ alisis Matem´ atico Universidad de La Laguna 38271 La Laguna - S/C de Tenerife Spain E-mail address: r[email protected] 24