Borderline weighted estimates for commutators of singular integrals

arXiv:1507.08568v2 [math.CA] 22 Jan 2016 BORDERLINE WEIGHTED ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS CARLOS PÉREZ AND ISRAEL P. RIVERA-RÍOS Abstract. In this paper we establish the following estimate w({x∈Rn:|[b, T ]f(x)|> λ})≤cT ε2ˆRn ΦkbkBMO |f(x)| λML(log L)1+εw(x)dx where w≥0,0< ε < 1and Φ(t) = t(1 + log+(t)). This inequality relies upon the following sharp Lpestimate k[b, T ]fkLp(w)≤cT(p′)2p2p−1 δ1 p′ kbkBMO kfkLp(ML(log L)2p−1+δw) where 1< p < ∞, w ≥0and 0< δ < 1.As a consequence we recover the following estimate essentially contained in [18]: w({x∈Rn:|[b, T ]f(x)|> λ})≤cT[w]A∞1 + log+[w]A∞2ˆRn ΦkbkBMO |f(x)| λMw(x)dx We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols. 1. Introduction Motivated by a classical inequality due to C. Fefferman and E. Stein for the HardyLittlewood maximal function, namely kMfkL1,∞(w)≤cˆRn|f|Mwdx where Mdenotes the Hardy-Littlewood maximal operator and wis a weight, i.e. a locally non negative integrable function, B. Muckenhoupt and R. Wheeden conjectured that kHfkL1,∞(w)≤cˆRn|f|Mwdx where His the Hilbert transform. This conjecture was recently disproved by M. C. Reguera and C. Thiele [26] (see also [4] for the result in higher dimensions). The failure of this conjecture was suggested by the first author in [20] where the following positive result was obtained (1) kT fkL1,∞(w)≤cε,T ˆRn|f|ML(log L)ε(w)dx w ≥0, where Tis a Calderón-Zygmund operator (CZO). In the recent work [10], T. Hytönen and the first author improved the control on the cε,T constant and were able to consider the 2010 Mathematics Subject Classification. 42B35,46E30. Key words and phrases. Commutators, Rubio de Francia Extrapolation; Apweights; Hardy-Littlewood maximal function. The first author was supported by Grant MTM2014-53850-P and the second author was supported by Grant MTM2012-30748, Spanish Government. 1 2 CARLOS PÉREZ AND ISRAEL P. RIVERA-RÍOS maximal singular operator T∗obtaining the following estimate (see [6] for an improvement of this result) (2) kT∗fkL1,∞(w)≤cT εˆRn|f(x)|ML(log L)ε(w)(x)dx w ≥0 which implies (3) kT∗fkL1,∞(w)≤cTlog (e+ [w]A∞)ˆRn|f|Mwdx, when w∈A∞. This result improves the main theorem from [15], namely (4) kT∗kL1(w)→L1,∞(w)≤cT[w]A1log (e+ [w]A∞). It seemed that the logarithmic factor was superfluous and that it could be removed. However, this is not the case by a very impressive negative result obtained by F. Nazarov, A. Reznikov, V. Vasyunin and A. Volberg in [17]. In this work the authors disprove the so called A1 conjecture, namely they prove sup w∈A1 kHkL1(w)→L1,∞(w) [w]A1 =∞ where His the classical Hilbert transform. Furthermore, the same conclusion holds if the linear constant [w]A1is replaced by [w]A1log(e+ [w]A1)αfor a positive α < 1 5. This is indicating that most probably (4) is fully optimal. The main purpose of this paper is to prove estimates similar to (2) for commutators of CZOs Twith BMO functions b, usually called the symbol. These operators are defined formally by the expression [b, T ]f=bT (f)−T(b f), These commutators were introduced by Coifman, Rochberg and Weiss in [3] in connection with the classical factorization theorem for Hardy spaces. However, many other applications were found much later, specially in the theory of elliptic operators [13], [2]. Another interesting aspect of the theory is its connection with the following nonlinear commutators introduced by R. Rochberg and G. Weiss in [27]: f→Nf =T(flog |f|)−T f log |T f|. This operator is interesting due to its relationship with the Jacobian mapping and with nonlinear P.D.E. as shown in [12] and [8]. The main result from [3] states that [b, T ]is a bounded operator on Lp(Rn),1< p < ∞, when bis a BMO function and Tis a singular integral operator. In fact, the BMO condition of bis also a necessary condition for the Lp-boundedness of the commutator when Tis the Hilbert transform. From the theoretical point of view, these commutators are of interest because they are more singular than CZOs. For instance, the first author proved in [21] that these commutators are not of weak type (1,1) obtaining a suitable replacement, namely the following Llog L endpoint estimate: w({x∈Rn:|[b, T ]f(x)|> λ})≤cˆRn Φ|f| λkbkBMOwdx where Φ(t) = t1 + log+(t),w∈A1and the constant cdepends upon the A1constant of the weight. The approach to prove this result was based on an appropriate non-standard good-λinequality using the Fefferman-Stein “sharp” maximal function. However, this method BORDERLINE WEIGHTED ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS 3 does not produce good results for further developments and in particular when considering non-A∞weights. Later on, the first author together with G. Pradolini ([23]) established the following estimate for arbitrary weights w≥0, w({x∈Rn:|[b, T ]f(x)|> λ})≤cT,ε ˆRn Φ|f| λkbkBMOML(log L)1+εwdx for any ε > 0. The aim of this paper is to give a quantitative version of this estimate with a good control on the constant CT,ε in terms of ε. In the simplest situation our estimate can be stated as follows (5) w({x∈Rn:|[b, T ]f(x)|> λ})≤cT ε2ˆRn Φ|f| λkbkBMOML(log L)1+εwdx. See Theorem 2for the general situation. In fact, we will obtain a wider class of results since we will be considering symbol-multilinear commutators with symbols in Oscexp Lsclasses which are subspaces of the BMO space (cf. Section 3.2). We will see that this choice of symbols will be reflected in the maximal operator on the right hand side of the inequality. As a consequence of these type of estimates we can recover, among other results, the following endpoint A1result from C. Ortiz [18] (see Corollary 1): w({x∈Rn:|[b, T ]f(x)|> λ})≤cTΦ ([w]A1)2ˆRn ΦkbkBMO |f(x)| λw(x)dx. Estimate (5) should be compared with the case of CZOs (2). It is not clear whether is possible or not to establish (5) using techniques based on sparse operators as in [6]. Another open question is the analogue of the Muckenhoupt-Wheeden conjecture for the commutator, namely whether w({x∈Rn:|[b, T ]f(x)|> λ})≤cwˆRn ΦkbkBMO |f(x)| λM2(w)(x)dx. holds for every weight wor not. Techniques used in [26] and [4] rely upon an endpoint extrapolation result, firstly established in [5], or upon variations of it as in [6]. It is not clear how to perform a similar extrapolation from the Llog Lestimate that commutators satisfy. This paper is organized as follows: Section 2contains the statements of our main results and the proof of Corollary 1. Section 3contains precise definitions and facts which will be used throughout the paper. In section 4we give the proof of the strong type theorem, namely Theorem 1, and all the needed technical results. The proof of the endpoint estimate (Theorem 2) is presented in Section 5. Acknowledgments The authors are very grateful to Carmen Ortiz-Caraballo for her interest in this work and for suggesting some improvement in the presentation of the paper. We also wish to thank the referee for valuable comments on the paper. 2. Main results To state our main results we need to introduce some notation. Let bi∈Oscexp Lsisi≥1, i= 1,···, k (cf. section 3.2 after Lemma 1) and Ta CZO with associated kernel K. We 4 CARLOS PÉREZ AND ISRAEL P. RIVERA-RÍOS define the symbol-multilinear commutator with respect to the symbol ~ b= (b1,...,bk)as follows T~ bf(x) = ˆRn k Y i=1 (bi(x)−bi(y)) K(x, y)f(y)dy We also denote 1 s= k X i=1 1 si and k~ bk= k Y i=1 kbikOscexp Lsi. Our main results are the following. Theorem 1. Let be T~ bdefined as above and let wbe a weight. Then kT~ bfkLp(w)≤cT(p′)k+1 p1+ 1 sp−1 δ1 p′ k~ bkkfkLp(M L(log L)(1+ 1 s)p−1+δw) for every δ∈(0,1) and p∈(1,∞). This result can be applied to derive the following endpoint estimate. Theorem 2. Let be T~ band was above. Then wx∈Rn:T~ bf> λ≤cT εk+1 ˆRn Φ1 sk~ bk|f(x)| λML(log L)1 s+εw(x)dx for every ε∈(0,1) where Φρ(t) = t(1 + log+(t))ρ, ρ > 0. There is an interesting application of Theorem 2from which we can recover one of the main results of [18]. Corollary 1. Let Tbe a CZO and let b∈BMO. (1) If w∈A∞then w({x∈Rn:|[b, T ]f(x)|> λ})≤c[w]A∞1 + log+[w]A∞2ˆRn ΦkbkBMO |f(x)| λMw(x)dx, (2) If w∈A1then w({x∈Rn:|[b, T ]f(x)|> λ}) ≤c[w]A1[w]A∞1 + log+[w]A∞2ˆRn ΦkbkBMO |f(x)| λw(x)dx ≤cΦ ([w]A1)2ˆRn ΦkbkBMO |f(x)| λw(x)dx where Φ(t) = t(1 + log+t). Proof. For the proof of the corollary we follow the arguments in [10]. First observe that for every α > 0we have that log t≤tα α. Then we can write log(t)1+ε≤tα(1+ε) α1+ε, and hence ML(log L)1+εw≤1 α1+εML1+α(1+ε). BORDERLINE WEIGHTED ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS 5 Let us take α=1 cn[w]A∞(1+ε). Then, using the reverse Hölder inequality (Theorem 3), 1 ε2ML(log L)1+ε≤1 ε2[cn[w]A∞(1 + ε)]1+εML1+α(1+ε)w ≤2 ε2[cn[w]A∞(1 + ε)]1+εMw. If we choose ε=1 1+log+([w]A∞)we obtain the desired results just recalling that [w]A∞≤[w]A1.  3. Preliminaries and notation In this section we gather some definitions and properties which will be used throughout the paper. 3.1. Apweights. We recall that a weight wbelongs to the class Ap,1< p < ∞, if [w]Ap= sup Q1 |Q|ˆQ w 1 |Q|ˆQ w−1 p−1p−1 <∞. A weight wbelongs to the class A1if there is a finite constant Csuch that 1 |Q|ˆQ w(y)dy ≤Cinf Qw, and the infimum of these constants Cis called the A1constant of wdenoted by [w]A1. Since the Apclasses are increasing with respect to p, the A∞class of weights is defined in a natural way by A∞=∪p>1Ap. These classes of weights were introduced by B. Muckenhoupt in [16] where it was shown that for 1< p < ∞ w∈Ap⇐⇒ M:Lp(w)−→ Lp(w) and also w∈A1⇐⇒ M:L1(w)−→ L1,∞(w). From the definition of A∞it is not clear how to define an appropriate constant. However, Fujii proved essentially in [7] another characterization: w∈A∞⇐⇒ [w]A∞= sup Q 1 w(Q)ˆQ M(χQw)dx < ∞ which was also rediscovered later on by Wilson in [29]. Recently, this quantity was defined as the A∞constant in [9] since it was proved to be the most suitable one. In particular, the following optimal reverse Hölder’s inequality obtained in [9] (see also [11] for a better proof and [28] for some other related results) was used in the proof of Corollary 1. Theorem 3. Let w∈A∞, then there exists a dimensional constant τnsuch that 1 |Q|ˆQ wrw1 rw≤2 |Q|ˆQ w. where rw= 1 + 1 τn[w]A∞ 6 CARLOS PÉREZ AND ISRAEL P. RIVERA-RÍOS 3.2. Orlicz maximal functions. We recall that Φis a Young function if it is a continuous, nonnegative, strictly increasing and convex function defined on [0,∞)such that Φ(0) = 0 and limt→∞ Φ(t) = ∞. We define the localized Luxembourg norm of a function fwith respect to a Young function Φas follows kfkΦ,Q =kfkΦ(L),Q = inf λ > 0 : 1 |Q|ˆQ Φ|f(x)| λdx ≤1 which is equivalent to the following kfk′ Φ,Q = inf µ>0(µ+µ |Qj|ˆQj Φ|f(x)| µdx). This result is due to Krasnosel’ski˘ı, M. A. and Ruticki˘ı, Ja. B. [14, p. 92] (see also [25, p. 69]). In fact, kfkΦ,Q ≤ kfk′ Φ,Q ≤2kfkΦ,Q which will be quite useful for our purposes. Observe that the case Φ(t) = tcorresponds to the usual average and we can see these localized norms as a “different” way of taking averages. We can also define the maximal function associated to Φas MΦf(x) = sup x∈QkfkΦ,Q. Some useful examples that will be quite useful in the sequel are Llog Lfunctions Φρ(t) = t(1 + log+(t))ρwith t≥0 where log+(t) = χ(1,∞)(t) log(t)and ρ > 0. For such Φwe shall denote kfkΦ,Q =kfkL(log L)ρ,Q. Another useful property that makes interesting these “non-standard averages” is the following generalized Hölder inequality. Lemma 1. Let Φ0,Φ1,Φ2,...,Φkbe Young functions. If (6) Φ−1 1(t)Φ−1 2(t)...Φ−1 k(t)≤κΦ−1 0(t). then for all functions f1, . . . , fmand all cubes Qwe have that kf1f2. . . fkkΦ0,Q ≤kκkf1kΦ1,Qkf2kΦ2,Q ...kfkkΦk,Q. A particular case of interest, an especially in this paper, are the spaces defined by kfkOscexpLs= sup Qkf−fQkΨs,Q where Ψs(t) = ets−1t≥0, with s > 0, is a Young function. Then the space Oscexp Lsis defined as Oscexp Ls=f∈L1 loc(Rn) : kfkOscexpLs<∞. We observe that John-Nirenberg’s theorem yields BMO =Oscexp L. It’s also clear that for every s > 1 Oscexp Ls(BMO. Now we state a result borrowed from [24] that will be used in the proof of Theorem 2. BORDERLINE WEIGHTED ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS 7 Lemma 2. Let Φ0,...,Φkbe continuous, nonnegative, strictly increasing functions on [0,∞) with Φ(0) = 0 and limt→∞ Φ(t) = ∞such that Φ−1 1(t)Φ−1 2(t)...Φ−1 k(t)≤Φ−1 0(t)t≥0, then for all 0≤x1, x2, . . . , xk<∞ Φ0(x1x2. . . xk)≤Φ1(x1) + Φ2(x2) + ···+ Φk(xk). To close this section we provide a proof of Lemma 1and also a corollary of it that will be quite useful in the proof of Theorem 2. Proof. Fix (x1,...,xk)and consider t0= Φ1(x1) + Φ2(x2) + ···+ Φk(xk). Combining (6) and the fact that each Φiis increasing it readily follows that Φ0Φ−1 1(t0)Φ−1 2(t0)...Φ−1 k(t0) κ≤t0 and also that Φ−1 i(t0)≥Φ−1 i(Φi(xi)) = xi. Then we have that (7) Φ0x1x2. . . xk κ≤Φ1(x1) + Φ2(x2) + ···+ Φk(xk) We observe that this argument gives us a proof of Lemma 2. Coming back to our proof, let us consider now ti>kfikΦi,Q. We have that using 7, 1 m 1 |Q|ˆQ Φ0|f1. . . fk| κt1. . . tk ≤1 m1 |Q|ˆQ Φ1|f1| t1+···+1 |Q|ˆQ Φk|fk| tk <1 Consequently kf1. . . fkkΦ0,Q ≤κt1. . . tk and it is enough to take the infimum on each tito finish the proof of the lemma.  As a particular case, the following corollary holds which will be used several times in this paper. Corollary 2. Let s1,...,sk≥1and denote Pk i=1 1 si. Then 1 |Q|ˆQ|f1. . . fkg| ≤ cskf1kexp Ls1,Q ...kfkkexp Ls k,QkgkL(log L)1 s,Q Proof. We denote ϕη(t) = etη−1. Then ϕ−1 η(t) = log(x+ 1)1 ηand we have that ϕ−1 s1(t). . . ϕ−1 sk(t)Φ−1 1 s (t)≃ϕ−1 s1(t). . . ϕ−1 sk(t)x log (x+ 1)1 s≤x and Lemma 1gives the desired result.  8 CARLOS PÉREZ AND ISRAEL P. RIVERA-RÍOS 3.3. Symbol-multilinear commutators. We recall that an operator Tinitially defined on the Schwartz spaces and taking values into the space of tempered distributions T:S(Rn)→ S′(Rn)is a CZO if, (1) Tis bounded on L2(Rn). (2) For each smooth and compactly supported function f,T f admits the following representation T f(x) = ˆRn K(x, y)f(y)dy x 6∈ supp f where Kis a standard kernel. Recall that a kernel K:Rn×Rn\∆−→ R, where ∆is the diagonal in Rn×Rn, is a locally integrable function such that for some constants C1, C2, γ > 0the following conditions hold: (a) Size condition |K(x, y)| ≤ C1 1 |x−y|nif x6=y. (b) Regularity condition |K(x, y)−K(x′, y)|+|K(y, x′)−K(y, x)| ≤ C2|x−x′|γ |x−y|n+γ provided that |x−x′| ≤ 1 2|x−y|. The symbol-multilinear commutator T~ bwith vector symbol~ b= (b1,···, bk),bi∈Oscexp Lsi, i = 1,···, k, and CZO Twith kernel Kis defined for smooth functions fas follows T~ bf(x) = ˆRn k Y i=1 (bi(x)−bi(y)) K(x, y)f(y)dy x 6∈ supp f. Let b={b1, b2,...,bk}be a set of symbols with bi∈Oscexp Lsi, i = 1,···, k. Also, let b=σ∪σ′where σand σ′are pairwise disjoint sets be a splitting of b. If we identify iand bi we can introduce the following notation (b(x)−λ)σ=Y i∈σ (bi(x)−λi) where λ= (λ1, λ2,...,λk)and also to write Pi∈σ 1 si. By Cj(b)we refer the family of all the subsets σof bsuch that #σ=j. We shall also omit the set of symbols and write just Ck j. Finally if σis a subset of bwe write T~σf(x) = ˆRnY i∈σ (bi(x)−bi(y)) K(x, y)f(y)dy =ˆRn (b(x)−b(y))σK(x, y)f(y)dy x 6∈ supp f. We end this section with some further notation. We write k~ bk=Y bi∈bkbikOscexp Lsi and similarly k~σk=Y bi∈σkbikOscexp Lsi. We will denote by #σthe cardinal of the set of symbols σ. BORDERLINE WEIGHTED ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS 9 3.4. Some estimates involving the sharp function. In this paper we will use two classical operators and some of their variations. The first one is the Hardy-Littlewood maximal operator, Mf(x) = sup x∈Q 1 |Q|ˆQ|f(y)|dy, where each Qis a cube with sides parallel to the axis. Also, Mdwill denote its dyadic version, where the supremum is taken over dyadic cubes. We will also use the following variants, Mε(f) = M(|f|ε)1 ε, and similarly for Md εwhere ε∈(0,∞). The second operator is the Fefferman-Stein sharp maximal function, namely M♯f(x) = sup x∈Q 1 |Q|ˆQ|f(y)−fQ|dy, and its dyadic counterpart M♯,d. Similarly as above we define the following useful variation M♯ δ(f) = M♯(|f|δ)1 δ with δ∈(0,∞). The first result that we state in this section is borrowed from [19]. Lemma 3. Let 0< p < ∞,0< δ < 1and let w∈A∞. Then kfkLp(w)≤cp[w]A∞  M♯,d δf  Lp(w) for any function fsuch that |{x:|f(x)|> t}| <∞for all t > 0. Using the preceding lemma and following the proof of Lemma 3.1 in [18] we can derive the following improvement. Lemma 4. Let 0< p < ∞,0< ε ≤1and w∈A∞. Suppose that |{x:|f(x)|> t}| <∞ for all t > 0. Then there is a constant c=cn,ε such that  Md εf Lp(w)≤cp[w]A∞ M♯,d εf Lp(w) Proof. Applying previous lemma with δ=ε0and 0< ε0< ε < 1  Md εf Lp(w)≤cp[w]A∞ M♯,d ε0Md εf Lp(w). Now it suffices to prove that M♯,d ε0Md εf(x)≤cM♯,d εf(x). But this was done in Lemma 3.1 of [18].  The reason why it is important to deal with M♯ εfor small εwill be clear after the following pointwise estimate proved in [24]. Lemma 5. Let T~ bbe the symbol-multilinear commutator defined above and let 0< δ < ε < 1. Then there exists a constant c > 0, depending only on δand εsuch that M♯ δT~ bf(x)≤cδ,ε  k~ bkML(log L)1 s(f) + k X j=1 X σ∈Ck j k~σkMεT~ bσ′f(x)  for any bounded function fwith compact support. 16 CARLOS PÉREZ AND ISRAEL P. RIVERA-RÍOS Let us choose t=eλand tρ=eλρ. Then 1 + λ−ρlog 1 + log eλ eλρ 1 + log eλ eλρ= 1 + λρ−ρlog (1 + λ−λρ) 1 + λ−λρ = 1 + gρ(λ) Now we minimize gρ(λ). It’s easy to check that gρreaches its minimum when λ=e1+ λρ ρ+ λρ−1. We observe that gρe1+ λρ ρ+λρ−1=−ρ e1+ λρ ρ and since tρ=ρρ −ρ e1+ λρ ρ =−1 e and we obtain the desired lower bound. To finish the proof we focus on the bound. If t∈(0,1),then Aρ(˜ Xρ(t)) = tand there’s nothing to prove. If t∈[1, tρ]then we have that Aρ(˜ Xρ(t)) = t(1 + log t)ρ≤t(1 + log tρ)ρ=t(1 + ρlog ρ)ρ. Finally if t∈(tρ,∞)then it’s easy to check that Aρ(˜ Xρ(t)) ≤t(1 + log (tρ))ρ.  Finally, with the precise control of the inverses at our disposal we are ready to give the proof of lemma 6. Proof of Lemma 6.Proving (11) is equivalent to prove that ˆRn MLlog L1 sfw 1 pp′M L(log L)(1+ 1 s)p−1+δw1−p′ ≤cp′ np1+ 1 sp′p−1 δˆRn|f|p′ Using now the notation of Lemma 7, we can write A1 s(t) = t(1 + log+t)1 sand X1 s(t) = t (1+log+t)1 sand we have that A−1 1 s (t)≥X1 s(t) We observe now that X1 s(t) = t 1 + log+t1 s =t1 p 1 + log+t1 s+p−1+δ p·t1 p′1 + log+tp−1+δ p =  t 1 + log+t(1+ 1 s)p−1+δ  1 p t1 + log+t1+δ(p′−1)1 p′=F1(t)1 p·F2(t)1 p′ Using again the notation of Lemma 7, F1(t) = X(1+ 1 s)p−1+δ(t) = t 1 + log+t(1+ 1 s)p−1+δ. From that lemma it readily follows that F1(t)1 p≥ 1 1 + 1 sp+δ!(1+ 1 s)p−1+δ p A−1 (1+ 1 s)p−1+δ(t)1 p. BORDERLINE WEIGHTED ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS 17 Analogously, following the notation of Lemma 8 F2(t) = A1+δ(p′−1)(t) = t1 + log+t1+δ(p′−1) From that lemma it follows that F2(t)1 p′≥e−1 e1+δ(p′−1) p′˜ X−1 1+δ(p′−1)(t)1 p′. Taking into account (4.3) and (4.3) we obtain the following estimate A−1 1 s (t) (e′)1+δ(p′−1) p′1 + 1 sp+δ(1+ 1 s)p−1+δ p ≥A−1 (1+ 1 s)p−1+δ(t)1 p˜ X−1 1+δ(p′−1)(t)1 p′t > 0. Using now generalized Hölder inequality (Lemma 1) and taking into account that, since δ∈(0,1), (e′)1+δ(p′−1) p′(2p+δ)(1+ 1 s)p−1+δ p≤cp1+ 1 s and also that kwkΨ(L)=kwpk 1 p ΨL1 pif Ψis a Young function, we have that   fw1 p  L(log L)1 s,Q ≤cp1+ 1 skfk˜ X1+δ(p′−1)(Lp′),Q kwk 1 p A(1+ 1 s)p−1+δ(L),Q and consequently ML(log L)1 sfw 1 p≤cp1+ 1 sM˜ X1+δ(p′−1)(Lp′)(f)M L(log L)(1+ 1 s)p−1+δ(w)1 p. Using this estimate we have that ˆRn ML(log L)1 sfw1 pp′M L(log L)(1+ 1 s)p−1+δw1−p′ dx ≤ˆRncp(1+ 1 s)M˜ X1+δ(p′−1)(Lp′)(f)M L(log L)(1+ 1 s)p−1+δ(w)1 pp′M L(log L)(1+ 1 s)p−1+δw1−p′ dx =cp1+ 1 sp′ˆRn M˜ X1+δ(p′−1)(Lp′)(f)p′dx Lemma 2.1 of [10] yields ˆRn M˜ X1+δ(p′−1)(Lp′)f(x)p′dx1 p′ ≤cp−1 δ1 p′ˆRn|f|p′(x)dx1 p′ , since ˆ∞ 1 ˜ X1+δ(p′−1)(tp′) tp′ dt t!1 p′ =(1 + (p′−1)δ) p′log (1 + (p′−1)δ) + 1 (p′−1)δ1 p′ and 0< δ < 1allows us to write (1 + (p′−1)δ) p′log (1 + (p′−1)δ) + 1 (p′−1)δ1 p′ ≤cp−1 δ1 p′ . Consequently we have that   ML(log L)1 sfw 1 p  Lp′(v1−p′)≤cp1+ 1 sp−1 δ1 p′ kfkLp′(Rn). 18 CARLOS PÉREZ AND ISRAEL P. RIVERA-RÍOS This concludes the proof of (4.3).  5. Proof of Theorem 2 5.1. Case k= 1. Proof. By homogeneity we shall suppose that kbkOscexpLs= 1. We consider the CalderónZygmund decomposition of fat height λ. That decomposition allows us to obtain a family of dyadic cubes {Qj}which are pairwise disjoint such that λ≤1 |Qj|ˆQj|f| ≤ 2nλ. Let us denote Ω = [ j Qj As usual, we write f=g+hwhere g, the “good” part of f, is defined as g(x) = (f(x)x∈Ωc fQjx∈Qj and verifies that |g(x)| ≤ 2nλa.e. and h=Phjwhere hj=f−fQjχQjand fQj= 1 |Qj|´Qjf(x)dx. We denote w∗(x) = w(x)χRn\˜ Ω(x)and wj(x) = w(x)χRn\˜ Qjwhere ˜ Qj= 5√nQjand ˜ Ω = Sj˜ Qj. Using that decomposition we can write w({x∈Rn:|[b, T ]f(x)|> λ})≤wx∈Rn\˜ Ω : |[b, T ]g(x)|>λ 2+w(˜ Ω) +wx∈Rn\˜ Ω : |[b, T ]h(x)|>λ 2 =I+II +III To end the proof we have to estimate I, II and III. Let us begin with I. If p > 0, Chebyschev’s inequality gives wx∈Rn\˜ Ω : |[b, T ]g(x)|>λ 2≤2p λpˆRn|[b, T ]g(x)|pw∗(x)dx. Let us choose 1 + ε 3(1+ 1 s)< p < 1 + ε 2(1+ 1 s)yδ=ε−1 + 1 s(p−1). For that choice of pand δ, is easy to check that (p′)2pp(1+ 1 s)pp−1 δp p′ ≤cs 1 ε2and 1 + 1 sp−1 + δ=1 s+ε. Using now Theorem 1, we have that 2p λpˆRn|[b, T ]g(x)|pw∗(x)dx ≤c(p′)2pp(1+ 1 s)pp−1 δp p′ˆRn|g(x)|pM L(log L)(1+ 1 s)p−1+δw∗(x)dx ≤c1 ε2 2p λpˆRn|g(x)|pML(log L)1 s+εw∗(x)dx ≤c1 ε2 1 λˆRn|g(x)|ML(log L)1 s+εw∗(x)dx ≤c1 ε2 1 λˆRn\Ω|f(x)|ML(log L)1 s+εw(x)dx +ˆΩ|g(x)|ML(log L)1 s+εw∗(x)dx BORDERLINE WEIGHTED ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS 19 and it suffices to estimate last integral. Indeed, ˆΩ|g(x)|ML(log L)1 s+εw∗(x)dx ≤X j|f|QjˆQj ML(log L)1 s+εwj(x)dx ≤cX j|Qj|1 |Qj|ˆQj|f(y)|dy inf z∈Qj ML(log L)1 s+εwj(z) =cX jˆQj|f(y)|inf z∈Qj ML(log L)1 s+εwj(z)dy ≤cX jˆQj|f(y)|ML(log L)1 s+εwj(y)dy ≤cˆΩ|f(y)|ML(log L)1 s+εw(y)dy. Summarizing, we obtain that I≤c1 ε2ˆRn |f(y)| λML(log L)1 s+εw(y)dy. For II we have the following standard estimate II =w(˜ Ω) ≤X jˆ5√nQj w(x)dx =X j|5√nQj|1 |5√nQj|ˆ5√nQj w(x)dx ≤X j5√nn|Qj|inf z∈Qj Mw(z)≤5√nnX j 1 λˆQj f(y)dy inf z∈Qj Mw(z) ≤5√nnX j 1 λˆQj Mw(y)f(y)dy ≤5√nnˆRn f(y) λMw(y)dy To estimate III we split the operator as follows [b, T ]h=X j [b, T ]hj=X j (bT (hj)−T(bhj)) = X jb−bQjT(hj)−X j Tb−bQjhj. Then we continue with III ≤w (x∈Rn\˜ Ω : X jb(x)−bQjT hj(x) >λ 4)! +w (x∈Rn\˜ Ω : X j Tb−bQjhj(x) >λ 4)! =A+B 20 CARLOS PÉREZ AND ISRAEL P. RIVERA-RÍOS To estimate Awe use standard computations based on the smoothness property of the kernel Kand the cancellation of each hj, A≤c λˆRn\˜ ΩX j|b(x)−bQj||T hj(x)|w(x)dx ≤c λX jˆRn\˜ Qj|b(x)−b|w(x)ˆQj|hj(y)|K(x, y)−K(x, xQj)dydx ≤c λX jˆQj|hj(y)|ˆRn\˜ Qj|K(x, y)−K(x, xQj)||b(x)−bQj|wj(x)dxdy ≤c λX jˆQj|hj(y)|ˆRn\˜ Qjy−xQj γ x−xQj n+γ|b(x)−bQj|wj(x)dxdy ≤c λX jˆQj|hj(y)|∞ X k=1 ˆ2kl(Qj)≤|x−xQj|<2k+1l(Qj)y−xQj γ x−xQj n+γ|b(x)−bQj|wj(x)dxdy ≤c λX j ˆQj|hj(y)|dy!∞ X k=1 2−γk |2k+1Qj|ˆ2k+1Qj|b(x)−bQj|wj(x)dx We now fix one term of the sum. Using generalized Hölder inequality, Lemma 1, we have ∞ X k=0 2−γk |2k+1Qj|ˆ2k+1Qj|b(x)−bQj|wj(x)dx ≤∞ X k=0 2−γk |2k+1Qj|ˆ2k+1Qj|b(x)−b2k+1Qj|wj(x)dx +∞ X k=0 2−γk |2k+1Qj|ˆ2k+1Qj|b2k+1Qj−bQj|wj(x)dx ≤∞ X k=1 2−γkkb−b2k+1Qjkexp Ls,2k+1QjkwjkLlog L1 s,2k+1Qj +∞ X k=1 2−γk(k+ 1)kbkOscexpLsinf z∈Qj Mwj(z) ≤∞ X k=1 2−γkkbkOscexpLsinf z∈Qj MLlog L1 swj(z) +∞ X k=1 2−γk(k+ 1)kbkOscexpLsinf z∈Qj Mwj(z) ≤c inf z∈Qj MLlog L1 swj(z)∞ X k=1 2−γk + inf z∈Qj Mwj(z)∞ X k=1 2−γk(k+ 1)! ≤cinf z∈Qj MLlog L1 swj(z) BORDERLINE WEIGHTED ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS 21 Consequently, A≤c λX jˆQj|hj(y)|dy inf y∈Qj MLlog L1 s(wj)(y) ≤c λX jˆQj MLlog L1 s(wj)(y)|hj(y)|dy ≤c λ ˆRn|f(y)|MLlog L1 s(wj)(y)dy +X jˆQj MLlog L1 s(wj)(y)|fQj|dy! ≤c λ ˆRn|f(y)|MLlog L1 s(wj)(y)dy +X jˆQj f(y)dy inf z∈Qj MLlog L1 s(wj)(z)dy! ≤c λˆRn|f(y)|MLlog L1 s(wj)(y)dy To end the proof we estimate B. Theorem 1.1 from [10] gives B=w∗ (x∈Rn:X j Tb−bQjhj(x) >λ 4)! ≤c1 ε 1 λˆRnX jb(x)−bQjhj (x)ML(log L)ε(w∗)(x)dx ≤c1 ε 1 λX jˆQjb(x)−bQjf(x)−fQjML(log L)ε(wj)(x)dx ≤c1 ε 1 λX j inf z∈Qj ML(log L)ε(wj)(z) ˆQjb(x)−bQj|f(x)|dx +ˆQjb(x)−bQjfQjdx! =1 ε(B1+B2) For B2 B2=c λX j inf z∈Qj ML(log L)ε(wj)(z)ˆQjb(x)−bQjfQjdx ≤c λX j 1 |Qj|ˆQjb(x)−bQjdx ˆQj|f(y)|ML(log L)ε(wj)(y)dx ≤c λX jkbkOscexpLsˆQj|f(y)|ML(log L)ε(wj)(y)dx ≤cX jˆQj |f(y)| λML(log L)ε(wj)(y)dy ≤cˆRn |f(x)| λML(log L)εw(x)dx. 22 CARLOS PÉREZ AND ISRAEL P. RIVERA-RÍOS For B1we use the generalized Hölder inequality Lemma 1and we obtain B1=c λX j inf z∈Qj ML(log L)ε(wj)(z)ˆQjb(x)−bQj|f(x)|dx ≤cX j inf z∈Qj ML(log L)ε(wj)(z)1 λ|Qj|kbkOscexpLskfkL(log L)1 sL,Qj =cX j inf z∈Qj ML(log L)ε(wj)(z)1 λ|Qj|kfkL(log L)1 s,Qj. (12) Now we see that 1 λ|Qj|kfkL(log L)1 s,Qj≤1 λ|Qj|inf µ>0(µ+µ |Qj|ˆQj Φ1 s|f(x)| µdx) ≤1 λ|Qj| λ+λ |Qj|ˆQj Φ1 s|f(x)| λdx!=|Qj|+ˆQj Φ1 s|f(x)| λdx ≤1 λˆQj|f(x)|dx +ˆQj Φ1 s|f(x)| λdx ≤2ˆQj Φ1 s|f(x)| λdx. (13) Consequently B1≤cX j inf z∈Qj ML(log L)ε(wj)(z)ˆQj Φ1 s|f(x)| λdx ≤cX jˆQj Φ1 s|f(x)| λML(log L)ε(wj)(x)dx ≤cˆRn Φ1 s|f(x)| λML(log L)ε(w)(x)dx.  5.2. Case k > 1. Proof. Let us suppose that the desired inequality holds for l≤k−1symbols. By homogeneity we may assume that kbkOscexpLs1=···=kbkOscexpLsk= 1. Using the Calderón-Zygmund decomposition with the same notation used in the case k= 1 we can write wx∈Rn:|T~ bf(x)|> λ≤wx∈Rn\˜ Ω : |T~ bg(x)|>λ 2+w(˜ Ω) +wx∈Rn\˜ Ω : |T~ bh(x)|>λ 2 =I+II +III We consider now each term separately. To estimate Iwe use Chebyschev’s inequality for p > 1that will be chosen appropriately, wx∈Rn\˜ Ω : |T~ bg(x)|>λ 2≤2p λpˆR|T~ bg(x)|pw∗(x)dx. BORDERLINE WEIGHTED ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS 23 Let us choose, as we did in the case k= 1,psuch that 1 + ε 3(1+ 1 s)< p < 1 + ε (1+ 1 s)2and δ=ε−1 + 1 sp. For this choice of pand δwe have that (p′)(k+1)pp(1+ 1 s)pp−1 δ1 p′ ≤cs 1 εk+1 and 1 + 1 sp−1 + δ=1 s+ε Using now theorem 1and the choice of δand pwe have that 2p λpˆR|T~ bg(x)|pw∗(x)dx ≤cn(p′)(k+1)pp(1+ 1 s)pp−1 δ1 p′ˆR|g(x)|pM L(log L)(1+ 1 s)p−1+δw∗(x)dx ≤c1 εk+1 2p λpˆR|g(x)|pML(log L)1 s+εw∗(x)dx Arguing as in the case k= 1 we obtain that I≤cn 1 εk+1 ˆRn |f(y)| λML(log L)1 s+εw(y)dy. For II, as in the case k= 1, we have the following estimate II ≤3nˆRn f(y) λMw(y)dy It remains to estimate III. Following the computations of page 684 of [24] we can write T~ bf(x) = (b1(x)−λ1)...(bk(x)−λk)T f(x) + (−1)kT((b1−λ1)...(bk−λk)f) (x) + k−1 X i=1 X σ∈Ci(b) (−1)k−ib(x)−~ λσˆRnb(y)−~ λσ′K(x, y)f(y)dy. (14) Now we work on the last double summation. We observe that for each term we can write b(x)−~ λσˆRnb(y)−~ λσ′K(x, y)f(y)dy =ˆRnb(y)−~ λσ′[b(x)−b(y)] + hb(y)−~ λiσK(x, y)f(y)dy τ∪τ′=σ =ˆRnb(y)−~ λσ′ #σ X j=0 X τ∈Cj(σ) (b(x)−b(y))τb(y)−~ λτ′K(x, y)f(y)dy = #σ X j=0 X τ∈Cj(σ)ˆRn (b(x)−b(y))τb(y)−~ λσ′∪τ′K(x, y)f(y)dy = #σ X j=0 X τ∈Cj(σ) T~τ b−~ λσ′∪τ′f =T((b1−λ1)...(bk−λk)f) (x) + #σ X j=1 X τ∈Cj(σ) T~τ b−~ λσ′∪τ′f. 24 CARLOS PÉREZ AND ISRAEL P. RIVERA-RÍOS Plugging this into the double summation of (14), since τ∪τ′∪σ′=bwe can write, k−1 X i=1 X σ∈Ci(b) (−1)k−ib(x)−~ λσˆRnb(y)−~ λσ′K(x, y)f(y)dy =ckT((b1−λ1)...(bk−λk)f) (x) + k−1 X i=1 X σ∈Ci(b) cσT~σ b−~ λσ′f where cσis a constant that counts the number of repetitions of each T~σ. Summarizing T~ bf(x) = (b1(x)−λ1)...(bk(x)−λk)T f(x) +ckT((b1−λ1)...(bk−λk)f) (x) + k−1 X i=1 X σ∈Ci(b) cσT~σ b(y)−~ λσ′f(x) Using this for each hjand summing on j, X j T~ bhj(x) = X j (b1(x)−λ1)...(bk(x)−λk)T hj(x) +X j ckT((b1−λ1)...(bk−λk)hj) (x) +X j k−1 X i=1 X σ∈Ci(b) cσT~σ b−~ λσ′hj(x) Then we can estimate III as follows III ≤w (y∈Rn\˜ Ω : X jb1(x)−(b1)Qj...bk(x)−(bk)QjT hj(x) >λ 6)! +w (y∈Rn\˜ Ω : X j ckTb1−(b1)Qj...bk−(bk)Qjhj(x) >λ 6)! +w    y∈Rn\˜ Ω : X j k−1 X i=1 X σ∈Ck i cσT~σ b−~ bQjσ′hj(x) >λ 6    =L1+L2+L3 BORDERLINE WEIGHTED ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS 25 To estimate L1we denote wj=χRn\5√nQjwand B(x) = Qk i=1 bi(x)−(bi)Qj. Then L1≤c λˆRn\˜ ΩX jb1(x)−(b1)Qj...bk(x)−(bk)QjT hj(x) w(x)dx ≤X j c λˆRn\˜ Ω B(x)|T hj(x)|w(x)dx = ≤X j c λˆRn\˜ Ω B(x)w(x) ˆQj|hj(y)||K(x, y)−K(x, xQj)|dy!dx ≤X j c λˆQj|hj(y)|ˆRn\5√nQj B(x)wj(x)|K(x, y)−K(x, xQj)|dxdy A standard computation using the smoothness condition of Kyields that the latter is bounded by X j c λˆQj|hj(y)|X mˆ2ml(Qj)≤|x−xQj|≤2m+1l(Qj) B(x)wj(x)|y−xQj|γ |x−xQj|n+γdxdy ≤X j c λˆQj|hj(y)|X m 2−mγ (2m+1l(Qj))nˆ|x−xQj|≤2m+1l(Qj) B(x)wj(x)dxdy = (15) Let us estimate the inner sum. We have that X m 2−mγ (2m+1l(Qj))nˆ|x−xQj|≤2m+1l(Qj) B(x)wj(x)dx ≤X m 2−mγ (2m+1l(Qj))nˆ2m+1Qj k Y i=1 bi(x)−(bi)Qjwj(x)dx =X m 2−mγ (2m+1l(Qj))nˆ2m+1Qj k Y i=1 bi(x)−(bi)2m+1Qj+(bi)2m+1Qj−(bi)Qjwj(x)dx =X m 2−mγ (2m+1l(Qj))nˆ2m+1Qj k X l=0 X σ∈Ck l Y i∈σbi(x)−(bi)2m+1Qj! Y i∈σ′(bi)2m+1Qj−(bi)Qj!wj(x)dx =X m k X l=0 X σ∈Cl(b) Y i∈σ′(bi)2m+1Qj−(bi)Qj!2−mγ (2m+1l(Qj))nˆ2m+1Qj Y i∈σbi(x)−(bi)2m+1Qj!wj(x)dx ≤X m k X l=0 X σ∈Cl(b) Y i∈σ′kbikOscexpLsi!2−mγ (2m+1l(Qj))nˆ2m+1Qj Y i∈σbi(x)−(bi)2m+1Qj!wj(x)dx Applying Corollary 2we have that 1 (2m+1l(Qj))nˆ2m+1Qj Y i∈σbi(x)−(bi)2m+1Qj!wj(x)dx ≤c Y i∈σkbikOscexpLsi!inf z∈2m+1Qj ML(log L)Pi∈σ1 si(wj)(x)