Continuity properties for the maximal operator associated with the commutator of the Bochner-Riesz operator

Publ. Mat. 47 (2003), 45–69 CONTINUITY PROPERTIES FOR THE MAXIMAL OPERATOR ASSOCIATED WITH THE COMMUTATOR OF THE BOCHNER-RIESZ OPERATOR Zongguang Liu, Guozhen Lu and Shanzhen Lu∗ Abstract In this paper, we obtain some strong and weak type continuity properties for the maximal operator associated with the commutator of the Bochner-Riesz operator on Hardy spaces, Hardy type spaces and weak Hardy type spaces. 1. Introduction Let b∈BMO(Rn) and Tbeastandard Calder´on-Zygmund singular integral operator, the commutator [b, T ]isdefined by [b, T]f(x)=T((b(x)−b(·))f)(x). Many authors have investigated the properties for [b, T]. A celebrated result of Coifman, Rochberg and Weiss [5] states that the commutator [b, T]isbounded on Lp(1 <p<∞). Subsequently, Coifman and Meyer [4] observed that the weighted Lp(1 <p<∞)boundedness for [b, T] can be obtained by the weighted Lpestimate with Muckenhoupt Apweight for T. Later, ´ Alvarez, Bagby, Kurtz and P´erez [2] extended the idea of Coifman and Meyer and proved the following result: For a general linear operator T,if1<p,q<∞and Tis bounded on Lp(w) for all w∈Aq, then [b, T]isbounded on Lp(u) for all u∈Aq.Inthe case of p=1,itisawell-known fact that Calder´on-Zygmund singular integral operator Tis a weak type (1,1) operator and a bounded operator from the standard Hardy space H1to L1.Fairly recently, P´erez [13] observed the fact that [b, T]isneither a weak type (1,1) operator nor a bounded operator from H1to L1.Heobtained a weak type Llog Linequality and 2000 Mathematics Subject Classification. 42B20, 35J05. Key words. Bochner-Riesz operator, commutator, Hardy space, Hardy type space, weak Hardy type space, BMO(Rn). ∗G. Lu was partly supported by US NSF grant DMS9970352 and S. Lu was partly supported by the National 973 Project Foundation of China (G19990751). 46 Z. Liu, G. Lu, S. Lu the boundedness from a certain modified Hardy space H1 bto L1for [b, T]. In this paper, we will consider the commutator of Bochner-Riesz operator, let λand rbe twopositive numbers, the Bochner-Riesz operator Tr λ in Rn(n≥2) is defined in terms of Fourier transforms by  Tr λf(ξ)=1−|ξ|2 r2λ + ˆ f(ξ), where ˆ fdenotes the Fourier transforms of f.Itcan be written as a convolution operator Tr λf(x)=p.v. Rn Br λ(x−y)f(y)dy, where Br λ(x)isthe kernel of Tr λand Br λ(x)=r−nBλ(x r), it is well-known that Bλ(x) satisfies the following inequality:  ∂β ∂xβBλ(x)≤C(1 + |x|)−(λ+n+1 2), for any x∈Rnand r>0 and any multi-index β∈Zn +. Let b∈BMO(Rn), the commutator generated by band Tr λis defined by Tr λ,bf(x)=Tr λ((b(x)−b(·))f)(x),f∈S(Rn), or Tr λ,bf(x)=p.v. Rn Br λ(x−y)(b(x)−b(y))f(y)dy, f ∈S(Rn). The maximal operator associated with Tr λ,b is defined by T∗ λ,bf(x)=sup r>0Tr λ,bf(x). If λ≥n−1 2, Shi and Sun [14] showed that the maximal Bochner-Riesz operator, T∗ λf(x)=sup r>0 |Tr λf(x)|,isbounded on Lp(w) provided that 1<p<∞and w∈Ap. Combining the above result due to ´ Alvarez, Bagby, Kurtz and P´erez with the result due to Shi and Sun [14], we can easily observe that T∗ λ,b is bounded on Lp(Rn). Hu and Lu [7] further discussed the Lpboundedness for T∗ λ,b in the case of λ>(n−1) 1 p−1 2 and proved the following result The Commutator of Bochner-Riesz Operator 47 Theorem A. If b∈BMO(Rn),1<p<∞and λ>(n−1) 1 p−1 2, then T∗ λ,b is bounded on Lp(Rn)with bound C(n, p)b∗. We notice the fact that (n−1) 1 p−1 2<n−1 2whenever 1 <p<∞ and in this case T∗ λis not bounded on Lp(w) for w∈Ap.Itisnatural to investigate the properties for T∗ λ,b with 0 <p≤1. In Section 2 of this paper, we consider the case of p=1and obtain the boundedness from H1 to weak L1for T∗ λ,b.InSection 3 of this paper, we get some strong and weak type boundedness estimates for T∗ λ,b on a certain modified Hardy space, Hp b, and a certain modified weak Hardy space, Hp,∞ b, where 0<p≤1. However, we do not know whether the operator T∗ λ,b satisfies the weak type Llog Linequality. After we submitted the paper, we learned that Jiang, Tang and Yang [9] proved, independently of us, the similar results in Hp band Hp,∞ bwhen n n+1 <p≤1. Our range in this paper allows to have all 0 <p<1. Now, let us recall some notations and definitions. Most of the notations we use are standard. Qdenotes a cube with sides parallel to the axes and λQ (λ>0) denotes the cube Qdilated by λ.Foralocally integrable function f,fQdenotes the average fQ=1 |Q|Qf(y)dy. Sometimes aQdenotes an atom in certain Hardy spaces with compact support included in cube Q.Forb∈BMO(Rn), b∗denotes the norm of bon BMO(Rn). Definition 1. Let 0 <p≤1 and bbe alocally integrable function. Given a bounded function a,wesay that aisa(p, b, ∞) atom, if (1) supp a⊂Q=Q(xQ,r Q); (2) aL∞(Rn)≤|Q|−1/p; (3) Rna(x)xβdx= Rna(x)b(x)xβdx =0,for |β|≤[n(1/p −1)], where [x] denotes the integer part of x.Atempered distribution fis said to belong to the Hardy type space Hp b(Rn) if, in the S-sense, it can be written as f= ∞  j=1 λjaj, where ajare (p, b, ∞) atoms and ∞  j=1 |λj|p<∞.Asusual, we define on Hp b(Rn) the quasinorm as fHp b(Rn)= inf λjaj=f  ∞  j=1 |λj|p  1/p . Definition 2. Let bbe alocally integrable function. We say that a tempered distribution fbelongs to the weak Hardy type space Hp,∞ b(Rn), if there exists a sequence {fk}∞ k=−∞ ⊂L∞(Rn) such that 48 Z. Liu, G. Lu, S. Lu (1) f= ∞  k=−∞ fk,inthe S-sense; (2) Each function fkcan be decomposed as fk= ∞  j=1 bk jin L∞∩Hp, where the functions bk jsatisfy the following properties: (2i) supp bk j⊂Qk jwith sup k ∞  j=1 χQk j<∞and sup k 2kp ∞  j=1 |Qk j|<∞. (2ii) There exists a constant C=C(n, p)>0 such that bk jL∞(Rn)≤ C2k, for any kand j. (2iii) Rnbk j(x)xβdx =Rnbk j(x)b(x)xβdx =0,for |β|≤[n(1/p − 1)]. We define on the space Hp,∞ b(Rn) the following quasinorm fp Hp,∞ b(Rn)= inf kjbk j=f sup k∈Z 2kp ∞  j=1 |Qk j|. For brevity, we will sometimes denote C1= sup k∈Z 2kp ∞  j=1 |Qk j|. 2. Weak type (H1,L 1) estimate In this section, we establish the weak type (H1,L 1) estimate for T∗ λ,b, where H1(Rn)isawell-known standard Hardy space. Our main result is the following theorem. Theorem 1. Let b∈BMO(Rn)and λ>n−1 2, then T∗ λ,b is a weak type (H1,L 1)bounded operator, i.e. there exists a constant C>0, such that {x∈Rn:T∗ λ,bf(x)>α}≤C αfH1(Rn), for any α>0and any f∈H1(Rn). Remark. After the paper is accepted for publication, it has been proved using similar method that T∗ λ,b is bounded from H1to L1(see [11]). However, we still do not know if T∗ λ,b is weak type (1,1). To prove our Theorem 1, we first recall the following lemma due to M. Christ [3]. The Commutator of Bochner-Riesz Operator 49 Lemma 1. For any α>0and any finite collection of dyadic cubes Q and associated positive scalars λQ, there exists a collection of pairwise disjoint dyadic cubes Ssuch that (1)  Q⊂S λQ≤8α|S|for all S; (2)  S |S|≤α−1 Q λQ; (3)  Q⊂S λQ|Q|−1χQL∞(Rn)≤α. Now we begin to prove Theorem 1. It is easy to see that the result of Theorem 1 follows from the inequality {x∈Rn:|Tr λ,bf(x)|>α}≤C αfH1(Rn), where Cis independent of r,fand α. For any given f∈H1(Rn), we have the well-known atomic decomposition f= ∞  j=1 λjaj,inthe S-sense, where each ajbe a (1,∞,0) atom with fH1(Rn)= inf ∞  j=1 |λj|. We may assume that fis a finite sum QλQaQwith Q|λQ|≤ 2fH1(Rn). Once Theorem 1 is proved for such f.For general f,it is the limit of this kind of fk(in H1norm or almost everywhere sense) where fkare finite sums having forms of QλQaQ, and then Theorem 1 follows by a limiting argument. It is convenient for us to assume that each Q(the supporting cube of aQ)inthe given atomic decomposition of fis dyadic and λQ>0. For fixed α>0 and the finite collection of dyadic cube Qand associated positive scalars λQ>0inthe given atomic decomposition of f, by Lemma 1, there exists a collection of pairwise disjoint dyadic cube S such that (1)  Q⊂S λQ≤8α|S|, for all S; (2)  S |S|≤α−1 Q λQ; (3)  Q⊂S λQ|Q|−1χQL∞(Rn)≤α. Denote E=S2S, then |E|≤C αfH1(Rn). 50 Z. Liu, G. Lu, S. Lu Set h(x)= S Q⊂S λQaQand g(x)=f(x)−h(x). By (3), we easily know that gL∞(Rn)≤α. Using the L2(Rn)boundedness of T∗ λ,b we get that x∈Rn\E:|Tr λ,bg(x)|>α 4≤x∈Rn\E:|T∗ λ,bg(x)|>α 4 ≤C α2T∗ λ,bg2 L2(Rn) ≤C α2g2 L2(Rn) ≤C αgL1(Rn) ≤C αfL1(Rn) ≤C αfH1(Rn). Thus, we only need to prove the following inequality x∈Rn\E:|Tr λ,bh(x)|>α 4≤C αfH1(Rn). For fixed cube Q=Q(xQ,r Q), by the vanishing moments of aQwe have Tr λ,baQ(x)=Rn (Br λ(x−y)−Br λ(x−xQ))(b(x)−bQ)aQ(y)dy +Rn Br λ(x−y)(bQ−b(y))aQ(y)dy =IQ(x)+Tr λ((bQ−b)aQ)(x). Now, we first estimate IQ(x). When 0 <r≤rQand any x∈Rn\Eand any y∈Q, since x∈Rn\E implies x∈Rn\2Qfor any Q,wehave that |x−y|≥|x−xQ|−|xQ−y|≥|x−xQ|−rQ≥|x−xQ| 2, this implies that |Br λ(x−y)|≤Cr−n1+|x−xQ| r−(λ+n+1 2) ≤Crλ−n−1 2|x−xQ|−(λ+n+1 2). The Commutator of Bochner-Riesz Operator 51 Similarly we can get |Br λ(x−xQ)|≤Crλ−n−1 2|x−xQ|−(λ+n+1 2). These above estimates imply the following inequality |IQ(x)|≤Crλ−n−1 2|x−xQ|−(λ+n+1 2)|b(x)−bQ|. Thus   x∈Rn\E: S Q⊂S λQ|IQ(x)|>α    ≤C α S Q⊂S λQRn\2Q |IQ(x)|dx ≤C α S Q⊂S λQ ∞  l=1 2l+1Q\2lQ rλ−n−1 2|x−xQ|−(λ+n+1 2)|b(x)−bQ|dx ≤C α S Q⊂S λQrλ−n−1 2 ∞  l=1 (2l+1rQ)n (2lrQ)λ+n+1 2 1 |2l+1Q|2l+1Q |b(x)−bQ|dx ≤Cb∗ α S Q⊂S λQr rQλ−n−1 2∞  l=1 l2−l(λ−n−1 2) ≤Cb∗ α S Q⊂S λQ≤Cb∗ αfH1(Rn). When r>rQ,wecan choose λ0satisfying n−1 2<λ0<minλ,n+1 2. By the mean value theorem and the boundedness of Br λ,wehave |IQ(x)|≤Cr−n−1|b(x)−bQ|Q |aQ(y)||y−xQ|1+|x−xQ| r−(λ+n+1 2) dy ≤Cr−n−1|b(x)−bQ|Q |aQ(y)||y−xQ|1+|x−xQ| r−(λ0+n+1 2) dy ≤CrQrλ0−n+1 2|b(x)−bQ||x−xQ|−(λ0+n+1 2). 52 Z. Liu, G. Lu, S. Lu This implies the following estimate   x∈Rn\E: S Q⊂S λQ|IQ(x)|>α    ≤C α S Q⊂S λQ ∞  l=12l+1Q\2lQ rQrλ0−n+1 2|x−xQ|−(λ0+n+1 2)|b(x)−bQ|dx ≤C α S Q⊂S λQ ∞  l=1 rQrλ0−n+1 22l rQ−(λ0+n+1 2)+n1 |2l+1Q|2l+1Q |b(x)−bQ|dx ≤Cb∗ α S Q⊂S λQr rQλ0−n+1 2∞  l=1 l2−l(λ0−n−1 2) ≤Cb∗ α S Q⊂S λQ≤Cb∗ αfH1(Rn). We notice the fact that T∗ λf(x)≤CMf(x) with λ>n−1 2, where M is the well-known Hardy-Littlewood maximal operator. Thus T∗ λis of weak type (1,1) and Tr λis weak type (1,1) uniformly associated with r. We get   x∈Rn\E: S Q⊂S λQTr λ((bQ−b)aQ)(x)>α    ≤C α S Q⊂S λQ(bQ−b)aQL1(Rn) ≤C α S Q⊂S λQ 1 |Q|Q |b(y)−bQ|dy ≤Cb∗ α S Q⊂S λQ ≤Cb∗ αfH1(Rn). This finishes the proof of Theorem 1. The Commutator of Bochner-Riesz Operator 53 3. (H p b ,L p )type estimate In this section we will establish the (Hp b,L p)type estmate for T∗ λ,b. Our main result is the following theorem. Theorem 2. Let b∈BMO(Rn)and 0<p≤1.Ifλ>n p−n+1 2, then T∗ λ,b is a bounded operator from Hp b(Rn)to Lp(Rn). Proof: By the definition of space Hp b(Rn), we only need to prove the following inequality for any (p, b, ∞) atom aQ, Rn |Tr λ,baQ(x)|pdx ≤C, where Cis a constant independent on aQand r. We easily get the following decomposition Rn |Tr λ,baQ(x)|pdx=2Q |Tr λ,baQ(x)|pdx+ Rn\2Q |Tr λ,baQ(x)|pdx=I1+I2. By the Lq(Rn)(1<q<∞)boundedness of T∗ λ,b and the H¨older inequality, we get I1≤|2Q|1−p/q 2Q |T∗ λ,baQ(x)|qdxp/q ≤C|Q|1−p/qaQp Lq(Rn) ≤C|Q|1−p/q|Q|p/q−1=C. Since 0 <p≤1, we have I2≤ ∞  j=1 2j+1Q\2jQ |b(x)−bQ|p|Tr λaQ(x)|pdx +2j+1Q\2jQ |Tr λ,b((bQ−b)aQ)(x)|pdx ≤ ∞  j=1 (J1j+J2j). 60 Z. Liu, G. Lu, S. Lu Similar to proof of Theorem 2, there exists a nonnegative integer m such that n n+m+1 <p≤n n+mfor fixed 0 <p≤1. Now we estimate K21 K21 ≤CRn\Bk0,N |b(x)−c|p|Tr λ(F2)(x)|pdx ≤C N  k=k0+1 ∞  j=1 Rn\AkQk j |b(x)−c|p|Tr λ(bk j)(x)|pdx. When n p−n+1 2<λ<m+n+1 2,if0<r<r k jfor fixed kand j,we have |Tr λ(bk j)(x)|=Qk j Br λ(x−y)bk j(y)dy ≤Cr−nQk j1+|x−xk j| r−(λ+n+1 2) |bk j(y)|dy ≤Crλ−n−1 2|x−xk j|−(λ+n+1 2)2k|Qk j| ≤C(rk j)λ−n−1 2|x−xk j|−(λ+n+1 2)2k|Qk j|, and if r≥rk j,bythe vanishing moments of bk jand the (m+1)-order Taylor formula of Br λ(x−y)atx−xQ,wehave |Tr λ(bk j)(x)|≤Cr−(n+m+1) Qk j |y−xk j|m+1 1+|x−xk j| r−(λ+n+1 2) |bk j(y)|dy ≤Crλ−m−n+1 2(rk j)m+1|x−xk j|−(λ+n+1 2)2k|Qk j| ≤C(rk j)λ−n−1 2|x−xk j|−(λ+n+1 2)2k|Qk j|. The Commutator of Bochner-Riesz Operator 61 These imply that K21 ≤C N  k=k0+1 ∞  j=1 ∞  l=1 2l+1AkQk j\2lAkQk j |b(x)−c|p(rk j)p(λ−n−1 2) ×|x−xk j|−p(λ+n+1 2)2kp|Qk j|pdx ≤C N  k=k0+1 ∞  j=1 ∞  l=1 2kp|Qk j|2l(n−p(λ+n+1 2))An−p(λ+n+1 2) k ×1 |2l+1AkQk j|2l+1AkQk j |b(x)−c|pdx ≤CB1 N  k=k0+1 2kp   ∞  j=1 |Qk j| An−p(λ+n+1 2) k ∞  l=1 2l(n−p(λ+n+1 2)) ≤CC1B1 N  k=k0+1 An−p(λ+n+1 2) k, where B1= sup Q 1 |Q|Q|b(x)−c|pdx. When λ≥m+n+1 2,if0<r<r k jfor a fixed kand j,wealso have following estimate, |Tr λ(bk j)(x)|≤Crλ−n−1 2|x−xk j|−(λ+n+1 2)2k|Qk j|, and if r≥rk,j for a fixed kand j, |Tr λ(bk j)(x)|≤Cr−(n+m+1) Qk j |y−xk j|m+1 1+|x−xk j| r−(λ+n+1 2) |bk j(y)|dy ≤Cr−(n+m+1) Qk j |y−xk j|m+1 1+|x−xk j| r−(n+m+1) |bk j(y)|dy ≤C(rk j)m+1|x−xk j|−(n+m+1)2k|Qk j|. 62 Z. Liu, G. Lu, S. Lu Thus we can get K21 ≤C N  k=k0+1 ∞  j=1 ∞  l=1 2l+1AkQk j\2lAkQk j |b(x)−c|p(rk j)p(λ−n−1 2) ×|x−xk j|−p(λ+n+1 2)2kp|Qk j|pdx +C N  k=k0+1 ∞  j=1 ∞  l=1 2l+1AkQk j\2lAkQk j |b(x)−c|p(rk j)p(m+1) ×|x−xk j|−p(n+m+1)2kp|Qk j|pdx =P1+P2. Because P2≤C N  k=k0+1 ∞  j=1 ∞  l=1 2lAkrk j−p(n+m+1) 2kp|Qk j|p(rk j)p(m+1) ×2l+1AkQk j\2lAkQk j |b(x)−c|pdx ≤C N  k=k0+1 2kp ∞  j=1 |Qk j|An−p(n+m+1) k ∞  l=1 2l(n−p(n+m+1)) 1 |2l+1AkQk j| ×2l+1AkQk j |b(x)−c|pdx ≤CC1B1 N  k0+1 An−p(n+m+1) k, and P1≤CC1B1 N  k=k0+1 An−p(λ+n+1 2) k. We obtain that K21 ≤CC1B1N  k=k0+1 An−p(λ+n+1 2) k+ N  k=k0+1 An−p(n+m+1) k. The Commutator of Bochner-Riesz Operator 63 Let Ak=2 (k−k0) max1 n+m+1 ,1 λ+n+1 2, then K21 ≤CC1B1 =CC1sup Q 1 |Q|Q |b(x)−c|pdx ≤CC1sup Q 1 |Q|Q |b(x)−c|dxp . By taking the infimum with c∈Cin the right-hand side of above inequality, we obtain that K21 ≤CC1bp ∗. Now let us estimate K22. Let Γ1={(k,j):k0+1≤k≤N, j ≥1 and 0 <r<r k j} and Γ2={(k,j):k0+1≤k≤N, j ≥1,r≥rk j}. Write K22 ≤αp  x∈Rn\Bk0,N : (k,j)∈Γ1 |Tr λ((b−c)bk j)(x)|>α 4   +αp  x∈Rn\Bk0,N : (k,j)∈Γ2 |Tr λ((b−c)bk j)(x)|>α 4   =M1+M2. 64 Z. Liu, G. Lu, S. Lu For fix (k,j)∈Γ1,wehave |Tr λ((b−c)bk j)(x)|≤Qk j |Br λ(x−y)(b(y)−c)bk j(y)|dy ≤Cr−nQk j1+|x−xk j| r−(λ+n+1 2) |b(y)−c||bk j(y)|dy ≤Crλ−n−1 2|x−xk j|−(λ+n+1 2)2kQk j |b(y)−c|dy ≤C(rk j)λ−n−1 2|x−xk j|−(λ+n+1 2)2kB2|Qk j| ≤C2kB2(rk j)λ+n+1 2 |x−xk j|λ+n+1 2 , where B2= sup Q 1 |Q|Q|b(y)−c|dy. Because of the facts that x∈Rn\Bk0,N :1 |x−xk j|λ+n+1 2 >α 4≤Cα−n λ+n+1 2 and  (k,j)∈Γ1C2kB2(rk j)λ+n+1 2n λ+n+1 2= (k,j)∈Γ1C2kB2n λ+n+1 2(rk j)n ≤C N  k=k0+1 B n λ+n+1 2 22 kn λ+n+1 2 ∞  j=1 |Qk j| ≤CC1B n λ+n+1 2 2 N  k=k0+1 2k(n λ+n+1 2 −p) ≤CC1B n λ+n+1 2 22k0(n λ+n+1 2 −p), The Commutator of Bochner-Riesz Operator 65 by Lemma 2, we obtain M1≤αpCC1B n λ+n+1 2 22k0(n λ+n+1 2 −p)α−n λ+n+1 2 ≤CC1B n λ+n+1 2 2. For (k,j)∈Γ2,ifn p−n−1 2<λ<m+n+1 2,wehave the following estimate by the vanishing moments of bk jand Taylor formula, Tr λ((b−c)bk j)(x)≤Cr−(n+m+1)Qk j |y−xk j|m+1 1+|x−xk j| r−(λ+n+1 2) ×|b(y)−c||bk j(y)|dy ≤Crλ−m−n+1 2(rk j)m+12k|x−xk j|−(λ+n+1 2) Qk j |b(y)−c|dy ≤C2kB2(rk j)λ+n+1 2 |x−xk j|λ+n+1 2 . Similar to the estimate of M1,weget M2≤CC1B n λ+n+1 2 2. 66 Z. Liu, G. Lu, S. Lu For (k,j)∈Γ2,ifλ≥m+n+1 2,wealso obtain Tr λ((b−c)bk j)(x)≤Cr−(n+m+1) Qk j |y−xk j|m+11+|x−xk j| r−(λ+n+1 2) ×|b(y)−c||bk j(y)|dy ≤Cr−(n+m+1) Qk j |y−xk j|m+11+|x−xk j| r−(n+m+1) ×|b(y)−c||bk j(y)|dy ≤C(rk j)m+1|x−xk j|−(n+m+1)2kQk j |b(y)−c|dy ≤CB22k(rk j)n+m+1 |x−xk j|n+m+1 . Since that x∈Rn\Bk0,N :1 |x−xk j|n+m+1 >α 4≤Cα−n n+m+1 , and  (k,j)∈Γ2C2kB2(rk j)n+m+1n n+m+1 =C (k,j)∈Γ2 B n n+m+1 22kn n+m+1 |Qk j| ≤CC1B n n+m+1 2 N  k=k0+1 2k(n n+m+1 −p) ≤CC1B n n+m+1 22k0(n n+m+1 −p), using Lemma 2 we obtain M2≤αpCC1B n n+m+1 22k0(n n+m+1 −p)α−n n+m+1 ≤CC1B n n+m+1 2. The Commutator of Bochner-Riesz Operator 67 Thus we get that K22 ≤CC1B n λ+n+1 2 2+B n n+m+1 2 =CC1sup Q 1 |Q|Q |b(y)−c|dyn λ+n+1 2 +sup Q 1 |Q|Q |b(y)−c|dyn n+m+1 . By taking the infimum with c∈Cin each term on the right hand of above inequality, we obtain that K22 ≤CC1b n λ+n+1 2 ∗+b n n+m+1 ∗. Finally let us estimate K1, K1≤αp|Bk0,N |≤αp N  k=k0+1 An k ∞  j=1 |Qk j| ≤CC1αp N  k=k0+1 An k2−kp ≤CC1αp2−k0p N  k=k0+1 An k2−(k−k0)p ≤CC1 N  k=k0+1 2(k−k0)(max( n n+m+1 ,n λ+n+1 2 )−p) ≤CC1. Following above estimates, we have that αpx∈Rn:Tr λ,b N  k=−N fk(x)>λ  ≤CC11+bp ∗+b n n+m+1 ∗+b n λ+n n+1 ∗, where Cis dependent on r,Nand the atomic decomposition of f. 68 Z. Liu, G. Lu, S. Lu Finally, taking the limit as N→∞, the infimum of C1over all possible representations  k j bk j=fand the supremum for r>0inthe left-hand side of above inequality, we complete the proof of Theorem 3. References [1] J. ´ Alvarez, Continuity properties for linear commutators of Calder´on-Zygmund operators, Collect. Math. 49(1) (1998), 17–31. [2] J. ´ Alvarez, R. J. Bagby, D. S. Kurtz and C. P´ erez,Weighted estimates for commutators of linear operator, Studia Math. 104(2) (1993), 195–209. [3] M. Christ,Weak type (1,1) bounds for rough operators, Ann. of Math. 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[11] Zongguang Liu and Shanzhen Lu,Anote on maximal commutators of Bocher-Riesz operators, Panamer. Math. J. (to appear). [12] Shanzhen Lu,“Four lectures on real Hpspaces”,World Scientific Publishing Co., Inc., River Edge, NJ, 1995. [13] C. P´ erez, Endpoint estimates for commutators of singular integral operators, J. Funct. Anal. 128(1) (1995), 163–185. [14] Xianliang Shi and Qiyu Sun,Weighted norm inequalities for Bochner-Riesz operators and singular integral operators, Proc. Amer. Math. Soc. 116(3) (1992), 665–673. [15] E. M. Stein,“Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals”, Princeton Mathematical Series 43, The Commutator of Bochner-Riesz Operator 69 Monographs in Harmonic Analysis III, Princeton University Press, Princeton, NJ, 1993. Zongguang Liu: Department of Mathematics Beijing Normal University Beijing 100875 P. R. China E-mail address:[email protected] Current address: Department of Mathematics China University of Mining and Technology (Beijing) Beijing 100083 P. R. China Guozhen Lu: Department of Mathematics Wayne State University Detroit, MI 48202 U.S.A. E-mail address:[email protected] Shanzhen Lu: Department of Mathematics Beijing Normal University Beijing 100875 P. R. China E-mail address:[email protected] Primera versi´o rebuda el 26 d’octubre de 2001, darrera versi´o rebuda el 22 d’octubre de 2002.