Endpoint estimates for commutators of singular integral operators

Journal of functional analysis (1) 128 (1995), 163–185. ENDPOINT ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRAL OPERATORS Dedicated to Professor Guido L. Weiss Carlos P´erez Departmento de Matem´aticas Universidad Aut´onoma de Madrid 28049 Madrid, Spain e–mail: cp[email protected] work partially supported by DGICYT grant PB90187, Spain 1 1 Introduction and statements of the results 1.1 The commutator of Coifman, Rochberg and Weiss Let bbe a locally integrable function on Rnand let Tbe a Calder´on–Zygmund singular integral operator. Consider the commutator operator [b, T] defined for smooth functions fby [b, T]f=b T(f)−T(bf). A by now classical result of R. Coifman, R. Rochberg and G. Weiss [1] states that [b, T] is a bounded operator on Lp(Rn), 1 < p < ∞, when bis a BMO function. Unlike the classical theory of singular integral operators, the proof of this theorem does not rely on a weak type (1,1) inequality for [b, T]. In fact, simple examples show (cf. section 5) that these operators fail to be of weak type (1,1) when b∈BMO. The purpose of this paper is to provide an endpoint theory for these operators. Our first theorem is the following: THEOREM 1.1 Let Tbe a Calder´on–Zygmund singular integral operator and let bbe a function in BMO. Then, there exists a positive constant Csuch that for each smooth function with compact support fand for all λ > 0 |{y∈Rn:|[b, T]f(y)|> λ}| ≤ CkbkBMO ZRn |f(y)| λ(1 + log+(|f(y)| λ)) dy. (1) By imposing a condition on bstronger than BMO we can sharpen this estimate getting as close as we want to the L1norm. Indeed, we may replace LlogL by the smaller function L(logL), > 0. To be precise we shall say that a function bbelongs to the space osc(expL)r, 1 ≤r < ∞, if there is a positive constant c=cbsuch that sup Q 1 |Q|ZQexp(|b(y)−bQ c|r)dy < ∞, where the supremum is taken over all the cubes Qin Rn. The infimum of all these cis denoted by kbkosc(expL)r. The case r= 1 corresponds to the BMO space by the John–Nirenberg estimate (cf. section 3). Other interesting examples are provided by Trudinger’s inequality for Riesz potentials (cf. [11] or [4]). For 0 < α < n let Iαfbe the Riesz potential or fractional integral of order α. Let fbe an arbitary function on Ln/α(Rn). Trudinger’s estimate says that Iαfbelongs to osc(expL)(n/α)0. 2 Other examples are given by the following fact [6]: let 0 <  ≤1 and suppose that Fis a H¨older continuous function of order , then if b∈BMO F(b)∈osc(expL)1/ . The same method of proof of Theorem 1.1 gives the following. THEOREM 1.2 Let Tbe a Calder´on–Zygmund singular integral operator. For 0<  < 1we let bbe a function in osc(expL)1/ . Then, there exists a positive constant Csuch that for each smooth function with compact support fand for all λ > 0 |{y∈Rn:|[b, T]f(y)|> λ}| ≤ Ckbkosc(expL)1/ ZRn |f(y)| λ(1 + log+(|f(y)| λ))dy. Also, we are going to consider endpoint estimates related to Hardy type spaces. It is well known that any singular integral operator maps H1(Rn) into L1(Rn). However, it was observed in [8] that the corresponding result for [b, H] is false when bis a BMO function. Here, we introduce a subspace of H1(Rn) from which [b, H] is a bounded operator. DEFINITION 1.3 A function ais a b–atom if there is a cube Qfor which (i) supp(a)⊂Q (ii) kakL∞≤1/|Q| (iii) RQa(y)dy = 0 (iv) RQa(y)b(y)dy = 0. The space H1 b(Rn)consists of the subspace of L1(Rn)of functions fwhich can be written as f=Pjλjajwhere ajare b-atoms, and λjare complex numbers with Pj|λj|<∞. THEOREM 1.4 Let Tbe a Calder´on–Zygmund singular integral operator and a let bbe a function in BMO. Then, [b, T] : H1 b(Rn)→L1(Rn) Let us give an outline of the proof of Theorem 1.1. Our approach exhibits the close connection existing between commutators and iterations of the Hardy– Littlewood maximal operator. The proof is based on two lemmas, and the first one makes explicit this relationship. 3 LEMMA 1.5 Let Φ(t) = t(1 + log+t). Then, there exists a positive constant C such that for any smooth function with compact support f sup t>0 1 Φ(1 t)|{y∈Rn:|[b, T]f(y)|> t}| ≤ CkbkBMO sup t>0 1 Φ(1 t)|{y∈Rn:M2f(y)> t}|, where M2=M◦M The size of M2is given by the following. LEMMA 1.6 There exists a positive constant Csuch that for any function fand for all λ > 0 |{y∈Rn:M2f(y)> λ}| ≤ CZRn |f(y)| λ(1 + log+(|f(y)| λ)) dy. (2) The proof of Lemma 1.5 is based on a good–λtype argument as follows. For δ > 0 we denote by Lδthe functional Lδ(f) = sup t>0 1 Φ(1 t)|{y∈Rn:Mδ([b, T]f)(y)> t}|. Then the operator [b, T] satisfies the following inequality for 0 < δ < 1 and  > 0: Lδ(f)≤ C Lδ(f) + C()kbkBMO sup t>0 1 Φ(1 t)|{y∈Rn:M2f(y)> t}|. By choosing small enough we get Lemma 1.5 easily. 1.2 Higher order commutators In section 7 we shall extend all these results to higher order commutators Tm b,m, m= 0,1,2,· · · as consider for instance in [10] and [3]. These operators are defined by Tm bf(x) = Z(b(x)−b(y))mK(x−y)f(y)dy. Observe that T0 b=T,Tm b= [b, Tm−1 b], m= 1,2,· · ·. We shall show the following. 4 THEOREM 1.7 Let m= 0,1,2,· · ·, and let bbe a function in BMO. Then, there exists a positive constant Csuch that for each λ > 0 |{y∈Rn:|Tm bf(y)|> λ}| ≤ Ckbkm BMO ZRn |f(y)| λ(1 + log+(|f(y)| λ))mdy. The proof of this theorem uses an induction argument from the case m= 1. It would be interesting to give a direct proof of this result which would avoid the induction argument. The analog of H1 b(Rn) space that we need is the following. DEFINITION 1.8 A function ais a b–atom of order mif there is a cube Qfor which (i) supp(a)⊂Q (ii) kakL∞≤1/|Q| (iii) Ra(y)b(y)mdy = 0,m= 0,1,2,· · ·. The space H1 b,m(Rn)consists of the subspace of L1(Rn)of functions fwhich can be written as f=Pjλjajwhere ajare b-atoms of order m, and λjare complex numbers with Pj|λj|<∞. THEOREM 1.9 Let Tbe a Calder´on–Zygmund singular integral operator and a let bbe a function in BMO. Then, Tm b:H1 b,m(Rn)→L1(Rn) with constant less or equal than a multiple of kbkm BMO. 2 Some preliminaries and notation In this section we shall collect some known results or variants of them and set some notation. The basic tool we are going to use is a modification of the sharp maximal operator M#of C. Fefferman and E. Stein: for δ > 0 we define the δ–sharp maximal operator M# δas M# δ(f) = M#(|f|δ)1/δ. 5 Recall that M#is defined by M#f(x) = sup x∈Q inf c 1 |Q|ZQ|f(y)−c|dy ≈sup x∈Q 1 |Q|ZQ|f(y)−fQ|dy The idea of relating commutators with the sharp maximal operator is due to J. O. Str¨omberg (cf. [5]). The basic estimate is contained in the following lemma of C. Fefferman and E. Stein. Recall that Mδf=M(fδ)1/δ LEMMA 2.1 Let M=Mdbe the dyadic Hardy–Littlewood maximal operator. Then, there exists a positive dimensional constant Cfor which the following good–λ inequality holds {y∈Rn:Mf(y)> λ, M#f(y)≤λ}≤C {y∈Rn:Mf(y)>λ 2}(3) for all λ,  > 0. As a consequence we have the following estimate: Let ϕ: (0,∞)→(0,∞)be a doubling function. Then, there exists a positive constant Csuch that sup λ>0 ϕ(λ)|{y∈Rn:Mδf(y)> λ}| ≤ Csup λ>0 ϕ(λ){y∈Rn:M# δf(y)> λ} for all functions fsuch that the left side is finite. Then, there exists a positive constant Csuch that sup λ>0 ϕ(λ)|{y∈Rn:Mδf(y)> λ}| ≤ Csup λ>0 ϕ(λ){y∈Rn:M# δf(y)> λ} for all functions fsuch that the left side is finite. A simple proof of (3) can be found in [7] p. 42. Throughout the paper we will assume that the singular integral operators we are working with Tf(x) = v.p. ZRnK(x−y)f(y)dy are regular (following [2] p. 204), which means that the kernel satisfies (i) |K(x)| ≤ c/|x|nx6= 0 (ii) |K(x−y)−K(x)| ≤ |y|/|x|n+1 |x|>2|y|>0. 6 All the results still hold for the more general class of Calder´on–Zygmund operators [7]. Most the notation we use is standard. Qwill always denote a cube with sides parallel to the axes. λQ,λ > 0 denotes the cube Qdilated by λ. For a locally integrable function f,fQdenotes the average fQ=1 |Q|RQf(y)dy. Also B=B(x, r) will denote a ball centered at xwith radius rand corresponding notation stands for λB and fB. As usual, a function A: [0,∞)→[0,∞) is a Young function if it is continuous, convex and increasing satisfying A(0) = 0 and A(t)→ ∞ as t→ ∞. We define the A–average of a function fover a cube Qby means of the following Luxemburg norm kfkA,Q = inf{λ > 0 : 1 |Q|ZQA |f(y)| λ!dy ≤1}.(4) The following generalized H¨older’s inequality holds: 1 |Q|ZQ|f(y)g(y)|dy ≤ kfkA,Q kgk¯ A,Q ,(5) where ¯ Ais the complementary Young function associated to A. The main example that we are going to be using is A(t) = t(1 + log+t)m,m= 1,2,3,· · ·, with maximal function denoted by ML(logL)m. The complementary Young fuction is given by ¯ A(t)≈et1/m with corresponding maximal function denoted by M(expL)1/m . 3 Pointwise estimates In this section we shall prove the following pointwise inequality for commutators which is the key estimate for the proof of Lemma 1.5. LEMMA 3.1 Let b∈BMO and let 0< δ <  < 1. Then, there exists a positive constant C=Cδ, such that, M# δ([b, T]f)(x)≤CkbkBMO(M(Tf)(x) + M2f(x)) for all smooth functions f. 7 Proof Let B=B(x, r) be an arbitrary ball. Since 0 < δ < 1 implies ||α|δ− |β|δ| ≤ |α−β|δfor α, β ∈Rit is enough to show for some complex constant c=cBthat there exists C=Cδ>0 such that, 1 |B|ZB|[b, T]f(y)−c|δdy!1/δ ≤C Mf(x).(6) Let f=f1+f2, where f1=f χ2B. For arbitrary constant awe can write [b, T]f= (b−a)Tf −T((b−a)f1)−T((b−a)f2). If we pick c= (T((b−a)f2))B,a=b2Bwe can estimate the left hand side of (6) by a multiple of 1 |B|ZB|(b(y)−b2B)Tf(y)|δdy!1/δ + 1 |B|ZB|T((b−b2B)f1)(y)|δdy!1/δ + 1 |B|ZB|T((b−b2B)f2)−(T((b−b2B)f2))B|δdy!1/δ =I+II +III. To estimate I we use H¨older’s inequality with exponents rand r0where 1 < r <  δ: I≤ 1 |B|ZB|b(y)−b2B)|δr0dy!1/δr0 1 |B|ZB|Tf(y)|δr dy!1/δr ≤ ≤CkbkBMOMδr(Tf)(x)≤CkbkBMOM(Tf)(x). Since T:L1(Rn)→L1,∞(Rn) and 0 < δ < 1, Kolmogorov’s inequality (cf. [2] p. 485 for instance) yields II ≤C |B|ZB|(b(y)−b2B)f1(y)|dy =C |2B|Z2B|(b(y)−b2B)f(y)|dy ≤ ≤Ckb−b2BkexpL,2BkfkLlogL,2B, by the generalized H¨older’s inequality (5). We claim the following: there is a positive constant Csuch that for all ball B kb−bBkexpL,B ≤CkbkBMO.(7) 8 Indeed, this is equivalent to 1 |B|ZBexp(|b(y)−bB| CkbkBMO )dy ≤C0, which is the fundamental estimate of F. John and L. Nirenberg (see [2] p. 166). Then, II ≤CkbkBMO MLlogLf(x). To take care of III we observe that Jensen’s inequality together with Fubini’s theorem yield, III ≤1 |B|ZB|T((b−b2B)f2)(y)−(T((b−b2B)f2))B|dy ≤ ≤1 |B|2ZBZBZRn−2B|k(y−w)−k(z−w)||(b(w)−b2B)f(w)|dw dz dy ≤ ≤1 |B|2ZBZB ∞ X j=1 Z2jr≤|w−x|<2j+1r |y−z| |x−w|n+1 |b(w)−b2B||f(w)|dw dz dy ≤ ≤C ∞ X j=1 2−j (2j+1r)nZ2j+1B|b(w)−b2B||f(w)|dw ≤ ≤C ∞ X j=1 2−j (2j+1r)nZ2j+1B|b(w)−b2j+1B||f(w)|dw+ +C ∞ X j=1 2−j|b2j+1B−b2B|1 (2j+1r)nZ2j+1B|f(w)|dw ≤ ≤C ∞ X j=1 2−jkb−b2j+1BkexpL,2j+1BkfkLlogL,2j+1B+CkbkBMO ∞ X j=1 j 2jMf(x)≤ ≤CkbkBMOMLlogLf(x) + CkbkBMO Mf(x)≤ ≤CkbkBMOMLlogLf(x), where we have used that |b2j+1B−b2B| ≤ 2jkbkBMO, inequality (7) and Mf(x)≤ MLlogLf(x) by the generalized Jensen’s inequality. Finally, since M2f≈MLlogL (cf. (21) below) we are done. 2 9 6 The H1 bestimate Proof of Theorem 1.4 Let bbe a function in BMO. We need to prove that there exists a constant C such that for each function fin H1 b(Rn) ZRn|[b, T]f(y)|dy ≤CkbkBMOkfkH1 b(Rn).(17) By an standard argument, it is enough to show that there is a constant Csuch that ZRn|[b, T]a(y)|dy ≤CkbkBMO for each b–atom a. To prove this, suppose that supp(a)⊂Bfor some ball B. Then ZRn|[b, T]a(y)|dy =Z2B|[b, T]a(y)|dy +ZRn\2B|[b, T]a(y)|dy =I+II. The estimate for I follows from the boundedness of [b, T] on L2(Rn) ([1]) I≤C|B| 1 |2B|Z2B|[b, T]a(y)|2dy!1/2 ≤CkbkBMO |B| 1 |B|ZB|a(y)|2dy!1/2 ≤ ≤CkbkBMO|B| kakL∞(Rn)≤CkbkBMO, by the definition of b–atom. Now, to majorize II we split [b, T] in the following way [b, T]a= (b−bB)Ta − T((b−bB)a), then II ≤ZRn−2B|(b(x)−bB)Ta(x)|dx +ZRn−2B|T((b−bB)a)(x)|dx =III +IV. To estimate III we only use the cancellation of aRBa(y)dy = 0, so that if xB denotes the center of Band rthe radius of B III ≤ZB|a(y)|ZRn−2B|k(x−y)−k(x−xB)||b(x)−bB|dx dy = =ZB|a(y)| ∞ X j=1 Z2jr≤|x−xB|<2j+1r |y−xB| |x−xB|n+1 |b(x)−bB|dx dy ≤ 16 ≤ZB|a(y)|dy ∞ X j=1 2−j |2j+1B|Z2j+1B|b(x)−bB|dx ≤ ≤C ∞ X j=1 2−j"1 |2j+1B|Z2j+1B|b(x)−b2j+1B|dx +|b2j+1B−bB|# ≤C ∞ X j=1 2−j[kbkBMO + (j+ 1)kbkBMO] = CkbkBMO, since |b2j+1B−bB| ≤ 2(j+ 1) kbkBMO. Now, by the definition of a ZB(b(y)−bB)a(y)dy =ZBa(y)b(y)dy −bBZBa(y)dy = 0, and hence we estimate IV using Lemma 3.3 from [2] p. 413: IV ≤CZB|b(x)−bB||a(x)|dx ≤C1 |B|ZB|b(x)−bB|dx ≤CkbkBMO. This concludes the proof. 2 7 Commutator of order m In this section we extend the results of the previous sections to higher order commutators. Recall that for m= 0,1,2,· · · the commutator of order mis defined by Tm bf(x) = Z(b(x)−b(y))mK(x−y)f(y)dy. Observe that T0 b=T,T1 b= [b, T] and that Tm b= [b, Tm−1 b]. Proof of Theorem 1.7 As in the proof of Theorem 1.1, it is sufficient to consider the case λ= 1 |{y∈Rn:|Tm bf(y)|>1}| ≤ CZRn|f(y)|(1 + log+|f(y)|)mdy, (18) for each fsmooth with compact support. We need the following version of Lemma 3.1: 17 LEMMA 7.1 For each b∈BMO,0< δ <  < 1, there exists C=Cδ, >0such that, M# δ(Tm bf)(x)≤CkbkBMO m−1 X j=0 M(Tj b)(x) + kbkm BMO Mm+1f(x) (19) for all smooth functions f, and where M(f) = M(|f|)1/. Let Φm(t) = t(1 + log+t)m. Then it follows from (19) that there exists a constant Csuch that for each smooth function f sup t>0 1 Φm(1 t)|{y∈Rn:|Tm bf(y)|> t}| ≤ CkbkBMO sup t>0 1 Φm(1 t)|{y∈Rn:Mm+1f(y)> t}| (20) Let us postpone the proof of the lemma for the moment and continue with the proof of (18). We need the following analog of Lemma 1.6: |{y∈Rn:Mm+1f(y)> t}| ≤ CZRn |f(y)| t(1 + log+(|f(y)| t))mdy for all t > 0. The proof is the same as for the case m= 1 using that Mm+1f(x)≈ML(logL)mf(x) (21) for each x∈Rnand m= 0,1,2,· · ·. We shall omit it. Therefore, using the notation Φm(t) = t(1 + log+t)mwe have |{y∈Rn:Mm+1f(y)> t}| ≤ CZRnΦm(|f(y)| t)dy ≤CZRnΦm(|f(y)|)Φm(1 t)dy, since Φmis submultiplicative. Hence, sup t>0 1 Φm(1 t)|{y∈Rn:Mm+1f(y)> t}| ≤ CZRnΦm(|f(y)|)dy. This together with (20) yields the desired estimate since |{y∈Rn:|Tm bf(y)|>1}| ≤ sup t>0 1 Φm(1 t)|{y∈Rn:|Tm bf(y)|> t}|. 18 2 Proof of Lemma 7.1 Following [3], we expand b(x)−b(y) = (b(x)−λ)− (b(y)−λ), where λis an arbitrary constant as follows: Tm bf(x) = Z(b(x)−b(y))mK(x−y)f(y)dy = = m X j=0 Cj,m (b(x)−λ)jZ(b(y)−λ)m−jK(x−y)f(y)dy = = m X j=1 Cj,m (b(x)−λ)jZ(b(y)−λ)m−jK(x−y)f(y)dy +T((b−λ)mf)(x) = = m X j=1 Cj,m (b(x)−λ)jZ(b(y)−b(x)+b(x)−λ)m−jK(x−y)f(y)dy+T((b−λ)mf)(x) = = m X j=1 m−j X h=0 Cj,m,h (b(x)−λ)h+jZ(b(x)−b(y))m−j−hK(x−y)f(y)dy+T((b−λ)mf)(x) = = m−1 X α=0 Cα,m (b(x)−λ)m−αTα bf(x) + T((b−λ)mf)(x). For arbitrary ball Bcentered at xand with radius r, we split fin the usual way: f=f1+f2where f1=f χ2B. Then, with λ=b2Band c= (T((b−b2B)mf2))Bwe have 1 |B|ZB|Tmf(y)−c|δdy!1/δ = 1 |B|ZB|Pm−1 α=0 Cα,m (b(y)−λ)m−αTαf(y) + T((b−λ)mf(y)−c|δdy!1/δ ≤ ≤C m−1 X α=0 1 |B|ZB(b(y)−λ)(m−α)δ|Tαf(y)|δdy!1/δ + 1 |B|ZB|T((b−b2B)mf1)(y)|δdy!1/δ + + 1 |B|ZB|T((b−b2B)mf2)−(T((b−b2B)mf2))B|δdy!1/δ =I+II +III. 19 To estimate I we use H¨older’s inequality with exponents qand q0where 1 < q <  δ: I≤ m−1 X α=0 1 |B|ZB|b(y)−b2B)|(m−α)δq0dy!1/δq0 1 |B|ZB|Tαf(y)|δq dy!1/δq ≤ ≤C m−1 X α=0 kbkBMOMδq(Tαf)(x)≤C m−1 X α=0 kbkBMOM(Tαf)(x). Again, since T:L1(Rn)→L1,∞(Rn) and 0 < δ < 1, Kolmogorov’s inequality yields II ≤C |B|ZB|(b(y)−b2B)mf1(y)|dy =C |2B|Z2B|b(y)−b2B|m|f(y)|dy ≤ ≤Ck(b−b2B)mk(expL)1/m,2BkfkL(logL)m,2B=Ckb−b2Bkm expL,2BkfkL(logL)m,2B. Then, II ≤Ckbkm BMO ML(logL)mf(x)≈ kbkm BMO Mm+1f(x). The last term III is estimated as follows III ≤1 |B|ZB|T((b−b2B)mf2)(y)−(T(((b−b2B)mf2)B|dy ≤ ≤1 |B|2ZBZBZRn−2B|k(y−w)−k(z−w)||b(w)−b2B|m|f(w)|dw dz dy ≤ ≤1 |B|2ZBZB ∞ X j=1 Z2jr≤|w−x|<2j+1r |y−z| |x−w|n+1 |b(w)−b2B|m|f(w)|dw dz dy ≤ ≤C ∞ X j=1 2−j (2j+1r)nZ2j+1B|b(w)−b2B|m|f(w)|dw ≤ ≤C ∞ X j=1 2−j (2j+1r)nZ2j+1B|b(w)−b2j+1B|m|f(w)|dw+ +C ∞ X j=1 2−j|b2j+1B−b2B|m1 (2j+1r)nZ2j+1B|f(w)|dw ≤ ≤C ∞ X j=1 2−jkb−b2j+1Bkm expL,2j+1BkfkL(logL)m,2j+1B+Ckbkm BMO Mf(x) ∞ X j=1 2−jjm≤ 20 ≤Ckbkm BMOML(logL)mf(x) ∞ X j=1 2−j+Ckbkm BMO Mf(x)≤ ≤Ckbkm BMO ML(logL)mf(x)≈ kbkm BMO Mm+1f(x). The proof of (20) can be given by iterating the pointwise estimate (19) mtimes starting from the basic estimate of Fefferman–Stein Lemma 2.1 (notice that each time Φjis doubling); when the case m= 1 is reached we use Theorem 1.1. Since we iterate mtimes we obtain kbkm BMO as a constant. 2 Proof of Theorem 1.9 Let bbe a function in BMO. We need to prove that there exists a constant C such that for each function fin Hm b(Rn) ZRn|Tm bf(x)|dx ≤Ckbkm BMOkfkH1 b,m(Rn).(22) By an standard argument, it is enough to show that there is a costant Csuch that ZRn|Tm ba(x)|dx ≤Ckbkm BMO for each b–atom aof order m. To prove this, suppose that supp(a)⊂Bfor some ball B. Then ZRn|Tm ba(x)|dx =Z2B|Tm ba(x)|dx +ZRn\2B|Tm ba(x)|dx =I+II. The estimate for I follows from the boundedness of Tm bon L2(Rn) I≤C|B| 1 |2B|Z2B|Tm ba(x)|2dx!1/2 ≤Ckbkm BMO |B| 1 |B|ZB|a(x)|2dx!1/2 ≤ ≤Ckbkm BMO|B| kakL∞(Rn)≤Ckbkm BMO by the definition of b–atom of order m. Now, to majorize II we can write using that Ra(y)b(y)mdy = 0,for each m= 0,1,2,· · · Tm ba(x) = Z(b(x)−b(y))mK(x−y)a(y)dy = 21 =Z(b(x)−b(y))m[K(x−y)−K(x−xB)]a(y)dy, where xBdenotes the center of B. Then, if rdenotes the radius of Bwe have ZRn−2B|Tm ba(x)|dx ≤ZB|a(y)|ZRn−2B|b(x)−b(y)|m|k(x−y)−k(x−xB)|dx dy ≤ ≤ ∞ X j=1 ZB|a(y)|Z2jr≤|x−xB|<2j+1r |y−xB| |x−xB|n+1 |b(x)−b(y)|mdx dy ≤ C ∞ X j=1 ZB|a(y)|dy 2−j |2j+1B|Z2j+1B|b(x)−b(y)|mdx dy ≤ ≤C ∞ X j=1 2−jZB|a(y)|1 |2j+1B|Z2j+1B(|b(x)−bB|m+|bB−b(y)|m)dx dy ≤ ≤C ∞ X j=1 2−j1 |2j+1B|Z2j+1B|b(x)−bB|mdx + ∞ X j=1 2−jZB|a(y)||bB−b(y)|mdy ≤ ≤C ∞ X j=1 2−j"1 |2j+1B|Z2j+1B|b(x)−b2j+1B|mdx +|b2j+1B−bB|m#+ +C ∞ X j=1 2−j1 |B|ZB|bB−b(y)|mdy ≤Ckbkm BMO. 2 8 Remarks on weighted inequalities In this section we sketch how to extend the main result of the paper to the weighted case. We recall some definitions. A weight wis in the class A1if there is a positve constant Csuch that Mw(x)≤Cw(x), a.e. x ∈Rn. We denote by [w]A1the infimum of all these C’s. A weight wis in the class A∞if there are positve constants C,  such that w(E) w(Q)≤c(|E| |Q|), for all cubes Qand all measurable sets E⊂Q. We denote by [w]A∞the infimum of all these C’s. Recall that it is well known that A1⊂A∞. 22 THEOREM 8.1 Let m= 0,1,2,· · ·, and let bbe a function in BMO. Suppose that wis an A1weight. Then, there exists a positive constant Csuch that for each smooth function with compact support fand for all λ > 0 w({y∈Rn:|Tm bf(y)|> λ})≤Ckbkm BMO [w]A1ZRn |f(y)| λ(1+log+(|f(y)| λ))mw(y)dy. (23) The proof of this result goes along the same line as that of the unweighted case once we give the weighted version of the basic lemmata 1.5 and 1.6. Recall that we use the notation Φm(t) = t(1 + log+t)m. Then, we have LEMMA 8.2 Let wbe an A∞weight. Then, there exists a positive constant C such that for each smooth function f sup t>0 1 Φm(1 t)w({y∈Rn:|Tm bf(y)|> t})≤ Ckbkm BMO [w]A∞sup t>0 1 Φm(1 t)w({y∈Rn:Mm+1f(y)> t}) The proof of this is based on the pointwise estimate (19) and the weighted version of the basic estimate of the Fefferman– Stein Lemma 2.1: Let wbe an A∞weight, then w({y∈Rn:Mf(y)> λ, M#f(y)≤λ})≤cn[w]A∞w({y∈Rn:Mf(y)>λ 2}), (24) for all λ,  > 0. As a consequence we have the following estimates for 0 < δ and ϕ: (0,∞)→(0,∞) doubling. Then, there exists a positive constant Csuch that sup λ>0 ϕ(λ)w({y∈Rn:Mδf(y)> λ})≤C[w]A∞sup λ>0 ϕ(λ)w({y∈Rn:M# δf(y)> λ}) LEMMA 8.3 There exists a positive constant Csuch that for any weight wand all λ > 0 w({y∈Rn:Mm+1f(y)> λ} ≤ CZRn |f(y)| λ(1 + log+(|f(y)| λ))mMw(y)dy for every locally integrable function f The proof of this result is standard and we shall omit it. 23 References [1] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611–635. [2] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North Holland Math. Studies 116, North Holland, Amsterdam, (1985). [3] J. Garcia-Cuerva, E. Harboure, C. Segovia and J. L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J. 40, (1991), 1398–1420. [4] A. E. Gatto and S. V´agi, On the exponential integrability of fractional integrals on spaces of homogeneous type, preprint (1991). [5] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16, (1978), 263–270. [6] R. L. Johnson and C. J. Neugebauer, Properties of BMO functions whose reciprocals are also BMO , preprint (1991). [7] J. L. Journ´e, Calder´on–Zygmund operators, pseudo–differential operators and the Cauchy integral of Calder´on, Lect. Notes Math. 994, Springer Verlag, (1983). [8] M. Paluszy´nski, Characterization of Lipschitz spaces via commutator of Coifman, Rochberg and Weiss; a multiplier theorem for the semigroup of contractions, Thesis, Washington University (1992). [9] D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals and Radon transforms, Acta Math., 157, (1985), 99–157. [10] C. Segovia and J. L. Torrea, Higher order commutators for vector–valued Calder´on–Zygmund operators, Trans. Amer. Math. Soc. 336, (1993), 537– 556. [11] W. P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, Springer Verlag, 120 (1989). 24