ENDPOINT ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS RELATED TO SCHRODINGER OPERATORS

Abstract. Let L = −∆ + V be a Schr¨odinger operator on R d , d ≥ 3, where V is a nonnegative potential, V 6= 0, and belongs to the reverse H¨older class RHd2. In this paper, we study the commutators b, T for T in a class KL of sublinear operators containing the fundamental operators in harmonic analysis related to L. More precisely, when T ∈ KL, we prove that there exists a bounded subbilinear operator R = RT : H1 L (R d ) × BMO(R d ) → L 1 (R d ) such that (1) |T(S(f, b))| − R(f, b) ≤ |b, T(f)| ≤ R(f, b) + |T(S(f, b))|, where S is a bounded bilinear operator from H1 L (R d ) × BMO(R d ) into L 1 (R d ) which does not depend on T. The subbilinear decomposition (1) allows us to explain why commutators with the fundamental operators are of weak type (H1 L , L1 ), and when a commutator b, T is of strong type (H1 L , L1 ). Also, we discuss the H1 L estimates for commutators of the Riesz transforms associated with the Schr¨odinger operator L.

ENDPOINT ESTIMATES FOR COMMUTATORS OF SINGULAR ¨ INTEGRALS RELATED TO SCHRODINGER OPERATORS LUONG DANG KY Abstract. Let L = −∆ + V be a Schr¨odinger operator on Rd , d ≥ 3, where V is a nonnegative potential, V = 0, and belongs to the reverse H¨older class RHd/2 . In this paper, we study the commutators [b, T ] for T in a class KL of sublinear operators containing the fundamental operators in harmonic analysis related to L. More precisely, when T ∈ KL , we prove that there exists a bounded subbilinear operator R = RT : HL1 (Rd ) × BM O(Rd ) → L1 (Rd ) such that (1) |T (S(f, b))| − R(f, b) ≤ |[b, T ](f )| ≤ R(f, b) + |T (S(f, b))|, where S is a bounded bilinear operator from HL1 (Rd ) × BM O(Rd ) into L1 (Rd ) which does not depend on T . The subbilinear decomposition (1) allows us to explain why commutators with the fundamental operators are of weak type (HL1 , L1 ), and when a commutator [b, T ] is of strong type (HL1 , L1 ). Also, we discuss the HL1 -estimates for commutators of the Riesz transforms associated with the Schr¨ odinger operator L. Contents 1. Introduction 2. Some preliminaries and notations 3. Statement of the results 3.1. Two decomposition theorems 3.2. Hardy estimates for linear commutators 4. Some fundamental operators and the class KL 4.1. The Schr¨odinger-Calder´on-Zygmund operators 4.2. The L-maximal operators 4.3. The L-square functions 5. Proof of the main results 5.1. Proof of Theorem 3.1 and Theorem 3.2 5.2. Proof of Theorem 3.3 and Theorem 3.4 6. Proof of the key lemmas 7. Some applications 7.1. Atomic Hardy spaces related to b ∈ BM O(Rd ) 2 6 11 12 12 13 13 15 16 18 19 20 25 35 35 2010 Mathematics Subject Classification. Primary: 42B35, 35J10 Secondary: 42B20. Key words and phrases. Schr¨ odinger operator, commutator, Hardy space, Calder´on-Zygmund operator, Riesz transforms, BM O, atom. 1 COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 1 7.2. The spaces HL,b (Rd ) related to b ∈ BM O(Rd ) log 7.3. Atomic Hardy spaces HL,α (Rd ) 7.4. The Hardy-Sobolev space HL1,1 (Rd ) References 2 36 38 39 41 1. Introduction Given a function b locally integrable on Rd , and a (classical) Calder´on-Zygmund operator T , we consider the linear commutator [b, T ] defined for smooth, compactly supported functions f by [b, T ](f ) = bT (f ) − T (bf ). A classical result of Coifman, Rochberg and Weiss (see [12]), states that the commutator [b, T ] is continuous on Lp (Rd ) for 1 < p < ∞, when b ∈ BM O(Rd ). Unlike the theory of (classical) Calder´on-Zygmund operators, the proof of this result does not rely on a weak type (1, 1) estimate for [b, T ]. Instead, an endpoint theory was provided for this operator, see for example [37, 38]. A general overview about these facts can be found for instance in [28]. Let L = −∆+V be a Schr¨odinger operator on Rd , d ≥ 3, where V is a nonnegative potential, V = 0, and belongs to the reverse H¨older class RHd/2 . We recall that a nonnegative locally integrable function V belongs to the reverse H¨older class RHq , 1 < q < ∞, if there exists C > 0 such that 1/q 1 C (V (x))q dx ≤ V (x)dx |B| |B| B B d holds for every balls B in R . In [16], Dziuba´ nski and Zienkiewicz introduced 1 d the Hardy space HL (R ) as the set of functions f ∈ L1 (Rd ) such that f HL1 := ML f L1 < ∞, where ML f (x) := supt>0 |e−tL f (x)|. There, they characterized HL1 (Rd ) in terms of atomic decomposition and in terms of the Riesz transforms associated with L, Rj = ∂xj L−1/2 , j = 1, ..., d. In the recent years, there is an increasing interest on the study of commutators of singular integral operators related to Schr¨odinger operators, see for example [7, 10, 21, 32, 43, 44, 45]. In the present paper, we consider commutators of singular integral operators T related to the Schr¨odinger operator L. Here T is in the class KL of all sublinear operators T , bounded from HL1 (Rd ) into L1 (Rd ) and satisfying for any b ∈ BM O(Rd ) and a a generalized atom related to the ball B (see Definition 2.1), we have (b − bB )T a L1 ≤C b BM O , where bB denotes the average of b on B and C > 0 is a constant independent of b, a. The class KL contains the fundamental operators (we refer the reader to [28] for the COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 3 classical case L = −∆) related to the Schr¨odinger operator L: the Riesz transforms Rj , L-Calder´on-Zygmund operators (so-called Schr¨odinger-Calder´on-Zygmund operators), L-maximal operators, L-square operators, etc... (see Section 4). It should be pointed out that, by the work of Shen [39] and Definition 2.2 (see Remark 2.3), one only can conclude that the Riesz transforms Rj are Schr¨odinger-Calder´on-Zygmund operators whenever V ∈ RHd . In this work, we consider all potentials V which belong to the reverse H¨older class RHd/2 . Although Schr¨odinger-Calder´on-Zygmund operators map HL1 (Rd ) into L1 (Rd ) (see Proposition 4.1), it was observed in [32, 48] that, when b ∈ BM O(Rd ), the commutators [b, Rj ] do not map, in general, HL1 (Rd ) into L1 (Rd ). In the classical setting, it was derived by M. Paluszy´ nski [35] that the commutator of the Hilbert transform [b, H] does not map, in general, H 1 (R) into L1 (R). After, C. P´erez showed in [37] that if H 1 (Rd ) is replaced by a suitable atomic subspace Hb1 (Rd ) then commutators of the classical Calder´on-Zygmund operators are continuous from Hb1 (Rd ) into L1 (Rd ). Recall that (see [37]) a function a is a b-atom if i) supp a ⊂ Q for some cube Q, ii) a L∞ ≤ |Q|−1 , iii) Rd a(x)dx = Rd a(x)b(x)dx = 0. The space Hb1 (Rd ) consists of the subspace of L1 (Rd ) of functions f which can be written as f = ∞ j=1 λj aj where aj are b-atoms, and λj are complex numbers with ∞ d j=1 |λj | < ∞. Thus, when b ∈ BM O(R ), it is natural to ask for subspaces of HL1 (Rd ) such that all commutators of Schr¨odinger-Calder´on-Zygmund operators and the Riesz transforms map continuously these spaces into L1 (Rd ). In this paper, we are interested in the following two questions. 1 Question 1. For b ∈ BM O(Rd ). Find the largest subspace HL,b (Rd ) of HL1 (Rd ) such that all commutators of Schr¨odinger-Calder´on-Zygmund operators and the Riesz 1 transforms are bounded from HL,b (Rd ) into L1 (Rd ). 1 Question 2. Characterize the functions b in BM O(Rd ) so that HL,b (Rd ) ≡ HL1 (Rd ). Let X be a Banach space. We say that an operator T : X → L1 (Rd ) is a sublinear operator if for all f, g ∈ X and α, β ∈ C, we have |T (αf + βg)(x)| ≤ |α||T f (x)| + |β||T g(x)|. Obviously, a linear operator T : X → L1 (Rd ) is a sublinear operator. We also say that an operator T : HL1 (Rd ) × BM O(Rd ) → L1 (Rd ) is a subbilinear operator if for every (f, g) ∈ HL1 (Rd ) × BM O(Rd ), the operators T(f, ·) : BM O(Rd ) → L1 (Rd ) and T(·, g) : HL1 (Rd ) → L1 (Rd ) are sublinear operators. To answer Question 1 and Question 2, we study commutators of sublinear operators in KL . More precisely, when T ∈ KL is a sublinear operator, we prove (see Theorem 3.1) that there exists a bounded subbilinear operator R = RT : HL1 (Rd ) × BM O(Rd ) → L1 (Rd ) so that for all (f, b) ∈ HL1 (Rd ) × BM O(Rd ), (1.1) |T (S(f, b))| − R(f, b) ≤ |[b, T ](f )| ≤ R(f, b) + |T (S(f, b))|, COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 4 where S is a bounded bilinear operator from HL1 (Rd )×BM O(Rd ) into L1 (Rd ) which does not depend on T (see Proposition 5.2). When T ∈ KL is a linear operator, we prove (see Theorem 3.2) that there exists a bounded bilinear operator R = RT : HL1 (Rd ) × BM O(Rd ) → L1 (Rd ) such that for all (f, b) ∈ HL1 (Rd ) × BM O(Rd ), (1.2) [b, T ](f ) = R(f, b) + T (S(f, b)). The decompositions (1.1) and (1.2) give a general overview and explains why almost commutators of the fundamental operators are of weak type (HL1 , L1 ), and when a commutator [b, T ] is of strong type (HL1 , L1 ). Let b be a function in BM O(Rd ). We assume that b non-constant, otherwise 1 [b, T ] = 0. We define the space HL,b (Rd ) as the set of all f in HL1 (Rd ) such that [b, ML ](f )(x) = ML (b(x)f (·) − b(·)f (·))(x) belongs to L1 (Rd ), and the norm on 1 1 HL,b (Rd ) is defined by f HL,b = f HL1 b BM O + [b, ML ](f ) L1 . Then, using the subbilinear decomposition (1.1), we prove that all commutators of Schr¨odinger1 Calder´on-Zygmund operators and the Riesz transforms are bounded from HL,b (Rd ) 1 (Rd ) is the largest space having this property, and into L1 (Rd ). Furthermore, HL,b 1 HL,b (Rd ) ≡ HL1 (Rd ) if and only if b ∈ BM OLlog (Rd ) (see Theorem 7.2), that is,   b log BM OL 1 ρ(x)  = sup log e + r |B(x, r)| B(x,r)  |b(y) − bB(x,r) |dy  < ∞, B(x,r) 1 where ρ(x) = sup{r > 0 : rd−2 V (y)dy ≤ 1}. This space BM OLlog (Rd ) arises B(x,r) naturally in the characterization of pointwise multipliers for BM OL (Rd ), the dual space of HL1 (Rd ), see [3, 33]. The above answers Question 1 and Question 2. As another interesting application of the subbilinear decomposition (1.1), we find subspaces of HL1 (Rd ) which do not depend on b ∈ BM O(Rd ) and T ∈ KL , such that [b, T ] maps continuously these spaces into L1 (Rd ) (see Section 7). For instance, when L = −∆ + 1, Theorem 7.4 state that for every b ∈ BM O(Rd ) and T ∈ KL , the commutator [b, T ] is bounded from HL1,1 (Rd ) into L1 (Rd ). Here HL1,1 (Rd ) is the (inhomogeneous) Hardy-Sobolev space considered by Hofmann, Mayboroda and McIntosh in [23], defined as the set of functions f in HL1 (Rd ) such that ∂x1 f, ..., ∂xd f ∈ HL1 (Rd ) with the norm d f 1,1 HL = f 1 HL + ∂xj f 1. HL j=1 Recently, similarly to the classical result of Coifman-Rochberg-Weiss, Gou et al. proved in [21] that the commutators [b, Rj ] are bounded on Lp (Rd ) whenever b ∈ dq BM O(Rd ) and 1 < p < d−q where V ∈ RHq for some d/2 < q < d. Later, in [7], Bongioanni et al. generalized this result by showing that the space BM O(Rd ) can be replaced by a larger space BM OL,∞ (Rd ) = ∪θ≥0 BM OL,θ (Rd ), where BM OL,θ (Rd ) COMMUTATORS OF SINGULAR INTEGRAL OPERATORS is the space of locally integrable functions f satisfying  f BM OL,θ  = sup  B(x,r) 1 1+ r ρ(x) θ 1 |B(x, r)| 5   |f (y) − fB(x,r) |dy  < ∞. B(x,r) Rj∗ Let be the adjoint operators of Rj . Bongioanni et al. established in [6] that the operators Rj∗ are bounded on BM OL (Rd ), and thus from L∞ (Rd ) into BM OL (Rd ). Therefore, it is natural to ask for a class of functions b so that the commutators [b, Rj∗ ] map continuously L∞ (Rd ) into BM OL (Rd ). In [7], the authors found such a class of functions. More precisely, they proved in [7] that the commutators log (Rd ) = [b, Rj∗ ] map continuously L∞ (Rd ) into BM OL (Rd ) whenever b ∈ BM OL,∞ log log (Rd ) is the space of functions f ∈ L1loc (Rd ) such (Rd ). Here BM OL,θ ∪θ≥0 BM OL,θ that   ρ(x) 1  log e + r  f BM Olog = sup  |f (y) − fB(x,r) |dy  < ∞. θ L,θ |B(x, r)| r B(x,r) 1 + ρ(x) B(x,r) A natural question arises: can one replace the space L∞ (Rd ) by BM OL (Rd )? Question 3. Are the commutators [b, Rj∗ ], j = 1, ..., d, bounded on BM OL (Rd ) log whenever b ∈ BM OL,∞ (Rd )? Motivated by this question, we study the HL1 -estimates for commutators of the Riesz transforms. More precisely, given b ∈ BM OL,∞ (Rd ), we prove that the comlog mutators [b, Rj ] are bounded on HL1 (Rd ) if and only if b belongs to BM OL,∞ (Rd ) log (see Theorem 3.4). Furthermore, if b ∈ BM OL,θ (Rd ) for some θ ≥ 0, then there exists a constant C > 1, independent of b, such that d C −1 b log BM OL,θ ≤ b BM OL,θ + [b, Rj ] 1 →H 1 HL L ≤C b log . BM OL,θ j=1 As a consequence, we get the positive answer for Question 3. Now, an open question is the following: Open question. Find the set of all functions b such that the commutators [b, Rj ], j = 1, ..., d, are bounded on HL1 (Rd ). Let us emphasize the three main purposes of this paper. First, we prove the two decomposition theorems: the subbilinear decomposition (1.1) and the bilinear decomposition (1.2). Second, we characterize functions b in BM OL,∞ (Rd ) so that the commutators of the Riesz transforms are bounded on HL1 (Rd ), which answers 1 Question 3. Finally, we find the largest subspace HL,b (Rd ) of HL1 (Rd ) such that all commutators of Schr¨odinger-Calder´on-Zygmund operators and the Riesz transforms 1 are bounded from HL,b (Rd ) into L1 (Rd ). Besides, we find also the characterization COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 6 1 of functions b ∈ BM O(Rd ) so that HL,b (Rd ) ≡ HL1 (Rd ), which answer Question 1 and Question 2. Especially, we show that there exist subspaces of HL1 (Rd ) which do not depend on b ∈ BM O(Rd ) and T ∈ KL , such that [b, T ] maps continuously these spaces into L1 (Rd ), see Section 7. This paper is organized as follows. In Section 2, we present some notations and preliminaries about Hardy spaces, new atoms, BM O type spaces and Schr¨odingerCalder´on-Zygmund operators. In Section 3, we state the main results: two decomposition theorems (Theorem 3.1 and Theorem 3.2), Hardy estimates for commutators of Schr¨odinger-Calder´on-Zygmund operators and the commutators of the Riesz transforms (Theorem 3.3 and Theorem 3.4). In Section 4, we give some examples of fundamental operators related to L which are in the class KL . Section 5 is devoted to the proofs of the main theorems. Section 6 is devoted to the proofs of the key lemmas. Finally, in Section 7, we give some examples of subspaces of HL1 (Rd ) such that all commutators [b, T ], T ∈ KL , map continuously these spaces into L1 (Rd ). Throughout the whole paper, C denotes a positive geometric constant which is independent of the main parameters, but may change from line to line. The symbol f ≈ g means that f is equivalent to g (i.e. C −1 f ≤ g ≤ Cf ). In Rd , we denote by B = B(x, r) an open ball with center x and radius r > 0, and tB(x, r) := B(x, tr) whenever t > 0. For any measurable set E, we denote by χE its characteristic function, by |E| its Lebesgue measure, and by E c the set Rd \ E. 2. Some preliminaries and notations In this paper, we consider the Schr¨odinger differential operator L = −∆ + V on Rd , d ≥ 3, where V is a nonnegative potential, V = 0. As in the works of Dziuba´ nski et al [15, 16], we always assume that V belongs to the reverse H¨older class RHd/2 . Recall that a nonnegative locally integrable function V is said to belong to a reverse H¨older class RHq , 1 < q < ∞, if there exists C > 0 such that 1 |B| (V (x))q dx 1/q ≤ C |B| B V (x)dx B d holds for every balls B in R . By H¨older inequality, RHq1 ⊂ RHq2 if q1 ≥ q2 > 1. For q > 1, it is well-known that V ∈ RHq implies V ∈ RHq+ε for some ε > 0 (see [19]). Moreover, V (y)dy is a doubling measure, namely for any ball B(x, r) we have V (y)dy ≤ C0 (2.1) B(x,2r) V (y)dy. B(x,r) Let {Tt }t>0 be the semigroup generated by L and Tt (x, y) be their kernels. Namely, Tt f (x) = e−tL f (x) = Tt (x, y)f (y)dy, Rd f ∈ L2 (Rd ), t > 0. COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 7 We say that a function f ∈ L2 (Rd ) belongs to the space H1L (Rd ) if f H1L := ML f L1 < ∞, where ML f (x) := supt>0 |Tt f (x)| for all x ∈ Rd . The space HL1 (Rd ) is then defined as the completion of H1L (Rd ) with respect to this norm. In [15] it was shown that the dual of HL1 (Rd ) can be identified with the space BM OL (Rd ) which consists of all functions f ∈ BM O(Rd ) with f BM OL := f BM O 1 ρ(x)≤r |B(x, r)| |f (y)|dy < ∞, + sup B(x,r) where ρ is the auxiliary function defined as in [39], that is, (2.2) ρ(x) = sup r > 0 : 1 V (y)dy ≤ 1 , rd−2 B(x,r) x ∈ Rd . Clearly, 0 < ρ(x) < ∞ for all x ∈ Rd , and thus Rd = sets Bn are defined by (2.3) n∈Z Bn , where the Bn = {x ∈ Rd : 2−(n+1)/2 < ρ(x) ≤ 2−n/2 }. The following proposition plays an important role in our study. Proposition 2.1 (see [39], Lemma 1.4). There exist two constants κ > 1 and k0 ≥ 1 such that for all x, y ∈ Rd , κ−1 ρ(x) 1 + |x − y| ρ(x) −k0 ≤ ρ(y) ≤ κρ(x) 1 + |x − y| ρ(x) k0 k0 +1 . Throughout the whole paper, we denote by CL the L-constant (2.4) CL = 8.9k0 κ, where k0 and κ are defined as in Proposition 2.1. Given 1 < q ≤ ∞. Following Dziuba´ nski and Zienkiewicz [16], a function a is called a (HL1 , q)-atom related to the ball B(x0 , r) if r ≤ CL ρ(x0 ) and i) supp a ⊂ B(x0 , r), ii) a Lq ≤ |B(x0 , r)|1/q−1 , iii) if r ≤ C1L ρ(x0 ) then Rd a(x)dx = 0. A function a is called a classical (H 1 , q)-atom related to the ball B = B(x0 , r) if it satisfies (i), (ii) and Rd a(x)dx = 0. The following atomic characterization of HL1 (Rd ) is due to [16]. Theorem 2.1 (see [16], Theorem 1.5). Let 1 < q ≤ ∞. A function f is in HL1 (Rd ) 1 if and only if it can be written as f = j λj aj , where aj are (HL , q)-atoms and COMMUTATORS OF SINGULAR INTEGRAL OPERATORS j 8 |λj | < ∞. Moreover, f 1 HL ≈ inf |λj | : f = j λj aj . j Note that a classical (H 1 , q)-atom is not a (HL1 , q)-atom in general. In fact, there exists a constant C > 0 such that if f is a classical (H 1 , q)-atom, then it can be written as f = nj=1 λj aj , for some n ∈ Z+ , where aj are (HL1 , q)-atoms and n j=1 |λj | ≤ C, see for example [47]. In this work, we need a variant of the definition of atoms for HL1 (Rd ) which include classical (H 1 , q)-atoms and (HL1 , q)-atoms. This kind of atoms have been used in the work of Chang, Dafni and Stein [11, 13]. Definition 2.1. Given 1 < q ≤ ∞ and ε > 0. A function a is called a generalized (HL1 , q, ε)-atom related to the ball B(x0 , r) if i) supp a ⊂ B(x0 , r), ii) a Lq ≤ |B(x0 , r)|1/q−1 , iii) | Rd a(x)dx| ≤ r ρ(x0 ) ε . d 1 d The space H1,q,ε L,at (R ) is defined to be set of all functions f in L (R ) which can be 1 written as f = ∞ j=1 λj aj where the aj are generalized (HL , q, ε)-atoms and the λj 1,q,ε d are complex numbers such that ∞ j=1 |λj | < ∞. As usual, the norm on HL,at (R ) is defined by ∞ f H1,q,ε L,at ∞ |λj | : f = = inf j=1 λ j aj . j=1 k d The space H1,q,ε L,fin (R ) is defined to be set of all f = j=1 λj aj , where the aj are 1,q,ε 1 generalized (HL , q, ε)-atoms. Then, the norm of f in HL,fin (Rd ) is defined by k f H1,q,ε L,fin k |λj | : f = = inf j=1 λj aj . j=1 Remark 2.1. Let 1 < q ≤ ∞ and ε > 0. Then, a classical (H 1 , q)-atom is a generalized (HL1 , q, ε)-atom related to the same ball, and a (HL1 , q)-atom is CL ε times a generalized (HL1 , q, ε)-atom related to the same ball. Throughout the whole paper, we always use generalized (HL1 , q, ε)-atoms except in the proof of Theorem 3.4. More precisely, in order to prove Theorem 3.4, we need to use (HL1 , q)-atoms from Dziuba´ nski and Zienkiewicz (see above). The following gives a characterization of HL1 (Rn ) in terms of generalized atoms. d 1 d Theorem 2.2. Let 1 < q ≤ ∞ and ε > 0. Then, H1,q,ε L,at (R ) = HL (R ) and the norms are equivalent. In order to prove Theorem 2.2, we need the following lemma. COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 9 Lemma 2.1 (see [31], Lemma 2). Let V ∈ RHd/2 . Then, there exists σ0 > 0 depends only on L, such that for every |y − z| < |x − y|/2 and t > 0, we have |y − z| √ t |Tt (x, y) − Tt (x, z)| ≤ C σ0 d t− 2 e− |x−y|2 t ≤C |y − z|σ0 . |x − y|d+σ0 Proof of Theorem 2.2. As ML is a sublinear operator, by Remark 2.1 and Theorem 2.1, it is sufficient to show that ML (a) (2.5) L1 ≤C for all generalized (HL1 , q, ε)-atom a related to the ball B = B(x0 , r). Indeed, from the Lq -boundedness of the classical Hardy-Littlewood maximal operator M, the estimate ML (a) ≤ CM(a) and H¨older inequality, ML (a) (2.6) ≤ C M(a) L1 (2B) L1 (2B) ≤ C|2B|1/q M(a) Lq ≤ C, where 1/q + 1/q = 1. Let x ∈ / 2B and t > 0, Lemma 2.1 and (3.5) of [16] give |Tt (a)(x)| = Tt (x, y)a(y)dy Rd ≤ (Tt (x, y) − Tt (x, x0 ))a(y)dy + |Tt (x, x0 )| B a(y)dy B rε rσ0 . + C ≤ C |x − x0 |d+σ0 |x − x0 |d+ε Therefore, ML (a) L1 ((2B)c ) = sup |Tt (a)| L1 ((2B)c ) t>0 rσ0 dx + C |x − x0 |d+σ0 ≤C (2B)c rε dx |x − x0 |d+ε (2B)c ≤ C. (2.7) Then, (2.5) follows from (2.6) and (2.7). By Theorem 2.2, the following can be seen as a direct consequence of Proposition 3.2 of [47] and remark 2.1. Proposition 2.2. Let 1 < q < ∞, ε > 0 and X be a Banach space. Suppose that d T : H1,q,ε L,fin (R ) → X is a sublinear operator with sup{ T a X : a is a generalized (HL1 , q, ε) − atom} < ∞. Then, T can be extended to a bounded sublinear operator T from HL1 (Rd ) into X , moreover, T 1 →X HL ≤ C sup{ T a X : a is a generalized (HL1 , q, ε) − atom}. COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 10 Now, we turn to explain the new BM O type spaces introduced by Bongioanni, 1 Harboure and Salinas in [7]. Here and in what follows fB := |B| f (x)dx and B (2.8) M O(f, B) := 1 |B| |f (y) − fB |dy. B For θ ≥ 0, following [7], we denote by BM OL,θ (Rd ) the set of all locally integrable functions f such that   f BM OL,θ  = sup  B(x,r) 1 1+ r ρ(x) θ  M O(f, B(x, r)) < ∞, log (Rd ) the set of all locally integrable functions f such that and BM OL,θ   ρ(x)  log e + r  f BM Olog = sup  M O(g, B(x, r)) < ∞. θ L,θ r B(x,r) 1 + ρ(x) log When θ = 0, we write BM OLlog (Rd ) instead of BM OL,0 (Rd ). We next define BM OL,∞ (Rd ) = BM OL,θ (Rd ) θ≥0 and log BM OL,θ (Rd ). log BM OL,∞ (Rd ) = θ≥0 Observe that BM OL,0 (Rd ) is just the classical BM O(Rd ) space. Moreover, for any 0 ≤ θ ≤ θ ≤ ∞, we have (2.9) BM OL,θ (Rd ) ⊂ BM OL,θ (Rd ), log log BM OL,θ (Rd ) ⊂ BM OL,θ (Rd ) and (2.10) log log BM OL,θ (Rd ) = BM OL,θ (Rd ) ∩ BM OL,∞ (Rd ). Remark 2.2. The inclusions in (2.9) are strict in general. In particular: i) The space BM OL,∞ (Rd ) is in general larger than the space BM O(Rd ). Indeed, when V (x) ≡ |x|2 , it is easy to check that the functions bj (x) = |xj |2 , j = 1, ..., d, belong to BM OL,∞ (Rd ) but not to BM O(Rd ). log ii) The space BM OL,∞ (Rd ) is in general larger than the space BM OLlog (Rd ). Indeed, when V (x) ≡ 1, it is easy to check that the functions bj (x) = |xj |, j = 1, ..., d, log belong to BM OL,∞ (Rd ) but not to BM OLlog (Rd ). Next, let us recall the notation of Schr¨odinger-Calder´on-Zygmund operators. COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 11 Let δ ∈ (0, 1]. According to [33], a continuous function K : Rd × Rd \ {(x, x) : x ∈ Rd } → C is said to be a (δ, L)-Calder´on-Zygmund singular integral kernel if for each N > 0, C(N ) |x − y| −N (2.11) |K(x, y)| ≤ 1 + |x − y|d ρ(x) for all x = y, and (2.12) |K(x, y) − K(x , y)| + |K(y, x) − K(y, x )| ≤ C |x − x |δ |x − y|d+δ for all 2|x − x | ≤ |x − y|. As usual, we denote by Cc∞ (Rd ) the space of all C ∞ -functions with compact support, by S(Rd ) the Schwartz space on Rd . Definition 2.2. A linear operator T : S(Rd ) → S (Rd ) is said to be a (δ, L)Calder´on-Zygmund operator if T can be extended to a bounded operator on L2 (Rd ) and if there exists a (δ, L)-Calder´on-Zygmund singular integral kernel K such that / supp f , we have for all f ∈ Cc∞ (Rd ) and all x ∈ T f (x) = K(x, y)f (y)dy. Rd An operator T is said to be a Schr¨odinger-Calder´on-Zygmund operator associated with L (or L-Calder´on-Zygmund operator) if it is a (δ, L)-Calder´on-Zygmund operator for some δ ∈ (0, 1]. We say that T satisfies the condition T ∗ 1 = 0 if there are q ∈ (1, ∞] and ε > 0 so that Rd T a(x)dx = 0 holds for every generalized (HL1 , q, ε)-atoms a. Remark 2.3. i) Using Proposition 2.1, Inequality (2.11) is equivalent to C(N ) |x − y| −N |K(x, y)| ≤ 1 + |x − y|d ρ(y) for all x = y. ii) By Theorem 0.8 of [39] and Theorem 1.1 of [40], we see that the Riesz transforms Rj are L-Calder´on-Zygmund operators satisfying Rj∗ 1 = 0 whenever V ∈ RHd . iii) If T is a L-Calder´on-Zygmund operator then it is also a classical Calder´onZygmund operator, and thus T is bounded on Lp (Rd ) for 1 < p < ∞ and bounded from L1 (Rd ) into L1,∞ (Rd ). 3. Statement of the results Recall that KL is the set of all sublinear operators T bounded from HL1 (Rd ) into L (Rd ) and that there are q ∈ (1, ∞] and ε > 0 such that 1 (b − bB )T a d L1 ≤C b BM O (HL1 , q, ε)-atom for all b ∈ BM O(R ), any generalized C > 0 is a constant independent of b, a. a related to the ball B, where COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 12 3.1. Two decomposition theorems. Let b be a locally integrable function and T ∈ KL . As usual, the (sublinear) commutator [b, T ] of the operator T is defined by [b, T ](f )(x) := T (b(x) − b(·))f (·) (x). Theorem 3.1 (Subbilinear decomposition). Let T ∈ KL . There exists a bounded subbilinear operator R = RT : HL1 (Rd ) × BM O(Rd ) → L1 (Rd ) such that for all (f, b) ∈ HL1 (Rd ) × BM O(Rd ), we have |T (S(f, b))| − R(f, b) ≤ |[b, T ](f )| ≤ R(f, b) + |T (S(f, b))|, where S is a bounded bilinear operator from HL1 (Rd ) × BM O(Rd ) into L1 (Rd ) which does not depend on T . Using Theorem 3.1, we obtain immediately the following result. Proposition 3.1. Let T ∈ KL so that T is of weak type (1, 1). Then, the subbilinear operator T(f, g) = [g, T ](f ) maps continuously HL1 (Rd ) × BM O(Rd ) into L1,∞ (Rd ). As the Riesz transforms Rj = ∂xj L−1/2 are of weak type (1, 1) (see [30]), the following can be seen as a consequence of Proposition 3.1 (see also [32]). Corollary 3.1 (see [32], Theorem 4.1). Let b ∈ BM O(Rd ). Then, the commutators [b, Rj ] are bounded from HL1 (Rd ) into L1,∞ (Rd ). When T is linear and belongs to KL , we obtain the bilinear decomposition for the linear commutator [b, T ] of f , [b, T ](f ) = bT (f ) − T (bf ), instead of the subbilinear decomposition as stated in Theorem 3.1. Theorem 3.2 (Bilinear decomposition). Let T be a linear operator in KL . Then, there exists a bounded bilinear operator R = RT : HL1 (Rd ) × BM O(Rd ) → L1 (Rd ) such that for all (f, b) ∈ HL1 (Rd ) × BM O(Rd ), we have [b, T ](f ) = R(f, b) + T (S(f, b)), where S is as in Theorem 3.1. 3.2. Hardy estimates for linear commutators. Our first main result of this subsection is the following theorem. Theorem 3.3. i) Let b ∈ BM OLlog (Rd ) and T be a L-Calder´on-Zygmund operator satisfying T ∗ 1 = 0. Then, the linear commutator [b, T ] is bounded on HL1 (Rd ). ii) When V ∈ RHd , the converse holds. Namely, if b ∈ BM O(Rd ) and [b, T ] is bounded on HL1 (Rd ) for every L-Calder´on-Zygmund operator T satisfying T ∗ 1 = 0, then b ∈ BM OLlog (Rd ). Furthermore, d b log BM OL ≈ b BM O + [b, Rj ] 1 →H 1 . HL L j=1 Next result concerns the HL1 -estimates for commutators of the Riesz transforms. COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 13 Theorem 3.4. Let b ∈ BM OL,∞ (Rd ). Then, the commutators [b, Rj ], j = 1, ..., d, log are bounded on HL1 (Rd ) if and only if b ∈ BM OL,∞ (Rd ). Furthermore, if b ∈ log BM OL,θ (Rd ) for some θ ≥ 0, we have d b log BM OL,θ ≈ b BM OL,θ + [b, Rj ] 1 →H 1 . HL L j=1 Remark that the above constants depend on θ. log Note that BM OLlog (Rd ) is in general proper subset of BM OL,∞ (Rd ) (see Remark 2.2). When V ∈ RHd , although the Riesz transforms Rj are L-Calder´on-Zygmund operators satisfying Rj∗ 1 = 0, Theorem 3.4 cannot be deduced from Theorem 3.3. As a consequence of Theorem 3.4, we obtain the following interesting result. Corollary 3.2. Let b ∈ BM O(Rd ). Then, b belongs to LM O(Rd ) if and only if the vector-valued commutator [b, ∇(−∆ + 1)−1/2 ] maps continuously h1 (Rd ) into h1 (Rd , Rd ) = (h1 (Rd ), ..., h1 (Rd )). Furthermore, b LM O ≈ b BM O + [b, ∇(−∆ + 1)−1/2 ] h1 (Rd )→h1 (Rd ,Rd ) . Here h1 (Rd ) is the local Hardy space of D. Goldberg (see [20]), and LM O(Rd ) is the space of all locally integrable functions f such that f LM O := sup B(x,r) log e + 1 M O(f, B(x, r)) r < ∞. It should be pointed out that LM O type spaces appear naturally when studying the boundedness of Hankel operators on the Hardy spaces H 1 (Td ) and H 1 (Bd ) (where Bd is the unit ball in Cd and Td = ∂Bd ), characterizations of pointwise multipliers for BM O type spaces, endpoint estimates for commutators of singular integrals operators and their applications to PDEs, see for example [5, 9, 24, 25, 28, 36, 41, 42]. 4. Some fundamental operators and the class KL The purpose of this section is to give some examples of fundamental operators related to L which are in the class KL . 4.1. The Schr¨ odinger-Calder´ on-Zygmund operators. Proposition 4.1. Let T be any L-Calder´on-Zygmund operator. Then, T belongs to the class KL . Proposition 4.2. The Riesz transforms Rj are in the class KL . The proof of Proposition 4.2 follows directly from Lemma 5.7 and the fact that the Riesz transforms Rj are bounded from HL1 (Rd ) into L1 (Rd ). To prove Proposition 4.1, we need the following two lemmas. COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 14 Lemma 4.1. Let 1 ≤ q < ∞. Then, there exists a constant C > 0 such that for every ball B, f ∈ BM O(Rd ) and k ∈ Z+ , 1/q 1 q |f (y) − f | dy ≤ Ck f BM O . B |2k B| 2k B The proof of Lemma 4.1 follows directly from the classical John-Nirenberg inequality. See also Lemma 6.6 below. Lemma 4.2. Let 1 < q ≤ ∞ and ε > 0. Assume that T is a (δ, L)-Calder´onZygmund operator and a is a generalized (HL1 , q, ε)-atom related to the ball B = B(x0 , r). Then, T a Lq (2k+1 B\2k B) ≤ C2−kδ0 |2k B|1/q−1 for all k = 1, 2, ..., where δ0 = min{ε, δ}. Proof. Let x ∈ 2k+1 B \ 2k B, so that |x − x0 | ≥ 2r. Since T is a (δ, L)-Calder´onZygmund operator, we get (K(x, y) − K(x, x0 ))a(y)dy + |K(x, x0 )| |T a(x)| ≤ B a(y)dy Rd δ |y − x0 | 1 |x − x0 | |a(y)|dy + C 1+ d+δ d |x − x0 | |x − x0 | ρ(x0 ) ≤ C −ε r ρ(x0 ) ε B ≤ C rδ rε r δ0 + C ≤ C . |x − x0 |d+δ |x − x0 |d+ε |x − x0 |d+δ0 Consequently, Ta Lq (2k+1 B\2k B) r δ0 |2k+1 B|1/q ≤ C2−kδ0 |2k B|1/q−1 . (2k r)d+δ0 ≤C Proof of Proposition 4.1. Assume that T is a (δ, L)-Calder´on-Zygmund for some δ ∈ (0, 1]. Let us first verify that T is bounded from HL1 (Rd ) into L1 (Rd ). By Proposition 2.2, it is sufficient to show that Ta L1 ≤C for all generalized (HL1 , 2, δ)-atom a related to the ball B. Indeed, from the L2 boundedness of T and Lemma 4.2, we obtain that ∞ Ta L1 = Ta L1 (2B) + Ta L1 (2k+1 B\2k B) k=1 ∞ ≤ C|2B|1/2 T L2 →L2 a L2 |2k+1 B|1/2 2−kδ |2k B|−1/2 +C k=1 ≤ C. COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 15 Let us next establish that (f − fB )T a L1 ≤C f BM O for all f ∈ BM O(Rd ), any generalized (HL1 , 2, δ)-atom a related to the ball B = B(x0 , r). Indeed, by H¨older inequality, Lemma 4.1 and Lemma 4.2, we get = (f − fB )T a L1 (f − fB )T a L1 (2B) + (f − fB )T a L1 (2k+1 B\2k B) a f − fB k≥1 ≤ (f − fB )χ2B L2 T L2 →L2 L2 + L2 (2k+1 B) Ta L2 (2k+1 B\2k B) k≥1 ≤ C f BM O + C(k + 1) f k+1 B|1/2 2−kδ |2k B|−1/2 BM O |2 k≥1 ≤ C f BM O , which ends the proof. 4.2. The L-maximal operators. Recall that {Tt }t>0 is heat semigroup generated by L and Tt (x, y) are their kernels. Namely, Tt f (x) = e−tL f (x) = f ∈ L2 (Rd ), Tt (x, y)f (y)dy, t > 0. Rd Then the ”heat” maximal operator is defined by ML f (x) = sup |Tt f (x)|, t>0 and the ”Poisson” maximal operator is defined by MPL f (x) = sup |Pt f (x)|, t>0 where Pt f (x) = e √ −t L ∞ t f (x) = √ 2 π t2 e− 4u 3 0 u2 Tu f (x)du. Proposition 4.3. The ”heat” maximal operator ML is in the class KL . Proposition 4.4. The ”Poisson” maximal operator MPL is in the class KL . Here we just give the proof of Proposition 4.3. For the one of Proposition 4.4, we leave the details to the interested reader. Proof of Proposition 4.3. Obviously, ML is bounded from HL1 (Rd ) into L1 (Rd ). Now, let us prove that (f − fB )ML (a) L1 ≤C f BM O COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 16 for all f ∈ BM O(Rd ), any generalized (HL1 , 2, σ0 )-atom a related to the ball B = B(x0 , r), where the constant σ0 > 0 is as in Lemma 2.1. Indeed, by the proof of Theorem 2.2, for every x ∈ / 2B, rσ0 ML (a)(x) ≤ C . |x − x0 |d+σ0 Therefore, using Lemma 4.1, the L2 -boundedness of the classical Hardy-Littlewood maximal operator M and the estimate ML (a) ≤ CM(a), we obtain that = (f − fB )ML (a) (f − fB )ML (a) ≤ C f − fB L2 (2B) L1 L1 (2B) M(a) + (f − fB )ML (a) L2 L1 ((2B)c ) |f (x) − fB(x0 ,r) | +C r σ0 dx |x − x0 |d+σ0 |x−x0 |≥2r ≤ C f BM O , where we have used the following classical inequality r σ0 |f (x) − fB(x0 ,r) | dx ≤ C f |x − x0 |d+σ0 BM O , |x−x0 |≥2r which proof can be found in [17]. This completes the proof of Proposition 4.3. 4.3. The L-square functions. Recall (see [15]) that the L-square funcfions g and G are defined by ∞ 1/2 dt g(f )(x) =  |t∂t Tt (f )(x)|2  t 0 and  1/2 ∞ |t∂t Tt (f )(y)|2  G(f )(x) =  dydt   td+1 . 0 |x−y| 0 such that 2 c |x−y| |h| δ (4.1) |t∂t Tt (x, y + h) − t∂t Tt (x, y)| ≤ C √ t−d/2 e− 4 t , t COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 17 for all |h| < |x−y| , 0 < t. Here and in the proof of Proposition 4.5, the constants 2 δ, c ∈ (0, 1) are as in Proposition 4 of [15]. √ . Otherwise, (4.1) follows Proof. One only needs to consider the case t < |h| < |x−y| 2 directly√from (b) in Proposition 4 of [15]. For t < |h| < |x−y| . By (a) in Proposition 4 of [15], we get 2 |x−y−h|2 + Ct−d/2 e−c |t∂t Tt (x, y + h) − t∂t Tt (x, y)| ≤ Ct−d/2 e−c t 2 c |x−y| |h| δ ≤ C √ t−d/2 e− 4 t . t |x−y|2 t Proof of Proposition 4.5. The (HL1 − L1 ) type boundedness of g is well-known, see for example [15, 22]. Let us now show that (f − fB )g(a) L1 ≤C f BM O for all f ∈ BM O(Rd ), any generalized (HL1 , 2, δ)-atom a related to the ball B = B(x0 , r). Indeed, it follows from Lemma 4.3 and (a) in Proposition 4 of [15] that for every t > 0, x ∈ / 2B, |t∂t Tt (a)(x)| (t∂t Tt (x, y) − t∂t Tt (x, x0 ))a(y)dy + t∂t Tt (x, x0 ) = B r ≤ C √ r ≤ C √ a(y)dy B δ t t δ t −d/2 |x−x0 |2 − 4c t a e 2 c |x−x0 | t t−d/2 e− 4 −d/2 + Ct L1 √ √ t t 1+ + ρ(x) ρ(x0 ) |x−x0 |2 −c t e 2 c |x−x0 | t g(a)(x) ≤ C  ∞ 2 r t r ρ(x0 ) . Therefore, as 0 < δ < 1, using the estimate e− 2   −δ δ |x−x0 |2 − 2c t t−d e 0  2   |x−x0 | r2 ≤ C  t  0 rδ ≤ C . |x − x0 |d+δ ≤ C(c, d)( |x−xt 0 |2 )d+2 , 1/2 dt  t ∞ δ t−d t |x − x0 |2 d+2 dt t + |x−x0 |2 r2 t δ t−d 1/2  dt  t  δ COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 18 Therefore, the L2 -boundedness of g and Lemma 4.1 yield = (f − fB )g(a) (f − fB )g(a) ≤ f − fB L2 (2B) L1 L1 (2B) g(a) + (f − fB )g(a) L2 L1 ((2B)c ) |f (x) − fB(x0 ,r) | +C rδ dx |x − x0 |d+δ |x−x0 |≥2r ≤ C f BM O , which ends the proof. 5. Proof of the main results In this section, we fix a non-negative function ϕ ∈ S(Rd ) with supp ϕ ⊂ B(0, 1) and Rd ϕ(x)dx = 1. Then, we define the linear operator H by ψn,k f − ϕ2−n/2 ∗ (ψn,k f ) , H(f ) = n,k where ψn,k , n ∈ Z, k = 1, 2, ... is as in Lemma 2.5 of [16] (see also Lemma 6.2). Remark 5.1. When V (x) ≡ 1, we can define H(f ) = f − ϕ ∗ f . Let us now consider the set E = {0, 1}d \ {(0, · · · , 0)} and {ψ σ }σ∈E the wavelet with compact support as in Section 3 of [4] (see also Section 2 of [28]). Suppose that ψ σ is supported in the cube ( 12 − 2c , 12 − 2c )d for all σ ∈ E. As it is classical, for σ ∈ E and I a dyadic cube of Rd which may be written as the set of x such that 2j x − k ∈ (0, 1)d , we note ψIσ (x) = 2dj/2 ψ σ (2j x − k). In the sequel, the letter I always refers to dyadic cubes. Moreover, we note kI the cube of same center dilated by the coefficient k. Remark 5.2. For every σ ∈ E and I a dyadic cube. Because of the assumption on the support of ψ σ , the function ψIσ is supported in the cube cI. In [4] (see also [28]), Bonami et al. established the following. Proposition 5.1. The bounded bilinear operator Π, defined by f, ψIσ g, ψIσ (ψIσ )2 , Π(f, g) = I σ∈E is bounded from H 1 (Rd ) × BM O(Rd ) into L1 (Rd ). COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 19 5.1. Proof of Theorem 3.1 and Theorem 3.2. In order to prove Theorem 3.1 and Theorem 3.2, we need the following key two lemmas which proofs will given in Section 6. Lemma 5.1. The linear operator H is bounded from HL1 (Rd ) into H 1 (Rd ). Lemma 5.2. Let T ∈ KL . Then, the subbilinear operator U(f, b) := [b, T ](f − H(f )) is bounded from HL1 (Rd ) × BM O(Rd ) into L1 (Rd ). By Proposition 5.1 and Lemma 5.1, we obtain: Proposition 5.2. The bilinear operator S(f, g) := −Π(H(f ), g) is bounded from HL1 (Rd ) × BM O(Rd ) into L1 (Rd ). We recall (see [28]) that the class K is the set of all sublinear operators T bounded from H 1 (Rd ) into L1 (Rd ) so that for some q ∈ (1, ∞], (b − bB )T a L1 ≤C b BM O , for all b ∈ BM O(Rd ), any classical (H 1 , q)-atom a related to the ball B, where C > 0 a constant independent of b, a. Remark 5.3. By Remark 2.1 and as H 1 (Rd ) ⊂ HL1 (Rd ), we obtain that KL ⊂ K, which allows to apply the two classical decomposition theorems (Theorem 3.1 and Theorem 3.2 of [28]). This is a key point in our proofs. Proof of Theorem 3.1. As T ∈ KL ⊂ K, it follows from Theorem 3.1 of [28] that there exists a bounded subbilinear operator V : H 1 (Rd ) × BM O(Rd ) → L1 (Rd ) such that for all (g, b) ∈ H 1 (Rd ) × BM O(Rd ), we have (5.1) |T (−Π(g, b))| − V(g, b) ≤ |[b, T ](g)| ≤ V(g, b) + |T (−Π(g, b))|. Let us now define the bilinear operator R by R(f, b) := |U(f, b)| + V(H(f ), b) for all (f, b) ∈ HL1 (Rd )×BM O(Rd ), where U is the subbilinear operator as in Lemma 5.2. Then, using the subbilinear decomposition (5.1) with g = H(f ), |T (S(f, b))| − R(f, b) ≤ |[b, T ](f )| ≤ |T (S(f, b))| + R(f, b), where the bounded bilinear operator S : HL1 (Rd ) × BM O(Rd ) → L1 (Rd ) is given in Proposition 5.2. Furthermore, by Lemma 5.2 and Lemma 5.1, we get R(f, b) L1 ≤ U(f, b) L1 + V(H(f ), b) L1 ≤ C f HL1 b BM O + C H(f ) H 1 b ≤ C f 1 HL b BM O BM O , where we used the boundedness of V on H 1 (Rd ) × BM O(Rd ) into L1 (Rd ). This completes the proof. COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 20 Proof of Theorem 3.2. The proof follows the same lines except that now, one deals with equalities instead of inequalities. Namely, as T is a linear operator in KL ⊂ K, Theorem 3.2 of [28] yields that there exists a bounded bilinear operator W : H 1 (Rd ) × BM O(Rd ) → L1 (Rd ) such that for every (g, b) ∈ H 1 (Rd ) × BM O(Rd ), [b, T ](g) = W(g, b) + T (−Π(g, b)) Therefore, for every (f, b) ∈ HL1 (Rd ) × BM O(Rd ), [b, T ](f ) = R(f, b) + T (S(f, b)), where R(f, b) := U(f, b) + W(H(f ), b) is a bounded bilinear operator from HL1 (Rd ) × BM O(Rd ) into L1 (Rd ). This completes the proof. 5.2. Proof of Theorem 3.3 and Theorem 3.4. First, recall that V M OL (Rd ) is the closure of Cc∞ (Rd ) in BM OL (Rd ). Then, the following result due to Ky [29]. Theorem 5.1. The space HL1 (Rd ) is the dual of the space V M OL (Rd ). In order to prove Theorem 3.3, we need the following key lemmas, which proofs will be given in Section 6. log Lemma 5.3. Let 1 ≤ q < ∞ and θ ≥ 0. Then, for every f ∈ BM OL,θ (Rd ), B = B(x, r) and k ∈ Z+ , we have 1 |f (y) − fB |q dy |2k B| 1+ 1/q ≤ Ck 2k B 2k r ρ(x) (k0 +1)θ log e + ( ρ(x) )k0 +1 2k r f log , BM OL,θ where the constant k0 is as in Proposition 2.1. Lemma 5.4. Let 1 < q < ∞, ε > 0 and T be a L-Calder´on-Zygmund operator. Then, the following two statements hold: i) If T ∗ 1 = 0, then T is bounded from HL1 (Rd ) into H 1 (Rd ). ii) For every f, g ∈ BM O(Rd ), generalized (HL1 , q, ε)-atom a related to the ball B, (f − fB )(g − gB )T a L1 ≤C f BM O g BM O . Proof of Theorem 3.3. (i). Assume that T is a (δ, L)-Calder´on-Zygmund operator. We claim that, as, by Lemma 5.4, it is sufficient to prove that (5.2) (b − bB )a 1 HL ≤C b log BM OL and (5.3) (b − bB )T a 1 HL ≤C b log BM OL COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 21 hold for every generalized (HL1 , 2, δ)-atom a related to the ball B = B(x0 , r) with the constants are independent of b, a. Indeed, if (5.2) and (5.3) are true, then [b, T ](a) ≤ 1 HL (b − bB )T a ≤ C b ≤ C b log BM OL + C T ((b − bB )a) 1 HL +C T 1 →H 1 HL H1 (b − bB )a 1 HL log . BM OL Therefore, Proposition 2.2 yields that [b, T ] is bounded on HL1 (Rd ), moreover, [b, T ] ≤ C, 1 →H 1 HL L where the constant C is independent of b. The proof of (5.2) is similar to the one of (5.3) but uses an easier argument, we leave the details to the interested reader. Let us now establish (5.3). By Theorem 5.1, it is sufficient to show that (5.4) φ(b − bB )T a L1 ≤C b log BM OL φ BM OL for all φ ∈ Cc∞ (Rd ). Besides, from Lemma 5.4, (φ − φB )(b − bB )T a L1 ≤C b BM O φ BM O ≤C b log BM OL φ BM OL . This together with Lemma 2 of [15] allow us to reduce (5.4) to showing that (5.5) log e + ρ(x0 ) r (b − bB )T a L1 ≤C b log . BM OL Setting ε = δ/2, it is easy to check that there exists a constant C = C(ε) > 0 such that log(e + kt) ≤ Ck ε log(e + t) for all k ≥ 2, t > 0. Consequently, for all k ≥ 1, (5.6) log e + ρ(x0 ) ρ(x0 ) ≤ C2kε log e + k+1 r 2 r k0 +1 . COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 22 Then, by Lemma 4.2 and Lemma 5.3, we get ρ(x0 ) log e + (b − bB )T a L1 r ρ(x0 ) = log e + (b − bB )T a L1 (2B) + r ρ(x0 ) + (b − bB )T a L1 (2k+1 B\2k B) log e + r k≥1 ρ(x0 ) 2r ≤ C log e + k0 +1 ρ(x0 ) 2k+1 r 2kε log e + +C k≥1 ≤ C|2B|1/2 b log BM OL b − bB a L2 k0 +1 L2 (2B) Ta b − bB L2 + L2 (2k+1 B) Ta 2kε (k + 1)|2k+1 B|1/2 b +C L2 (2k+1 B\2k B) −kδ k |2 B|−1/2 log 2 BM OL k≥1 ≤ C b log , BM OL where we used δ = 2ε. This ends the proof of (i). (ii). By Remark 2.3, (ii) can be seen as a consequence of Theorem 3.4 that we are going to prove now. Next, let us recall the following lemma due to Tang and Bi [44]. Lemma 5.5 (see [44], Lemma 3.1). Let V ∈ RHd/2 . Then, there exists c0 ∈ (0, 1) such that for any positive number N and 0 < h < |x − y|/16, we have C(N ) V (z) 1 1 |Kj (x, y)| ≤ dz + N d−1 d−1 |x − y| |x − z| |x − y| 1 + |x−y| B(x,|x−y|) ρ(y) and |Kj (x, y+h)−Kj (x, y)| ≤ C(N ) 1+ |x−y| ρ(y) N hc0 |x − y|c0 +d−1 V (z) 1 dz+ , d−1 |x − z| |x − y| B(x,|x−y|) where Kj (x, y), j = 1, ..., d, are the kernels of the Riesz transforms Rj . In order to prove Theorem 3.4, we need also the following two technical lemmas, which proofs will be given in Section 6. Lemma 5.6. Let 1 < q ≤ d/2 and c0 be as in Lemma 5.5. Then, Rj (a) is C times a classical (H 1 , q, c0 )-molecule (e.g. [40]) for all generalized (HL1 , q, c0 )-atom a related to the ball B = B(x0 , r). Furthermore, for any N > 0 and k ≥ 4, we have C(N ) (5.7) Rj (a) Lq (2k+1 B\2k B) ≤ 2−kc0 |2k B|1/q−1 , N 2k r 1 + ρ(x0 ) COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 23 where C(N ) > 0 depends only on N . Lemma 5.7. Let 1 < q ≤ d/2 and θ ≥ 0. Then, for every f ∈ BM O(Rd ), g ∈ BM OL,θ (Rd ) and (HL1 , q)-atom a related to the ball B = B(x0 , r), we have (g − gB )Rj (a) L1 ≤C g BM OL,θ (f − fB )(g − gB )Rj (a) L1 ≤C f BM O and g BM OL,θ . log log Proof of Theorem 3.4. Suppose that b ∈ BM OL,∞ (Rd ), i.e. b ∈ BM OL,θ (Rd ) for some θ ≥ 0. By Proposition 3.2 of [47], in order to prove that [b, Rj ] are bounded on HL1 (Rd ), it is sufficient to show that [b, Rj ](a) HL1 ≤ C b BM Olog for all (HL1 , d/2)L,θ atom a. Similarly to the proof of Theorem 3.3, it remains to show (b − bB )a (5.8) 1 HL ≤C b log BM OL,θ and (b − bB )Rj (a) (5.9) 1 HL ≤C b log BM OL,θ hold for every (HL1 , d/2)-atom a related to the ball B = B(x0 , r), where the constants C in (5.8) and (5.9) are independent of b, a. As before, we leave the proof of (5.8) to the interested reader. Let us now establish (5.9). Similarly to the proof of Theorem 3.3, Lemma 5.7 allows to reduce (5.9) to showing that (5.10) log e + ρ(x0 ) r (b − bB )Rj (a) L1 ≤C b log . BM OL,θ Setting ε = c0 /2, there is a constant C = C(ε) > 0 such that for all k ≥ 1, (5.11) log e + ρ(x0 ) ρ(x0 ) ≤ C2kε log e + k+1 r 2 r k0 +1 . Note that r ≤ CL ρ(x0 ) since a is a (HL1 , d/2)-atom related to the ball B(x0 , r). In (5.7) of Lemma 5.6, we choose N = (k0 + 1)θ. Then, H¨older inequality, (5.11) and COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 24 Lemma 5.3 allow to conclude that ρ(x0 ) r ρ(x0 ) = log e + r log e + + log e + k≥4 ≤ C log e + (b − bB )Rj (a) L1 (b − bB )Rj (a) L1 (24 B) ρ(x0 ) r ρ(x0 ) 24 r (b − bB )Rj (a) k0 +1 ρ(x0 ) 2k+1 r 2kε log e + +C k≥4 ≤ C b log BM OL,θ +C b b − bB + L1 (2k+1 B\2k B) Rj (a) d L d−2 (24 B) k0 +1 b − bB Ld/2 d L d−2 (2k+1 B) + Rj (a) Ld/2 (2k+1 B\2k B) k2−kε log BM OL,θ k≥4 ≤ C b log BM OL,θ where we used c0 = 2ε. This proves (5.10), and thus [b, Rj ] are bounded on HL1 (Rd ). Conversely, assume that [b, Rj ] are bounded on HL1 (Rd ). Then, although b belongs log to BM OL,∞ (Rd ) from a duality argument and Theorem 2 of [7], we would also like to give a direct proof for completeness. As b ∈ BM OL,∞ (Rd ) by assumption, there exist θ ≥ 0 such that b ∈ BM OL,θ (Rd ). For every (HL1 , d/2)-atom a related to some ball B = B(x0 , r). By Remark 2.1 and Lemma 5.7, Rj ((b − bB )a) L1 ≤ (b − bB )Rj (a) ≤C b BM OL,θ L1 + C [b, Rj ](a) + C [b, Rj ] 1 HL 1 →H 1 HL L hold for all j = 1, ..., d. In addition, noting that r ≤ CL ρ(x0 ) since a is a (HL1 , d/2)atom related to some ball B = B(x0 , r), H¨older inequality and Lemma 1 of [7] (see also Lemma 6.6 below) give (b − bB )a L1 ≤ b − bB d L d−2 (B) a Ld/2 (B) ≤C b BM OL,θ . By the characterization of HL1 (Rd ) in terms of the Riesz transforms (see [16]), the above proves that (b − bB )a ∈ HL1 (Rd ), moreover, d (5.12) (b − bB )a 1 HL ≤C b BM OL,θ + [b, Rj ] j=1 where the constant C > 0 is independent of b, a. 1 →H 1 HL L COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 25 log Now, we prove that b ∈ BM OL,θ (Rd ). More precisely, the following log e + (5.13) 1+ ρ(x0 ) r r ρ(x0 ) θ d M O(b, B(x0 , r)) ≤ C b BM OL,θ + [b, Rj ] 1 →H 1 HL L j=1 holds for any ball B(x0 , r) in Rd . In fact, we only need to establish (5.13) for 0 < r < ρ(x0 )/2 since b ∈ BM OL,θ (Rd ). Indeed, in (5.12) we choose B = B(x0 , r) and a = (2|B|)−1 (f − fB )χB , where f = sign (b − bB ). Then, it is easy to see that a is a (HL1 , d/2)-atom related to the ball B. We next consider gx0 ,r (x) = χ[0,r] (|x − x0 |) log ρ(x0 ) ρ(x0 ) + χ(r,ρ(x0 )] (|x − x0 |) log . r |x − x0 | Then, thanks to Lemma 2.5 of [33], one has gx0 ,r BM OL ≤ C. Moreover, it is clear that gx0 ,r (b − bB )a ∈ L1 (Rd ). Consequently, (5.12) together with the fact that BM OL (Rd ) is the dual of HL1 (Rd ) allows us to conclude that log e + 1+ ρ(x0 ) r r ρ(x0 ) θ ρ(x0 ) M O(b, B(x0 , r)) r M O(b, B(x0 , r)) ≤ 3 log gx0 ,r (x)(b(x) − bB )a(x)dx = 6 Rd ≤ 6 gx0 ,r BM OL (b − bB )a 1 HL d ≤ C b BM OL,θ + [b, Rj ] 1 →H 1 HL L , j=1 where we used r < ρ(x0 )/2 and (b(x) − bB )a(x)dx = Rd 1 2|B(x0 , r)| |b(x) − bB(x0 ,r) |dx. B(x0 ,r) This ends the proof. 6. Proof of the key lemmas First, let us recall some notations and results due to Dziuba´ nski and Zienkiewicz in [16]. These notations and results play an important role in our proofs. COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 26 2 Let P (x) = (4π)−d/2 e−|x| /4 be the Gauss function. For n ∈ Z, the space h1n (Rd ) denotes the space of all integrable functions f such that Mn f (x) = sup |P√t ∗ f (x)| = 0 0 such that (b − bB )T a (6.2) L1 ≤C b BM O for all b ∈ BM O(Rd ) and generalized (HL1 , q, ε)-atom a related to the ball B. 1,q,ε d d d 1 From H1,q,ε L,fin (R ) is dense in HL,at (R ) = HL (R ) (see Theorem 2.2), we need only prove that U(f, b) L1 = [b, T ](f − H(f )) L1 ≤C f 1 HL b BM O d d holds for every (f, b) ∈ H1,q,ε L,fin (R ) × BM O(R ). For any (n, k) ∈ Z × Z+ . As xn,k ∈ Bn and ψn,k f ∈ h1n (Rd ), it follows from Lemma 6.5 and Remark 2.1 that there are generalized (HL1 , q, ε)-atoms an,k related to the j n,k n,k n,k n,k 2−n/2 balls B(xj , rj ) such that B(xj , rj ) ⊂ B(xn,k , 2 ) and (6.3) n,k λn,k j aj , ψn,k f = j |λn,k j | ≤ C ψn,k f j with a positive constant C independent of n, k and f . h1n COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 28 Clearly, supp ϕ2−n/2 ∗ an,k ⊂ B(xn,k , 5.2−n/2 ) since supp ϕ ⊂ B(0, 1) and supp j an,k ⊂ B(xn,k , 22−n/2 ); the following estimate holds j ϕ2−n/2 ∗an,k j Lq ≤ ϕ2−n/2 Lq an,k j L1 ≤ (2−n/2 )d(1/q−1) ϕ Lq ≤ C|B(xn,k , 5.2−n/2 )|1/q−1 . Moreover, as xn,k ∈ Bn , ϕ2−n/2 ∗ an,k j dx ≤ ϕ2−n/2 L1 an,k j L1 ≤C 5.2−n/2 ρ(xn,k ) ε . Rd These prove that ϕ2−n/2 ∗ an,k is C times a generalized (HL1 , q, ε)-atom related to j B(xn,k , 5.2−n/2 ). Consequently, (6.2) yields (6.4) (b − bB(xn,k ,5.2−n/2 ) )T (ϕ2−n/2 ∗ an,k j ) L1 ≤C b BM O . By an analogous argument, it is easy to check that (ϕ2−n/2 ∗an,k j )(b−bB(xn,k ,5.2−n/2 ) ) 1 q+1 is C b BM O times a generalized (HL , 2 , ε)-atom related to B(xn,k , 5.2−n/2 ). Hence, it follows from (6.3) and (6.4) that [b, T ](ϕ2−n/2 ∗ (ψn,k f )) L1 ≤ (b − bB(xn,k ,5.2−n/2 ) )T (ϕ2−n/2 ∗ (ψn,k f )) L1 + T (b − bB(xn,k ,5.2−n/2 ) )(ϕ2−n/2 ∗ (ψn,k f )) ≤ C ψn,k f (6.5) h1n b L1 BM O , where we used the fact that T is bounded from HL1 (Rd ) into L1 (Rd ) since T ∈ KL . d On the other hand, by f ∈ H1,q,ε L,fin (R ), there exists a ball B(0, R) such that supp f ⊂ B(0, R). As B(0, R) is a compact set, Lemma 6.1 allows to conclude that there is a finite set ΓR ⊂ Z × Z+ such that for every (n, k) ∈ / ΓR , B(xn,k , 21−n/2 ) ∩ B(0, R) = ∅. It follows that there are N, K ∈ Z+ such that N f= K ψn,k f = ψn,k f. n=−N k=1 n,k Therefore, (6.5) and Lemma 6.3 yield N U(f, b) L1 K [b, T ](ϕ2−n/2 ∗ (ψn,k f )) ≤ n=−N k=1 ≤ C b L1 ψn,k f BM O n,k which ends the proof. h1n ≤C f 1 HL b BM O , COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 29 Proof of Lemma 5.3. First, we claim that for every ball B0 = B(x0 , r0 ), 1 |B0 | (6.6) |f (y) − fB0 |q dy ≤C B0 (k0 +1)θ r0 ρ(x0 ) 1+ 1/q f 0 ) k0 +1 ) log e + ( ρ(x r0 log . BM OL,θ Assume that (6.6) holds for a moment. Then, 1 k |2 B| |f (y) − fB |q dy 1/q 2k B 1 k |2 B| ≤ |f (y) − f2k B |q dy k−1 1/q |f2j+1 B − f2j B | + j=0 2k B 1+ ≤ 2k r ρ(x) log e + 1+ k−1 ( ρ(x) )k0 +1 2k r 1+ ≤ Ck (k0 +1)θ 2k r ρ(x) f log BM OL,θ 2d + j=0 2j+1 r ρ(x) log e + θ ρ(x) 2j+1 r f log BM OL,θ (k0 +1)θ f )k0 +1 log e + ( ρ(x) 2k r log . BM OL,θ Now, it remains to prove (6.6). Let us define the function h on Rd as follows   x ∈ B0 , 1, 2r0 −|x−x0 | h(x) = , x ∈ 2B0 \ B0 , r  0, 0 x∈ / 2B0 , and remark that |h(x) − h(y)| ≤ (6.7) |x − y| . r0 Setting f := f − f2B0 . By the classical John-Nirenberg inequality, there exists a constant C = C(d, q) > 0 such that 1 |B0 | |f (y) − fB0 |q dy 1/q = 1 |B0 | B0 |h(y)f (y) − (hf )B0 |q dy B0 ≤ C hf BM O . Therefore, the proof of the lemma is reduced to showing that 1+ hf BM O ≤C r0 ρ(x0 ) (k0 +1)θ 0 ) k0 +1 log e + ( ρ(x ) r0 f log , BM OL,θ 1/q COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 30 namely, for every ball B = B(x, r), 1+ 1 |B| (6.8) |h(y)f (y) − (hf )B |dy ≤ C r0 ρ(x0 ) f 0 ) k0 +1 ) ( ρ(x r0 log e + B (k0 +1)θ log . BM OL,θ Now, let us focus on Inequality (6.8). Noting that supp h ⊂ 2B0 , Inequality (6.8) is obvious if B ∩ 2B0 = ∅. Hence, we only consider the case B ∩ 2B0 = ∅. Then, we have the following two cases: The case r > r0 : the fact B ∩ 2B0 = ∅ implies that 2B0 ⊂ 5B, and thus 1 |B| |h(y)f (y) − (hf )B |dy ≤ 2 1 |B| B |h(y)f (y)|dy B ≤ 2.5d 1 |2B0 | |f (y) − f2B0 |dy 2B0 1+ ≤ C 2r0 ρ(x0 ) log e + 1+ ≤ C θ ρ(x0 ) 2r0 r0 ρ(x0 ) f log BM OL,θ (k0 +1)θ f 0 ) k0 +1 log e + ( ρ(x ) r0 log . BM OL,θ The case r ≤ r0 : Inequality (6.7) yields 1 |B| |h(y)f (y) − (hf )B |dy ≤ 2 1 |B| B |h(y)f (y) − hB fB |dy B 1 ≤2 |B| |h(y)(f (y) − fB )|dy+ B + 2|fB | 1 |B| 1 |B| B (6.9) ≤2 1 |B| (h(x) − h(y))dy dx B |f (y) − fB |dy + 4 r |fB − f2B0 |. r0 B By r ≤ r0 , B = B(x, r) ∩ B(x0 , r0 ) = ∅, Proposition 2.1 gives r r0 r0 |x − x0 | 1+ ≤ ≤κ ρ(x) ρ(x) ρ(x0 ) ρ(x0 ) k0 ≤C 1+ r0 ρ(x0 ) k0 +1 . COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 31 Consequently, 1+ 1 |B| |f (y) − fB |dy ≤ B r ρ(x) ≤C f ρ(x) ) r log(e + log BM OL,θ (k0 +1)θ r0 ρ(x0 ) 1+ (6.10) θ f 0 ) k0 +1 log e + ( ρ(x ) r0 log , BM OL,θ and 1+ 1 |B(x, 23 r0 )| |f (y) − fB(x,23 r0 ) |dy ≤ B(x,23 r0 ) 23 r0 ρ(x) log(e + ≤C f ρ(x) ) 23 r0 1+ (6.11) θ log BM OL,θ (k0 +1)θ r0 ρ(x0 ) 0 ) k0 +1 ) log e + ( ρ(x r0 f log . BM OL,θ Noting that for every k ∈ N with 2k+1 r ≤ 23 r0 , |f2k+1 B − f2k B | ≤ 2d 1 1+ ≤ C 2k+1 B θ 23 r0 ρ(x) log(e + 1+ ≤ C |f (y) − f2k+1 B |dy |2k+1 B| ρ(x) ) 23 r0 r0 ρ(x0 ) f log BM OL,θ (k0 +1)θ 0 ) k0 +1 log e + ( ρ(x ) r0 f log , BM OL,θ allows us to conclude that (k0 +1)θ (6.12) r0 1 + ρ(x r0 0) |fB(x,r) − fB(x,23 r0 ) | ≤ C log e + r log e + ( ρ(x0 ) )k0 +1 r0 f log . BM OL,θ COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 32 Then, the inclusion 2B0 ⊂ B(x, 23 r0 ) together with the inequalities (6.9), (6.10), (6.11) and (6.12) yield 1 1 |h(y)f (y) − (hf )B |dy ≤ 2 |f (y) − fB |dy + |B| |B| B B r +4 |fB(x,r) − fB(x,23 r0 ) | + 4d M O(f, B(x, 23 r0 )) r0 1+ ≤ C 1+ r r0 log(e + ) r0 r log e + ( ρ(x0 ) )k0 +1 r0 1+ ≤ C we have used r r0 log(e + r0 ) r (k0 +1)θ r0 ρ(x0 ) r0 ρ(x0 ) f log BM OL,θ (k0 +1)θ 0 ) k0 +1 ) log e + ( ρ(x r0 f log , BM OL,θ ≤ supt≤1 t log(e + 1/t) < ∞. This ends the proof. By an analogous argument, we can also obtain the following, which was proved by Bongioanni et al (see Lemma 1 of [7]) through another method. Lemma 6.6. Let 1 ≤ q < ∞ and θ ≥ 0. Then, for every f ∈ BM OL,θ (Rd ), B = B(x, r) and k ∈ Z+ , we have 1 k |2 B| |f (y) − fB |q dy 1/q ≤ Ck 1 + 2k r ρ(x) (k0 +1)θ f BM OL,θ . 2k B Proof of Lemma 5.4. i) Assume that T is a (δ, L)-Calder´on-Zygmund operator for some δ ∈ (0, 1]. For every generalized (HL1 , 2, δ)-atom a related to the ball B, as T ∗ 1 = 0, Lemma 4.2 implies that T a is C times a classical (H 1 , 2, δ)-molecule (see for example [40]) related to B, and thus T a H 1 ≤ C. Therefore, Proposition 2.2 yields T maps continuously HL1 (Rd ) into H 1 (Rd ). ii) By Lemma 4.1, Lemma 4.2 and H¨older inequality, we get = (f − fB )(g − gB )T a L1 (f − fB )(g − gB )T a L1 (2B) (f − fB )(g − gB )T a + L1 (2k+1 B\2k B) k≥1 ≤ f − fB L2q (2B) f − fB + g − gB L2q (2k+1 B) L2q (2B) g − gB T (a) Lq + L2q (2k+1 B) T (a) Lq (2k+1 B\2k B) k≥1 ≤ C f BM O g BM O C(k + 1)2 f + k≥1 ≤ C f BM O g BM O , BM O g k+1 B|1/q BM O |2 2−kδ |2k B|1/q−1 COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 33 where 1/q + 1/q = 1. Proof of Lemma 5.6. It is well-known that the Riesz transforms Rj are bounded from HL1 (Rd ) into H 1 (Rd ), in particular, one has Rd Rj (a)(x)dx = 0. Moreover, by the Lq -boundedness of Rj (see [39], Theorem 0.5) one has Rj (a) Lq ≤ C|B|1/q−1 . Therefore, it is sufficient to verify (5.7). Thanks to Lemma 5.5, as a is a generalized (HL1 , q, c0 )-atom related to the ball B, for every x ∈ 2k+1 B \ 2k B, |Rj (a)(x)| ≤ (Kj (x, y) − Kj (x, x0 ))a(y)dy + |Kj (x, x0 )| B B c0 |y − x0 | |x − x0 |d+c0 −1 C(N ) ≤ 1+ B |x−x0 | ρ(x0 ) N +4N0 V (z) 1 |a(y)|dy dz + d−1 |x − z| |x − x0 | B(x,|x−x0 |) V (z) 1 dz + d−1 |x − z| |x − x0 | C(N ) + 1+ |x−x0 | ρ(x0 ) a(y)dy 1 N +4N0 +c0 |x − x0 |d−1 r ρ(x0 ) c0 B(x,|x−x0 |) (6.13)  C(N ) ≤ 1+ 2k r ρ(x0 ) N    2k+2 r ρ(x0 ) 1+ V (z) 2−kc0  dz + . |x − z|d−1 |2k B| r c0 N0 (2k r)d+c0 −1 1 B(x,|x−x0 |) Here and in what follows, the constants C(N ) depend only on N , but may change from line to line. Note that for every x ∈ 2k+1 B \ 2k B, one has B(x, |x − x0 |) ⊂ B(x, 2k+1 r) ⊂ B(x0 , 2k+2 r). The fact V ∈ RHd/2 , d/2 ≥ q > 1, and H¨older inequality yield V (z) dz |x − z|d−1 B(x,|x−x0 |) Lq (2k+1 B\2k B,dx) 2 ≤ C(2k+1 r)1− d      k+1 2 |V (z)|d/2 dz |x − z|d−1 B\2k B 1 2 2 ≤ C(2k r)1− d |2k+1 B| q − d B(x,2k+1 r)    dx   B(z,2k+1 r) (6.14) ≤ C2k r|2k B|1/q−1 B(x0 ,2k+2 r) V (z)dz. B(x0 ,2k+2 r) 2q d dx 1/q     2/d   |V (z)|d/2 dz |x − z|d−1   COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 34 Combining (6.13), (6.14) and Lemma 1 of [21], we obtain that Rj (a) Lq (2k+1 B\2k B)   C(N ) ≤ N 2k r ρ(x0 ) 1+ C(N ) ≤ N 2k r 1+  rc0 2k r|2k B|1/q−1  (2k r)d+c0 −1 1 1+ V (z)dz + N0 2k+2 r ρ(x0 ) 2−kc0 k+1 1/q  |2 B|  |2k B| B(x0 ,2k+2 r) 2−kc0 |2k B|1/q−1 , ρ(x0 ) where N0 = log2 C0 + 1 with C0 the constant in (2.1). This completes the proof. Proof of Lemma 5.7. Note that r ≤ CL ρ(x0 ) since a is a (HL1 , q)-atom related to the ball B = B(x0 , r); and a is CL c0 times a generalized (HL1 , q, c0 )-atom related to the ball B = B(x0 , r) (see Remark 2.1). In (5.7), we choose N = (k0 + 1)θ. Then, H¨older inequality and Lemma 6.6 give (g − gB )Rj (a) L1 ∞ = (g − gB )Rj (a) L1 (24 B) (g − gB )Rj (a) + L1 (2k+1 B\2k B) k=4 ∞ ≤ g − gB Lq (24 B) Rj Lq →Lq a Lq g − gB + Lq (2k+1 B\2k B) Rj (a) Lq (2k+1 B\2k B) k=4 ≤ C g BM OL,θ ∞ + (k + 1)|2k+1 B|1/q 1 + +C k=4 ≤ C g 2k+1 r ρ(x) 1 (k0 +1)θ g BM OL,θ 1+ (k0 +1)θ 2k r 2−kc0 |2k B|1/q−1 ρ(x) BM OL,θ , where 1/q + 1/q = 1. Similarly, we also obtain that (f − fB )(g − gB )Rj (a) L1 ∞ = (f − fB )(g − gB )Rj (a) ≤ f − fB L1 (24 B) (f − fB )(g − gB )Rj (a) + L1 (2k+1 B\2k B) k=4 L2q (24 B) g − gB L2q (24 B) Rj (a) Lq + ∞ f − fB + L2q (2k+1 B) k=4 ≤ C f BM O g BM OL,θ , which ends the proof. g − gB L2q (2k+1 B) Rj (a) Lq (2k+1 B\2k B) COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 35 7. Some applications The purpose of this section is to give some applications of the decomposition theorems (Theorem 3.1 and Theorem 3.2). To be more precise, we give some subspaces of HL1 (Rd ), which do not necessarily depend on b and T , such that all commutators [b, T ], for b ∈ BM O(Rd ) and T ∈ KL , map continuously these spaces into L1 (Rd ). Especially, using Theorem 3.1 and Theorem 3.2, we find the largest subspace 1 HL,b (Rd ) of HL1 (Rd ) so that all commutators of Schr¨odinger-Calder´on-Zygmund op1 (Rd ) into L1 (Rd ). Also, it erators and the Riesz transforms are bounded from HL,b d 1 allows to find all functions b in BM O(R ) so that HL,b (Rd ) ≡ HL1 (Rd ). 7.1. Atomic Hardy spaces related to b ∈ BM O(Rd ). Definition 7.1. Let 1 < q ≤ ∞, ε > 0 and b ∈ BM O(Rd ). A function a is called a 1 , q, ε)-atom related to the ball B = B(x0 , r) if a is a generalized (HL1 , q, ε)-atom (HL,b related to the same ball B and ε r . (7.1) a(x)(b(x) − bB )dx ≤ ρ(x0 ) Rd 1,q,ε d 1 As usual, the space HL,b (Rd ) is defined as H1,q,ε L,at (R ) with generalized (HL , q, ε)1 , q, ε)-atoms. atoms replaced by (HL,b 1,q,ε d d d 1 Obviously, HL,b (R ) ⊂ H1,q,ε L,at (R ) ≡ HL (R ) and the inclusion is continuous. Theorem 7.1. Let 1 < q ≤ ∞, ε > 0, b ∈ BM O(Rd ) and T ∈ KL . Then, the 1,q,ε commutator [b, T ] is bounded from HL,b (Rd ) into L1 (Rd ). Remark 7.1. The space Hb1 (Rd ) which has been considered by Tang and Bi [44] is 1,q,ε a strict subspace of HL,b (Rd ) in general. As an example, let us take 1 < q ≤ ∞, ε > 0, L = −∆ + 1, and b be a non-constant bounded function, then it is easy to 1,q,ε check that the function f = χB(0,1) belongs to HL,b (Rd ) but not to Hb1 (Rd ). Thus, Theorem 7.1 can be seen as an improvement of the main result of [44]. We should also point out that the authors in [44] proved their main result (see [44], Theorem 3.1) by establishing that [b, Rj ](a) L1 ≤C b BM O for all Hb1 -atom a. However, as pointed in [8] and [28], such arguments are not sufficient to conclude that [b, Rj ] is bounded from Hb1 (Rd ) into L1 (Rd ) in general. 1 Proof of Theorem 7.1. Let a be a (HL,b , q, ε)-atom related to the ball B = B(x0 , r). We first prove that (b−bB )a is C b BM O times a generalized (HL1 , (q +1)/2, ε)-atom, where q ∈ (1, ∞) will be defined later and the positive constant C is independent of b, a. Indeed, one has supp (b − bB )a ⊂ supp a ⊂ B. In addition, from H¨older inequality and John-Nirenberg (classical) inequality, (b − bB )a L(q+1)/2 ≤ (b − bB )χB Lq(q+1)/(q−1) a Lq ≤C b (−q+1)/(q+1) , BM O |B| COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 36 where q = q if 1 < q < ∞ and q = 2 if q = ∞. These together with (7.1) yield that (b − bB )a is C b BM O times a generalized (HL1 , (q + 1)/2, ε)-atom, and thus (b − bB )a HL1 ≤ C b BM O . We now prove that S(a, b) belongs to HL1 (Rd ). By Theorem 3.2, there exist d bounded bilinear operators Rj : HL1 (Rd )×BM O(Rd ) → L1 (Rd ), j = 1, ..., d, such that [b, Rj ](a) = Rj (a, b) + Rj (S(a, b)), since Rj is linear and belongs to KL (see Proposition 4.2). Consequently, for every j = 1, ..., d, as Rj ∈ KL , Rj (S(a, b)) L1 (b − bB )Rj (a) − Rj ((b − bB )a) − Rj (a, b) (b − bB )Rj (a) L1 + Rj HL1 →L1 (b − bB )a = ≤ ≤ C b L1 1 HL + Rj (a, b) L1 BM O . This together with Proposition 5.2 prove that S(a, b) ∈ HL1 (Rd ), and moreover that (7.2) S(a, b) ≤C b 1 HL BM O . 1,q,ε Now, for any f ∈ HL,b (Rd ), there exists an expansion f = ∞ k=1 λk ak where the ∞ 1 1,q,ε ak are (HL,b , q, ε)-atoms and k=1 |λk | ≤ 2 f H . Then, the sequence { nk=1 λk ak }n≥1 L,b 1,q,ε converges to f in HL,b (Rd ) and thus in HL1 (Rd ). Hence, Proposition 5.2 implies that the sequence S n k=1 λk ak , b λk ak , b ≤ converges to S(f, b) in L1 (Rd ). In addition, n≥1 by (7.2), n S n k=1 1 HL |λk | S(ak , b) 1 HL ≤C f 1,q,ε HL,b b BM O . k=1 We then use Theorem 3.1 and the weak-star convergence in HL1 (Rd ) (see [29]) to conclude that [b, T ](f ) L1 ≤ RT (f, b) ≤ C f ≤ C f 1 HL b 1,q,ε HL,b L1 + T BM O b 1 →L1 HL +C f S(f, b) 1,q,ε HL,b b 1 HL BM O BM O , which ends the proof. 1 7.2. The spaces HL,b (Rd ) related to b ∈ BM O(Rd ). In this section, we find 1 the largest subspace HL,b (Rd ) of HL1 (Rd ) so that all commutators of Schr¨odinger1 Calder´on-Zygmund operators and the Riesz transforms are bounded from HL,b (Rd ) 1 into L1 (Rd ). Also, we find all functions b in BM O(Rd ) so that HL,b (Rd ) ≡ HL1 (Rd ). COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 37 1 Definition 7.2. Let b be a non-constant BM O-function. The space HL,b (Rd ) consists of all f in HL1 (Rd ) such that [b, ML ](f )(x) = ML (b(x)f (·)−b(·)f (·))(x) belongs 1 to L1 (Rd ). We equipped HL,b (Rd ) with the norm f 1 HL,b = f 1 HL b BM O + [b, ML ](f ) L1 . Here, we just consider non-constant functions b in BM O(Rd ) since [b, T ] = 0 if b is a constant function. Theorem 7.2. Let b be a non-constant BM O-function. Then, the following statements hold: 1 (Rd ) into L1 (Rd ). i) For every T ∈ KL , the commutator [b, T ] is bounded from HL,b ii) Assume that X is a subspace of HL1 (Rd ) such that all commutators of the Riesz 1 transforms are bounded from X into L1 (Rd ). Then, X ⊂ HL,b (Rd ). 1 (Rd ) ≡ HL1 (Rd ) if and only if b ∈ BM OLlog (Rd ). iii) HL,b To prove Theorem 7.2, we need the following lemma. Lemma 7.1. Let b be a non-constant BM O-function and f ∈ HL1 (Rd ). Then, the following conditions are equivalent: 1 i) f ∈ HL,b (Rd ). ii) S(f, b) ∈ HL1 (Rd ). iii) [b, Rj ](f ) ∈ L1 (Rd ) for all j = 1, ..., d. Furthermore, if one of these conditions is satisfied, then f 1 HL,b = f ≈ f ≈ f 1 HL 1 HL b BM O + [b, ML ](f ) b BM O + S(f, b) b BM O + L1 1 HL d 1 HL [b, Rj ](f ) L1 , j=1 where the constants are independent of b and f . Proof. (i) ⇔ (ii). As ML ∈ KL (see Proposition 4.3), by Theorem 3.1, there is a bounded subbilinear operator R : HL1 (Rd ) × BM O(Rd ) → L1 (Rd ) such that ML (S(f, b)) − R(f, b) ≤ |[b, ML ](f )| ≤ ML (S(f, b)) + R(f, b). Consequently, [b, ML ](f ) ∈ L1 (Rd ) iff S(f, b) ∈ HL1 (Rd ), moreover, f 1 HL,b ≈ f 1 HL b BM O + S(f, b) 1. HL (ii) ⇔ (iii). As the Riesz transforms Rj are in KL (see Proposition 4.2), by Theorem 3.2, there are d bounded subbilinear operator Rj : HL1 (Rd ) × BM O(Rd ) → L1 (Rd ), j = 1, ..., d, such that [b, Rj ](f ) = Rj (f, b) + Rj (S(f, b)). COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 38 Therefore, S(f, b) ∈ HL1 (Rd ) iff [b, Rj ](f ) ∈ L1 (Rd ) for all j = 1, ..., d, moreover, d f 1 HL b BM O + S(f, b) 1 HL ≈ f 1 HL b BM O + [b, Rj ](f ) L1 . j=1 Proof of Theorem 7.2. By Theorem 3.1, there is a bounded subbilinear operator RT : HL1 (Rd ) × BM O(Rd ) → L1 (Rd ) such that |T (S(f, b))| − RT (f, b) ≤ |[b, T ](f )| ≤ |T (S(f, b))| + RT (f, b). 1 Applying Lemma 7.1 gives for every f ∈ HL,b (Rd ), [b, T ](f ) L1 ≤ T 1 →L1 HL ≤ C f 1 HL,b S(f, b) +C f 1 HL 1 HL b + RT (f, b) BM O ≤C f L1 1 . HL,b 1 (Rd ) into L1 (Rd ). This ends the proof of (i). Therefore, [b, T ] is bounded from HL,b The proof of (ii) follows directly from Lemma 7.1. The proof of (iii) follows directly from Theorem 3.4 and Lemma 7.1. log 7.3. Atomic Hardy spaces HL,α (Rd ). log Definition 7.3. Let α ∈ R. We say that the function a is a HL,α -atom related to the ball B = B(x0 , r) if i) supp a ⊂ B, ii) a iii) L2 Rd ≤ log(e + ρ(x0 ) ) r α |B|−1/2 , a(x)dx = 0. log 1 As usual, the space HL,α (Rd ) is defined as H1,q,ε L,at with generalized (HL , q, ε)-atoms log replaced by HL,α -atoms. log log log Clearly, HL,0 (Rd ) is just H 1 (Rd ) ⊂ HL1 (Rd ). Moreover, HL,α (Rd ) ⊂ HL,α (Rd ) for all α ≤ α . It should be pointed out that when L = −∆ + 1 and α ≥ 0, then log HL,α (Rd ) is just the space of all distributions f such that Rd Mf (x) λ α dx log(e + Mfλ(x) ) 0, moreover (see [27] for the details),  Mf (x)  λ f H log ≈ inf λ > 0 : L,α  log(e + Mfλ(x) ) d   . α dx ≤ 1  R Theorem 7.3. For every T ∈ KL and b ∈ BM O(Rd ), the commutator [b, T ] is log bounded from HL,−1 (Rd ) into L1 (Rd ). COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 39 log Proof. Let a be a HL,−1 -atom related to the ball B = B(x0 , r). Let us first prove 1 d that (b − bB )a ∈ HL (R ). As HL1 (Rd ) is the dual of V M OL (Rd ) (see Theorem 5.1), it is sufficient to show that for every g ∈ Cc∞ (Rd ), (b − bB )ag L1 ≤C b BM O g BM OL . Indeed, using the estimate |gB | ≤ C log e + ρ(xr 0 ) g BM OL (see Lemma 2 of [15]), H¨older inequality and classical John-Nirenberg inequality give (b − bB )ag L1 ≤ ≤ (g − gB )(b − bB )a L1 + |gB | (b − bB )a L1 (g − gB )χB L4 (b − bB )χB L4 a L2 + ρ(x0 ) +C log e + g BM OL (b − bB )χB L2 a r ≤ C b BM O g BM OL , which proves that (b − bB )a ∈ HL1 (Rd ), moreover, (b − bB )a Similarly to the proof of Theorem 7.1, we also obtain that S(f, b) 1 HL ≤C f log HL,−1 b 1 HL ≤C b L2 BM O . BM O log for all f ∈ HL,−1 (Rd ). Therefore, Theorem 3.1 allows to conclude that [b, T ](f ) L1 ≤C f log HL,−1 b BM O , which ends the proof. As a consequence of the proof of Theorem 7.3, we obtain the following result. Proposition 7.1. Let T ∈ KL . Then, T(f, b) := [b, T ](f ) is a bounded subbilinear log operator from HL,−1 (Rd ) × BM O(Rd ) into L1 (Rd ). 7.4. The Hardy-Sobolev space HL1,1 (Rd ). Following Hofmann et al. [23], we say that f belongs to the (inhomogeneous) Hardy-Sobolev HL1,1 (Rd ) if f, ∂x1 f, ..., ∂xd f ∈ HL1 (Rd ). Then, the norm on HL1,1 (Rd ) is defined by d f 1,1 HL = f 1 HL + ∂ xj f 1. HL j=1 1,1 It should be pointed out that the authors in [23] proved that the space H−∆ (Rd ) is just the classical (inhomogeneous) Hardy-Sobolev H 1,1 (Rd ) (see for example [1]), and can be identified with the (inhomogeneous) Triebel-Lizorkin space F11,2 (Rd ) (see [26]). More precisely, f belongs to H 1,1 (Rd ) if and only if 1/2 Wψ (f ) = | I f, ψIσ 2 −1/d 2 | (1 + |I| ) |I| χI σ∈E moreover, (7.3) f H 1,1 ≈ Wψ (f ) −1 L1 . ∈ L1 (Rd ), COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 40 Here {ψ σ }σ∈E is the wavelet as in Section 4. Theorem 7.4. Let L = −∆ + 1. Then, for every T ∈ KL and b ∈ BM O(Rd ), the commutator [b, T ] is bounded from HL1,1 (Rd ) into L1 (Rd ). Remark 7.2. When L = −∆ + 1, we can define H(f ) = f − ϕ ∗ f instead of H(f ) = n,k (ψn,k f − ϕ2−n/2 ∗ (ψn,k f )) as in Section 5. In other words, the bilinear operator S in Theorem 3.1 and Theorem 3.2 can be defined as S(f, g) = −Π(f − ϕ ∗ f, g). As H(f ) = f − ϕ ∗ f , it is easy to see that ∂xj (H(f )) = H(∂xj f ). Here and in what follows, for any dyadic cube Q = Q(y, r) := {x ∈ Rd : −r ≤ xj − yj < r for all j = 1, ..., d}, we denote by BQ the ball √ BQ := x ∈ Rd : |x − y| < 2 dr . To prove Theorem 7.4, we need the following lemma. Lemma 7.2. Let L = −∆ + 1. Then, the bilinear operator Π maps continuously H 1,1 (Rd ) × BM O(Rd ) into HL1 (Rd ). Proof. Note that ρ(x) = 1 for all x ∈ Rd since V (x) ≡ 1. We first claim that there exists a constant C > 0 such that (1 + |I|−1/d )−1 (ψIσ )2 (7.4) 1 HL ≤C for all dyadic I = Q[x0 , r) and σ ∈ E. Indeed, it follows from Remark 5.2 that supp (1 + |I|−1/d )−1 (ψIσ )2 ⊂ cI ⊂ cBI , and it is clear that (1 + |I|−1/d )−1 (ψIσ )2 L∞ ≤ |I|−1 ψ L∞ ≤ C|cBI |−1 . In addition, r (1 + |I|−1/d )−1 (ψIσ (x))2 dx = (1 + |I|−1/d )−1 ≤ C . ρ(x0 ) Rd Hence, (1 + |I|−1/d )−1 (ψIσ )2 is C times a generalized (HL1 , ∞, 1)-atom related to the ball cBI , and thus (7.4) holds. Now, for every (f, g) ∈ H 1,1 (Rd ) × BM O(Rd ), (7.4) implies that Π(f, g) 1 HL f, ψIσ g, ψIσ (ψIσ )2 = I 1 HL σ∈E | f, ψIσ |(1 + |I|−1/d ) | g, ψIσ | ≤ C I σ∈E ≤ C Wψ (f ) ≤ C f H 1,1 L1 g g 0,2 F˙∞ BM O , 0,2 where we have used the fact that BM O(Rd ) ≡ F˙ ∞ (Rd ) is the dual of H 1 (Rd ) ≡ 0,2 F˙ 1 (Rd ), we refer the reader to [18] for more details. COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 41 Proof of Theorem 7.4. Let (f, b) ∈ HL1,1 (Rd ) × BM O(Rd ). Thanks to Lemma 7.2, Remark 7.2 and Lemma 5.1, we get S(f, b) ≤ C H(f ) 1 HL ≤ C f H 1,1 1,1 HL b b BM O BM O . Then we use Theorem 3.1 to conclude that [b, T ](f ) L1 ≤ RT (f, b) ≤ C f 1,1 HL L1 b + T 1 →L1 HL S(f, b) 1 HL BM O , which ends the proof. As a consequence of the proof of Theorem 7.4, we obtain the following result. 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Department of Mathematics, University of Quy Nhon 170 An Duong Vuong, Quy Nhon, Binh Dinh, Viet Nam Email: dangky@math.cnrs.fr [...]... boundedness of Hankel operators on the Hardy spaces H 1 (Td ) and H 1 (Bd ) (where Bd is the unit ball in Cd and Td = ∂Bd ), characterizations of pointwise multipliers for BM O type spaces, endpoint estimates for commutators of singular integrals operators and their applications to PDEs, see for example [5, 9, 24, 25, 28, 36, 41, 42] 4 Some fundamental operators and the class KL The purpose of this section... operator ML is in the class KL Proposition 4.4 The ”Poisson” maximal operator MPL is in the class KL Here we just give the proof of Proposition 4.3 For the one of Proposition 4.4, we leave the details to the interested reader Proof of Proposition 4.3 Obviously, ML is bounded from HL1 (Rd ) into L1 (Rd ) Now, let us prove that (f − fB )ML (a) L1 ≤C f BM O COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 16 for. .. into L1,∞ (Rd ) 3 Statement of the results Recall that KL is the set of all sublinear operators T bounded from HL1 (Rd ) into L (Rd ) and that there are q ∈ (1, ∞] and ε > 0 such that 1 (b − bB )T a d L1 ≤C b BM O (HL1 , q, ε)-atom for all b ∈ BM O(R ), any generalized C > 0 is a constant independent of b, a a related to the ball B, where COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 12 3.1 Two decomposition... and [b, T ] is bounded on HL1 (Rd ) for every L-Calder´on-Zygmund operator T satisfying T ∗ 1 = 0, then b ∈ BM OLlog (Rd ) Furthermore, d b log BM OL ≈ b BM O + [b, Rj ] 1 →H 1 HL L j=1 Next result concerns the HL1 -estimates for commutators of the Riesz transforms COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 13 Theorem 3.4 Let b ∈ BM OL,∞ (Rd ) Then, the commutators [b, Rj ], j = 1, , d, log are bounded... where we used the boundedness of V on H 1 (Rd ) × BM O(Rd ) into L1 (Rd ) This completes the proof COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 20 Proof of Theorem 3.2 The proof follows the same lines except that now, one deals with equalities instead of inequalities Namely, as T is a linear operator in KL ⊂ K, Theorem 3.2 of [28] yields that there exists a bounded bilinear operator W : H 1 (Rd ) × BM O(Rd... the proof for Proposition 4.5 For the one of Proposition 4.6, we leave the details to the interested reader In order to prove Proposition 4.5, we need the following lemma Lemma 4.3 There exists a constant C > 0 such that 2 c |x−y| |h| δ (4.1) |t∂t Tt (x, y + h) − t∂t Tt (x, y)| ≤ C √ t−d/2 e− 4 t , t COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 17 for all |h| < |x−y| , 0 < t Here and in the proof of Proposition... which ends the proof g − gB L2q (2k+1 B) Rj (a) Lq (2k+1 B\2k B) COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 35 7 Some applications The purpose of this section is to give some applications of the decomposition theorems (Theorem 3.1 and Theorem 3.2) To be more precise, we give some subspaces of HL1 (Rd ), which do not necessarily depend on b and T , such that all commutators [b, T ], for b ∈ BM O(Rd )... an uniform constant C > 0 depends only on ϕ and d Proof of Lemma 5.1 It follows from Lemma 6.4 and Lemma 6.3 that H(f ) H1 (ψn,k f − ϕ2−n/2 ∗ (ψn,k f )) = n,k ψn,k f − ϕ2−n/2 ∗ (ψn,k f ) ≤ n,k ≤ C ψn,k f h1n ≤C f n,k for every f ∈ HL1 (Rd ) This completes the proof 1 HL H1 H1 COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 27 For 1 < q ≤ ∞ and n ∈ Z Recall (see [16]) that a function a is said to be a related. .. |2 2−kδ |2k B|1/q−1 COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 33 where 1/q + 1/q = 1 Proof of Lemma 5.6 It is well-known that the Riesz transforms Rj are bounded from HL1 (Rd ) into H 1 (Rd ), in particular, one has Rd Rj (a)(x)dx = 0 Moreover, by the Lq -boundedness of Rj (see [39], Theorem 0.5) one has Rj (a) Lq ≤ C|B|1/q−1 Therefore, it is sufficient to verify (5.7) Thanks to Lemma 5.5, as a is... section is to give some examples of fundamental operators related to L which are in the class KL 4.1 The Schr¨ odinger-Calder´ on-Zygmund operators Proposition 4.1 Let T be any L-Calder´on-Zygmund operator Then, T belongs to the class KL Proposition 4.2 The Riesz transforms Rj are in the class KL The proof of Proposition 4.2 follows directly from Lemma 5.7 and the fact that the Riesz transforms Rj ... Weighted estimates for commutators of some singular integrals related to Schr¨ odinger operators Bull Sci Math 138 (2014), no 2, 270–292 COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 42 [11] D-C... of singular integral operators related to Schr¨odinger operators, see for example [7, 10, 21, 32, 43, 44, 45] In the present paper, we consider commutators of singular integral operators T related. .. result concerns the HL1 -estimates for commutators of the Riesz transforms COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 13 Theorem 3.4 Let b ∈ BM OL,∞ (Rd ) Then, the commutators [b, Rj ], j = 1,