Endpoint estimates for commutators of singular integrals related to Schr\"odinger operators

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Endpoint estimates for commutators of singular integrals related to Schr’́odinger operators

Luong Dang Ky

To cite this version:

Luong Dang Ky. Endpoint estimates for commutators of singular integrals related to Schr’́odinger operators. Revista Matemática Iberoamericana, European Mathematical Society, 2015, 43 p. �hal- 01143641�

INTEGRALS RELATED TO SCHR ¨ODINGER OPERATORS

LUONG DANG KY

Abstract. LetL=−∆ +V be a Schr¨odinger operator onR^d, d≥3, whereV is a nonnegative potential,V 6= 0, and belongs to the reverse H¨older classRH_d/2. In this paper, we study the commutators [b, T] forT in a class KL of sublinear operators containing the fundamental operators in harmonic analysis related toL.

More precisely, when T ∈ K_L, we prove that there exists a bounded subbilinear operator R=R_T :H_L¹(R^d)×BM O(R^d)→L¹(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 H_L¹(R^d)×BM O(R^d) into L¹(R^d) which does not depend on T. The subbilinear decomposition (1) allows us to ex- plain why commutators with the fundamental operators are of weak type (H_L¹, L¹), and when a commutator [b, T] is of strong type (H_L¹, L¹).

Also, we discuss the H_L¹-estimates for commutators of the Riesz transforms associated with the Schr¨odinger operatorL.

Contents

1. Introduction 2

2. Some preliminaries and notations 6

3. Statement of the results 11

3.1. Two decomposition theorems 12

3.2. Hardy estimates for linear commutators 12

4. Some fundamental operators and the classK_L 13

4.1. The Schr¨odinger-Calder´on-Zygmund operators 13

4.2. The L-maximal operators 15

4.3. The L-square functions 16

5. Proof of the main results 18

5.1. Proof of Theorem 3.1 and Theorem 3.2 19

5.2. Proof of Theorem 3.3 and Theorem 3.4 20

6. Proof of the key lemmas 25

7. Some applications 35

7.1. Atomic Hardy spaces related to b ∈BM O(R^d) 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

7.2. The spaces H_L,b¹ (R^d) related tob ∈BM O(R^d) 36

7.3. Atomic Hardy spaces H_L,α^log(R^d) 38

7.4. The Hardy-Sobolev space H_L^1,1(R^d) 39

References 41

1. Introduction

Given a function b locally integrable on R^d, and a (classical) Calder´on-Zygmund operatorT, 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 com- mutator [b, T] is continuous onL^p(R^d) for 1< p <∞, whenb ∈BM O(R^d). 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].

LetL=−∆+V be a Schr¨odinger operator onR^d,d≥3, whereV is a nonnegative potential, V 6= 0, and belongs to the reverse H¨older class RH_d/2. We recall that a nonnegative locally integrable function V belongs to the reverse H¨older class RH_q, 1< q <∞, if there exists C > 0 such that

1

|B|

Z

B

(V(x))^qdx1/q

≤ C

|B| Z

B

V(x)dx

holds for every balls B in R^d. In [16], Dziuba´nski and Zienkiewicz introduced the Hardy space H_L¹(R^d) as the set of functions f ∈ L¹(R^d) such that kfk_H¹

L :=

kM_Lfk_L¹ < ∞, where M_Lf(x) := sup_t>0|e^−tLf(x)|. There, they characterized H_L¹(R^d) in terms of atomic decomposition and in terms of the Riesz transforms associated with L, R_j = ∂_xjL^−1/2, j = 1, ..., d. In the recent years, there is an in- creasing 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 K_L of all sublinear operatorsT, bounded fromH_L¹(R^d) intoL¹(R^d) and satisfying for anyb∈BM O(R^d) and a ageneralized atom related to the ballB (see Definition 2.1), we have

k(b−b_B)T ak_L¹ ≤Ckbk_{BM O},

whereb_B denotes the average ofb onB and C >0 is a constant independent of b, a.

The classK_L contains the fundamental operators (we refer the reader to [28] for the

classical case L=−∆) related to the Schr¨odinger operatorL: the Riesz transforms R_j,L-Calder´on-Zygmund operators (so-called Schr¨odinger-Calder´on-Zygmund oper- ators),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 transformsRj 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 RH_d/2.

Although Schr¨odinger-Calder´on-Zygmund operators mapH_L¹(R^d) intoL¹(R^d) (see Proposition 4.1), it was observed in [32, 48] that, whenb∈BM O(R^d), the commu- tators [b, R_j] do not map, in general, H_L¹(R^d) into L¹(R^d). 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, H¹(R) into L¹(R). After, C. P´erez showed in [37]

that if H¹(R^d) is replaced by a suitable atomic subspace H_b¹(R^d) then commuta- tors of the classical Calder´on-Zygmund operators are continuous from H_b¹(R^d) into L¹(R^d). Recall that (see [37]) a function a is a b-atom if

i) supp a⊂Q for some cube Q, ii) kak_L^∞ ≤ |Q|⁻¹,

iii) R

R^da(x)dx=R

R^da(x)b(x)dx= 0.

The space H_b¹(R^d) consists of the subspace of L¹(R^d) of functions f which can be written asf = P∞

j=1λ_ja_j where a_j are b-atoms, and λ_j are complex numbers with P∞

j=1|λ_j| < ∞. Thus, when b ∈ BM O(R^d), it is natural to ask for subspaces of H_L¹(R^d) such that all commutators of Schr¨odinger-Calder´on-Zygmund operators and the Riesz transforms map continuously these spaces intoL¹(R^d).

In this paper, we are interested in the following two questions.

Question 1. For b ∈ BM O(R^d). Find the largest subspace H¹_L,b(R^d) of H_L¹(R^d) such that all commutators of Schr¨odinger-Calder´on-Zygmund operators and the Riesz transforms are bounded fromH¹_L,b(R^d) into L¹(R^d).

Question 2. Characterize the functionsbinBM O(R^d)so thatH¹_L,b(R^d)≡H_L¹(R^d).

LetX be a Banach space. We say that an operatorT :X →L¹(R^d) 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(R^d) is a sublinear operator. We also say that an operator T:H_L¹(R^d)×BM O(R^d)→L¹(R^d) is a subbilinear operator if for every (f, g)∈H_L¹(R^d)×BM O(R^d), the operatorsT(f,·) :BM O(R^d)→L¹(R^d) and T(·, g) :H_L¹(R^d)→L¹(R^d) are sublinear operators.

To answer Question 1 and Question 2, we study commutators of sublinear op- erators in K_L. More precisely, when T ∈ K_L is a sublinear operator, we prove (see Theorem 3.1) that there exists a bounded subbilinear operator R = R_T : H_L¹(R^d)×BM O(R^d)→L¹(R^d) so that for all (f, b)∈H_L¹(R^d)×BM O(R^d),

(1.1) |T(S(f, b))| −R(f, b)≤ |[b, T](f)| ≤R(f, b) +|T(S(f, b))|,

whereSis a bounded bilinear operator fromH_L¹(R^d)×BM O(R^d) intoL¹(R^d) which does not depend on T (see Proposition 5.2). When T ∈ K_L is a linear operator, we prove (see Theorem 3.2) that there exists a bounded bilinear operator R = R_T : H_L¹(R^d)×BM O(R^d)→L¹(R^d) such that for all (f, b)∈H_L¹(R^d)×BM O(R^d), (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 (H_L¹, L¹), and when a commutator [b, T] is of strong type (H_L¹, L¹).

Let b be a function in BM O(R^d). We assume that b non-constant, otherwise [b, T] = 0. We define the space H¹_L,b(R^d) as the set of all f in H_L¹(R^d) such that [b,M_L](f)(x) = M_L(b(x)f(·)− b(·)f(·))(x) belongs to L¹(R^d), and the norm on H¹_L,b(R^d) is defined by kfk_H¹

L,b = kfk_H¹

Lkbk_{BM O} + k[b,M_L](f)k_L¹. Then, using the subbilinear decomposition (1.1), we prove that all commutators of Schr¨odinger- Calder´on-Zygmund operators and the Riesz transforms are bounded from H¹_L,b(R^d) into L¹(R^d). Furthermore, H¹_L,b(R^d) is the largest space having this property, and H¹_L,b(R^d)≡H_L¹(R^d) if and only ifb ∈BM O_L^log(R^d) (see Theorem 7.2), that is,

kbk_{BM O}^log

L = sup

B(x,r)





log

e+ ρ(x) r

1

|B(x, r)|

Z

B(x,r)

|b(y)−b_B(x,r)|dy





<∞, where ρ(x) = sup{r > 0 : _rd−2¹

R

B(x,r)V(y)dy ≤ 1}. This space BM O_L^log(R^d) arises naturally in the characterization of pointwise multipliers for BM O_L(R^d), the dual space ofH_L¹(R^d), see [3, 33].

The above answersQuestion 1andQuestion 2. As another interesting application of the subbilinear decomposition (1.1), we find subspaces of H_L¹(R^d) which do not depend on b ∈ BM O(R^d) and T ∈ KL, such that [b, T] maps continuously these spaces into L¹(R^d) (see Section 7). For instance, when L = −∆ + 1, Theorem 7.4 state that for every b ∈ BM O(R^d) and T ∈ K_L, the commutator [b, T] is bounded from H_L^1,1(R^d) into L¹(R^d). Here H_L^1,1(R^d) is the (inhomogeneous) Hardy-Sobolev space considered by Hofmann, Mayboroda and McIntosh in [23], defined as the set of functionsf inH_L¹(R^d) such that ∂x1f, ..., ∂x_df ∈H_L¹(R^d) with the norm

kfk_H^1,1

L =kfk_H¹

L+

d

X

j=1

k∂xjfk_H¹

L.

Recently, similarly to the classical result of Coifman-Rochberg-Weiss, Gou et al.

proved in [21] that the commutators [b, R_j] are bounded on L^p(R^d) whenever b ∈ BM O(R^d) and 1 < p < _d−q^dq where V ∈ RH_q for some d/2 < q < d. Later, in [7], Bongioanni et al. generalized this result by showing that the spaceBM O(R^d) can be replaced by a larger space BM OL,∞(R^d) = ∪θ≥0BM O_L,θ(R^d), where BM O_L,θ(R^d)

is the space of locally integrable functionsf satisfying kfk_{BM OL,θ} = sup

B(x,r)





 1

1 + _ρ(x)^r θ

1

|B(x, r)|

Z

B(x,r)

|f(y)−f_B(x,r)|dy





<∞.

LetR^∗_j be the adjoint operators ofR_j. Bongioanni et al. established in [6] that the operatorsR^∗_j are bounded on BM OL(R^d), and thus from L^∞(R^d) into BM OL(R^d).

Therefore, it is natural to ask for a class of functions b so that the commuta- tors [b, R^∗_j] map continuously L^∞(R^d) into BM O_L(R^d). In [7], the authors found such a class of functions. More precisely, they proved in [7] that the commutators [b, R^∗_j] map continuously L^∞(R^d) into BM O_L(R^d) whenever b ∈ BM O^log_L,∞(R^d) =

∪_θ≥0BM O_L,θ^log(R^d). Here BM O_L,θ^log(R^d) is the space of functions f ∈ L¹_loc(R^d) such that

kfk_{BM O}^log

L,θ = sup

B(x,r)





 log

e+ ^ρ(x)_r

1 + _ρ(x)^r θ

1

|B(x, r)|

Z

B(x,r)

|f(y)−f_B(x,r)|dy





<∞.

A natural question arises: can one replace the space L^∞(R^d) by BM O_L(R^d)?

Question 3. Are the commutators [b, R^∗_j], j = 1, ..., d, bounded on BM O_L(R^d) whenever b∈BM O^log_L,∞(R^d)?

Motivated by this question, we study the H_L¹-estimates for commutators of the Riesz transforms. More precisely, givenb ∈ BM O_L,∞(R^d), we prove that the com- mutators [b, R_j] are bounded on H_L¹(R^d) if and only if b belongs to BM O^log_L,∞(R^d) (see Theorem 3.4). Furthermore, if b ∈ BM O_L,θ^log(R^d) for some θ ≥ 0, then there exists a constantC > 1, independent of b, such that

C⁻¹kbk_{BM O}log L,θ

≤ kbk_{BM OL,θ} +

d

X

j=1

k[b, R_j]k_H¹

L→H_L¹ ≤Ckbk_{BM O}log L,θ

. As a consequence, we get the positive answer forQuestion 3.

Now, an open question is the following:

Open question. Find the set of all functions b such that the commutators [b, R_j], j = 1, ..., d, are bounded on H_L¹(R^d).

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,∞(R^d) so that the commutators of the Riesz transforms are bounded on H_L¹(R^d), which answers Question 3. Finally, we find the largest subspace H¹_L,b(R^d) of H_L¹(R^d) such that all commutators of Schr¨odinger-Calder´on-Zygmund operators and the Riesz transforms are bounded from H¹_L,b(R^d) into L¹(R^d). Besides, we find also the characterization

of functions b ∈ BM O(R^d) so that H¹_L,b(R^d) ≡ H_L¹(R^d), which answer Question 1 andQuestion 2. Especially, we show that there exist subspaces ofH_L¹(R^d) whichdo not dependonb ∈BM O(R^d) and T ∈ K_L, such that [b, T] maps continuously these spaces intoL¹(R^d), 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¨odinger- Calder´on-Zygmund operators. In Section 3, we state the main results: two decom- position theorems (Theorem 3.1 and Theorem 3.2), Hardy estimates for commuta- tors 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 classK_L. 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 H_L¹(R^d) such that all commutators [b, T],T ∈ K_L, map continuously these spaces into L¹(R^d).

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⁻¹f ≤ g ≤Cf). In R^d, 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 χ_E its characteristic function, by |E|its Lebesgue measure, and by E^c the setR^d\E.

Acknowledgements. The author would like to thank Aline Bonami, Sandrine Grellier and Fr´ed´eric Bernicot for many helpful suggestions and discussions. He would also like to thank Sandrine Grellier for many helpful suggestions, her carefully reading and revision of the manuscript. The author is deeply indebted to them.

2. Some preliminaries and notations In this paper, we consider the Schr¨odinger differential operator

L=−∆ +V

on R^d, d ≥ 3, where V is a nonnegative potential, V 6= 0. As in the works of Dziuba´nski et al [15, 16], we always assume that V belongs to the reverse H¨older classRH_d/2. Recall that a nonnegative locally integrable functionV is said to belong to a reverse H¨older class RH_q, 1< q <∞, if there exists C >0 such that

1

|B|

Z

B

(V(x))^qdx1/q

≤ C

|B| Z

B

V(x)dx

holds for every balls B in R^d. By H¨older inequality, RHq1 ⊂ RHq2 if q1 ≥ q2 > 1.

For q > 1, it is well-known thatV ∈RHq implies V ∈ RHq+ε for some ε > 0 (see [19]). Moreover,V(y)dyis a doubling measure, namely for any ball B(x, r) we have (2.1)

Z

B(x,2r)

V(y)dy≤C₀ Z

B(x,r)

V(y)dy.

Let{T_t}_t>0be the semigroup generated byLandT_t(x, y) be their kernels. Namely, T_tf(x) = e^−tLf(x) =

Z

R^d

T_t(x, y)f(y)dy, f ∈L²(R^d), t >0.

We say that a function f ∈L²(R^d) belongs to the space H¹L(R^d) if kfk_H¹

L :=kM_Lfk_L¹ <∞,

whereM_Lf(x) := sup_t>0|T_tf(x)| for all x ∈R^d. The space H_L¹(R^d) is then defined as the completion ofH¹L(R^d) with respect to this norm.

In [15] it was shown that the dual of H_L¹(R^d) can be identified with the space BM O_L(R^d) which consists of all functions f ∈BM O(R^d) with

kfk_{BM OL} :=kfk_{BM O}+ sup

ρ(x)≤r

1

|B(x, r)|

Z

B(x,r)

|f(y)|dy <∞,

whereρ is the auxiliary function defined as in [39], that is,

(2.2) ρ(x) = supn

r >0 : 1 r^d−2

Z

B(x,r)

V(y)dy≤1o ,

x∈ R^d. Clearly, 0 < ρ(x) <∞ for all x ∈ R^d, and thus R^d =S

n∈ZBn, where the sets Bn are defined by

(2.3) B_n ={x∈R^d : 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 constantsκ >1andk0 ≥1 such that for allx, y ∈R^d,

κ⁻¹ρ(x)

1 + |x−y|

ρ(x) −k0

≤ρ(y)≤κρ(x)

1 + |x−y|

ρ(x) _k^k0

0+1. Throughout the whole paper, we denote by C_L the L-constant

(2.4) CL= 8.9^k0κ

wherek₀ and κ are defined as in Proposition 2.1.

Given 1 < q ≤ ∞. Following Dziuba´nski and Zienkiewicz [16], a function a is called a (H_L¹, q)-atom related to the ball B(x₀, r) if r≤ C_Lρ(x₀) and

i) supp a⊂B(x₀, r), ii) kakL^q ≤ |B(x0, r)|^1/q−1, iii) if r≤ _C¹

Lρ(x0) thenR

R^da(x)dx= 0.

A function a is called a classical (H¹, q)-atom related to the ball B =B(x₀, r) if it satisfies (i), (ii) andR

R^da(x)dx= 0.

The following atomic characterization of H_L¹(R^d) is due to [16].

Theorem 2.1 (see [16], Theorem 1.5). Let 1< q ≤ ∞. A function f is in H_L¹(R^d) if and only if it can be written as f = P

jλ_ja_j, where a_j are (H_L¹, q)-atoms and P

j|λ_j|<∞. Moreover, kfk_H¹

L ≈inf (

X

j

|λ_j|:f =X

j

λ_ja_j )

.

Note that a classical (H¹, q)-atom is not a (H_L¹, q)-atom in general. In fact, there exists a constant C > 0 such that if f is a classical (H¹, q)-atom, then it can be written as f = Pn

j=1λ_ja_j, for some n ∈ Z⁺, where a_j are (H_L¹, q)-atoms and Pn

j=1|λ_j| ≤C, see for example [47]. In this work, we need a variant of the definition of atoms forH_L¹(R^d) which include classical (H¹, q)-atoms and (H_L¹, 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 (H_L¹, q, ε)-atom related to the ball B(x0, r) if

i) supp a⊂B(x0, r), ii) kak_L^q ≤ |B(x₀, r)|^1/q−1, iii) |R

R^da(x)dx| ≤

r ρ(x0)

ε

.

The spaceH^1,q,εL,at(R^d) is defined to be set of all functionsf inL¹(R^d) which can be written asf =P∞

j=1λjaj where theaj are generalized (H_L¹, q, ε)-atoms and the λj

are complex numbers such that P∞

j=1|λ_j|<∞. As usual, the norm onH^1,q,εL,at(R^d) is defined by

kfk

H^1,q,εL,at = inf nX^∞

j=1

|λj|:f =

∞

X

j=1

λjaj

o . The space H^1,q,εL,fin(R^d) is defined to be set of all f = Pk

j=1λ_ja_j, where the a_j are generalized (H_L¹, q, ε)-atoms. Then, the norm of f inH^1,q,εL,fin(R^d) is defined by

kfk

H^1,q,ε_L,fin = inf nX^k

j=1

|λj|:f =

k

X

j=1

λjaj

o .

Remark 2.1. Let 1 < q ≤ ∞ and ε > 0. Then, a classical (H¹, q)-atom is a generalized(H_L¹, q, ε)-atom related to the same ball, and a (H_L¹, q)-atom isC_L^ε times a generalized (H_L¹, q, ε)-atom related to the same ball.

Throughout the whole paper, we always use generalized (H_L¹, q, ε)-atoms except in the proof of Theorem 3.4. More precisely, in order to prove Theorem 3.4, we need to use (H_L¹, q)-atoms from Dziuba´nski and Zienkiewicz (see above).

The following gives a characterization of H_L¹(Rⁿ) in terms of generalized atoms.

Theorem 2.2. Let 1 < q ≤ ∞ and ε > 0. Then, H^1,q,εL,at(R^d) = H_L¹(R^d) and the norms are equivalent.

In order to prove Theorem 2.2, we need the following lemma.

Lemma 2.1(see [31], Lemma 2). LetV ∈RH_d/2. Then, there existsσ₀ >0depends only on L, such that for every |y−z|<|x−y|/2 and t >0, we have

|T_t(x, y)−T_t(x, z)| ≤C|y−z|

√t σ0

t^−d2e^−|x−y|

2

t ≤C |y−z|^σ0

|x−y|^d+σ0.

Proof of Theorem 2.2. As M_L is a sublinear operator, by Remark 2.1 and Theorem 2.1, it is sufficient to show that

(2.5) kM_L(a)k_L¹ ≤C

for all generalized (H_L¹, q, ε)-atoma related to the ball B =B(x₀, r).

Indeed, from the L^q-boundedness of the classical Hardy-Littlewood maximal op- eratorM, the estimate M_L(a)≤CM(a) and H¨older inequality,

(2.6) kM_L(a)k_L¹_(2B)≤CkM(a)k_L¹_(2B)≤C|2B|^1/q0kM(a)k_L^q ≤C, where 1/q⁰+ 1/q= 1. Let x /∈2B and t >0, Lemma 2.1 and (3.5) of [16] give

|T_t(a)(x)| =

Z

R^d

T_t(x, y)a(y)dy

≤

Z

B

(T_t(x, y)−T_t(x, x₀))a(y)dy

+|T_t(x, x₀)|

Z

B

a(y)dy

≤ C r^σ0

|x−x₀|^d+σ0 +C r^ε

|x−x₀|^d+ε. Therefore,

kM_L(a)k_L¹_((2B)^c₎ =ksup

t>0

|T_t(a)|k_L¹_((2B)^c₎

≤C Z

(2B)^c

r^σ0

|x−x0|^d+σ0dx+C Z

(2B)^c

r^ε

|x−x0|^d+εdx

≤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 T :H^1,q,εL,fin(R^d)→ X is a sublinear operator with

sup{kT ak_X :ais a generalized(H_L¹, q, ε)−atom}<∞.

Then, T can be extended to a bounded sublinear operator Te from H_L¹(R^d) into X, moreover,

kTek_H¹

L→X ≤Csup{kT akX :ais a generalized(H_L¹, q, ε)−atom}.

Now, we turn to explain the new BM O type spaces introduced by Bongioanni, Harboure and Salinas in [7]. Here and in what follows f_B := _|B|¹ R

Bf(x)dx and

(2.8) M O(f, B) := 1

|B|

Z

B

|f(y)−f_B|dy.

Forθ ≥0, following [7], we denote by BM O_L,θ(R^d) the set of all locally integrable functionsf such that

kfk_{BM OL,θ} = sup

B(x,r)





 1

1 + _ρ(x)^r

θM O(f, B(x, r))





<∞, and BM O_L,θ^log(R^d) the set of all locally integrable functions f such that

kfk_{BM O}^log

L,θ = sup

B(x,r)





 log

e+^ρ(x)_r

1 + _ρ(x)^r θ M O(g, B(x, r))





<∞.

When θ= 0, we write BM O^log_L (R^d) instead of BM O_L,0^log(R^d). We next define BM O_L,∞(R^d) = [

θ≥0

BM O_L,θ(R^d) and

BM O_L,∞^log (R^d) = [

θ≥0

BM O_L,θ^log(R^d).

Observe that BM O_L,0(R^d) is just the classical BM O(R^d) space. Moreover, for any 0≤θ ≤θ⁰ ≤ ∞, we have

(2.9) BM O_L,θ(R^d)⊂BM O_L,θ⁰(R^d), BM O^log_L,θ(R^d)⊂BM O_L,θ^log0(R^d) and

(2.10) BM O_L,θ^log(R^d) = BM O_L,θ(R^d)∩BM O^log_L,∞(R^d).

Remark 2.2. The inclusions in (2.9) are strict in general. In particular:

i) The space BM OL,∞(R^d) is in general larger than the space BM O(R^d). Indeed, when V(x) ≡ |x|², it is easy to check that the functions b_j(x) = |x_j|², j = 1, ..., d, belong toBM OL,∞(R^d) but not to BM O(R^d).

ii) The space BM O_L,∞^log (R^d) is in general larger than the space BM O^log_L (R^d). In- deed, when V(x)≡1, it is easy to check that the functionsbj(x) = |xj|, j = 1, ..., d, belong toBM O^log_L,∞(R^d) but not to BM O^log_L (R^d).

Next, let us recall the notation of Schr¨odinger-Calder´on-Zygmund operators.

Let δ ∈ (0,1]. According to [33], a continuous function K : R^d×R^d\ {(x, x) : x∈R^d} →C is said to be a (δ, L)-Calder´on-Zygmund singular integral kernel if for eachN >0,

(2.11) |K(x, y)| ≤ C(N)

|x−y|^d

1 + |x−y|

ρ(x) −N

for all x6=y, and

(2.12) |K(x, y)−K(x⁰, y)|+|K(y, x)−K(y, x⁰)| ≤C |x−x⁰|^δ

|x−y|^d+δ for all 2|x−x⁰| ≤ |x−y|.

As usual, we denote by C_c^∞(R^d) the space of all C^∞-functions with compact support, by S(R^d) the Schwartz space on R^d.

Definition 2.2. A linear operator T : S(R^d) → S⁰(R^d) is said to be a (δ, L)- Calder´on-Zygmund operator if T can be extended to a bounded operator on L²(R^d) and if there exists a (δ, L)-Calder´on-Zygmund singular integral kernel K such that for all f ∈C_c^∞(R^d) and all x /∈ supp f, we have

T f(x) = Z

R^d

K(x, y)f(y)dy.

An operator T is said to be a Schr¨odinger-Calder´on-Zygmund operator associ- ated withL (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 areq ∈ (1,∞] and ε > 0 so that R

R^dT a(x)dx= 0 holds for every generalized (H_L¹, q, ε)-atomsa.

Remark 2.3. i) Using Proposition 2.1, Inequality (2.11) is equivalent to

|K(x, y)| ≤ C(N)

|x−y|^d

1 + |x−y|

ρ(y) −N

for all x6=y.

ii) By Theorem 0.8 of [39] and Theorem 1.1 of [40], we see that the Riesz trans- formsR_j areL-Calder´on-Zygmund operators satisfyingR^∗_j1 = 0wheneverV ∈RH_d. iii) If T is a L-Calder´on-Zygmund operator then it is also a classical Calder´on- Zygmund operator, and thus T is bounded on L^p(R^d) for 1 < p < ∞ and bounded from L¹(R^d) into L^1,∞(R^d).

3. Statement of the results

Recall that KL is the set of all sublinear operatorsT bounded from H_L¹(R^d) into L¹(R^d) and that there are q∈(1,∞] and ε >0 such that

k(b−b_B)T ak_L¹ ≤Ckbk_{BM O}

for allb ∈BM O(R^d), any generalized (H_L¹, q, ε)-atoma related to the ballB, where C >0 is a constant independent of b, a.

3.1. Two decomposition theorems. Let b be a locally integrable function and T ∈ K_L. As usual, the (sublinear) commutator [b, T] of the operatorT is defined by [b, T](f)(x) := T

(b(x)−b(·))f(·) (x).

Theorem 3.1 (Subbilinear decomposition). Let T ∈ K_L. There exists a bounded subbilinear operator R = R_T : H_L¹(R^d)×BM O(R^d) → L¹(R^d) such that for all (f, b)∈H_L¹(R^d)×BM O(R^d), we have

|T(S(f, b))| −R(f, b)≤ |[b, T](f)| ≤R(f, b) +|T(S(f, b))|,

whereS is a bounded bilinear operator fromH_L¹(R^d)×BM O(R^d)into L¹(R^d)which does not depend onT.

Using Theorem 3.1, we obtain immediately the following result.

Proposition 3.1. LetT ∈ KLso thatT is of weak type(1,1). Then, the subbilinear operatorT(f, g) = [g, T](f) maps continuously H_L¹(R^d)×BM O(R^d) into L^1,∞(R^d).

As the Riesz transforms R_j = ∂_xjL^−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). Letb ∈BM O(R^d). Then, the commutators [b, R_j] are bounded from H_L¹(R^d) into L^1,∞(R^d).

When T is linear and belongs toKL, 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 : H_L¹(R^d)×BM O(R^d) → L¹(R^d) such that for all(f, b)∈H_L¹(R^d)×BM O(R^d), 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 O_L^log(R^d) and T be a L-Calder´on-Zygmund operator satisfying T^∗1 = 0. Then, the linear commutator [b, T] is bounded on H_L¹(R^d).

ii) When V ∈ RH_d, the converse holds. Namely, if b ∈ BM O(R^d) and [b, T] is bounded on H_L¹(R^d) for every L-Calder´on-Zygmund operator T satisfying T^∗1 = 0, then b∈BM O_L^log(R^d). Furthermore,

kbk_{BM O}^log

L ≈ kbk_{BM O}+

d

X

j=1

k[b, R_j]k_H¹

L→H_L¹.

Next result concerns the H_L¹-estimates for commutators of the Riesz transforms.

Theorem 3.4. Let b ∈ BM OL,∞(R^d). Then, the commutators [b, R_j], j = 1, ..., d, are bounded on H_L¹(R^d) if and only if b ∈ BM O^log_L,∞(R^d). Furthermore, if b ∈ BM O_L,θ^log(R^d) for some θ ≥0, we have

kbk_{BM O}^log

L,θ ≈ kbk_{BM OL,θ} +

d

X

j=1

k[b, R_j]k_H¹

L→H_L¹.

Remark that the above constants depend on θ.

Note that BM O_L^log(R^d) is in general proper subset of BM O^log_L,∞(R^d) (see Remark 2.2). When V ∈RH_d, although the Riesz transforms R_j are L-Calder´on-Zygmund operators satisfyingR^∗_j1 = 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(R^d). Then, b belongs to LM O(R^d) if and only if the vector-valued commutator [b,∇(−∆ + 1)^−1/2] maps continuously h¹(R^d) into h¹(R^d,R^d) = (h¹(R^d), ..., h¹(R^d)). Furthermore,

kbk_{LM O} ≈ kbk_{BM O}+k[b,∇(−∆ + 1)^−1/2]k_h1(R^d)→h¹(R^d,R^d).

Here h¹(R^d) is the local Hardy space of D. Goldberg (see [20]), and LM O(R^d) is the space of all locally integrable functions f such that

kfk_{LM O} := sup

B(x,r)

log

e+1 r

M O(f, B(x, r))

<∞.

It should be pointed out that LM O type spaces appear naturally when study- ing the boundedness of Hankel operators on the Hardy spaces H¹(T^d) and H¹(B^d) (where B^d is the unit ball in C^d and T^d = ∂B^d), 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 K_L

The purpose of this section is to give some examples of fundamental operators related toL which are in the class K_L.

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 R_j are in the class K_L.

The proof of Proposition 4.2 follows directly from Lemma 5.7 and the fact that the Riesz transformsR_j are bounded from H_L¹(R^d) into L¹(R^d).

To prove Proposition 4.1, we need the following two lemmas.

Lemma 4.1. Let 1 ≤ q < ∞. Then, there exists a constant C > 0 such that for every ball B, f ∈BM O(R^d) and k ∈Z⁺,

1

|2^kB| Z

2^kB

|f(y)−f_B|^qdy1/q

≤Ckkfk_{BM O}.

The proof of Lemma 4.1 follows directly from the classical John-Nirenberg in- equality. See also Lemma 6.6 below.

Lemma 4.2. Let 1 < q ≤ ∞ and ε > 0. Assume that T is a (δ, L)-Calder´on- Zygmund operator and a is a generalized (H_L¹, q, ε)-atom related to the ball B = B(x₀, r). Then,

kT ak_L^q₍₂^k+1_B\2^k_B)≤C2^−kδ0|2^kB|^1/q−1 for all k= 1,2, ..., where δ₀ = min{ε, δ}.

Proof. Let x ∈ 2^k+1B \2^kB, so that |x−x₀| ≥ 2r. Since T is a (δ, L)-Calder´on- Zygmund operator, we get

|T a(x)| ≤

Z

B

(K(x, y)−K(x, x0))a(y)dy

+|K(x, x0)|

Z

R^d

a(y)dy

≤ C Z

B

|y−x₀|^δ

|x−x₀|^d+δ|a(y)|dy+C 1

|x−x₀|^d

1 + |x−x₀| ρ(x₀)

−ε r ρ(x₀)

ε

≤ C r^δ

|x−x₀|^d+δ +C r^ε

|x−x₀|^d+ε ≤C r^δ0

|x−x₀|^d+δ0. Consequently,

kT ak_L^q₍₂^k+1_B\2^k_B)≤C r^δ0

(2^kr)^d+δ0|2^k+1B|^1/q ≤C2^−kδ0|2^kB|^1/q−1.

Proof of Proposition 4.1. Assume thatT is a (δ, L)-Calder´on-Zygmund for someδ∈ (0,1]. Let us first verify thatT is bounded fromH_L¹(R^d) intoL¹(R^d). By Proposition 2.2, it is sufficient to show that

kT ak_L¹ ≤C

for all generalized (H_L¹,2, δ)-atom a related to the ball B. Indeed, from the L²- boundedness of T and Lemma 4.2, we obtain that

kT ak_L¹ = kT ak_L¹_(2B)+

∞

X

k=1

kT ak_L¹₍₂^k+1_B\2^k_B)

≤ C|2B|^1/2kTk_L²→L²kak_L² +C

∞

X

k=1

|2^k+1B|^1/22^−kδ|2^kB|^−1/2

≤ C.

Let us next establish that

k(f −f_B)T ak_L¹ ≤Ckfk_{BM O}

for all f ∈ BM O(R^d), any generalized (H_L¹,2, 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

k(f −f_B)T ak_L¹

= k(f −f_B)T ak_L¹_(2B)+X

k≥1

k(f −f_B)T ak_L1(2^k+1B\2^kB)

≤ k(f −f_B)χ_2Bk_L²kTk_L²→L²kak_L² +X

k≥1

kf −f_Bk_L²₍₂^k+1_B)kT ak_L²₍₂^k+1_B\2^k_B)

≤ Ckfk_{BM O}+X

k≥1

C(k+ 1)kfk_{BM O}|2^k+1B|^1/22^−kδ|2^kB|^−1/2

≤ CkfkBM O, which ends the proof.

4.2. The L-maximal operators. Recall that{T_t}_t>0 is heat semigroup generated byL and T_t(x, y) are their kernels. Namely,

T_tf(x) = e^−tLf(x) = Z

R^d

T_t(x, y)f(y)dy, f ∈L²(R^d), t >0.

Then the ”heat” maximal operator is defined by M_Lf(x) = sup

t>0

|T_tf(x)|, and the ”Poisson” maximal operator is defined by

M^P_Lf(x) = sup

t>0

|Ptf(x)|, where

P_tf(x) =e^−t

√

Lf(x) = t 2√

π

∞

Z

0

e^−t

2 4u

u³² T_uf(x)du.

Proposition 4.3. The ”heat” maximal operator M_L is in the class K_L. Proposition 4.4. The ”Poisson” maximal operator M^P_L is in the class K_L.

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, M_L is bounded from H_L¹(R^d) into L¹(R^d).

Now, let us prove that

k(f −f_B)M_L(a)k_L¹ ≤Ckfk_{BM O}

for all f ∈ BM O(R^d), any generalized (H_L¹,2, 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 everyx /∈2B,

ML(a)(x)≤C r^σ0

|x−x₀|^d+σ0.

Therefore, using Lemma 4.1, the L²-boundedness of the classical Hardy-Littlewood maximal operatorMand the estimate M_L(a)≤CM(a), we obtain that

k(f −f_B)M_L(a)k_L¹

= k(f −f_B)M_L(a)k_L¹_(2B)+k(f −f_B)M_L(a)k_L¹_((2B)^c₎

≤ Ckf −f_Bk_L²_(2B)kM(a)k_L² +C Z

|x−x0|≥2r

|f(x)−f_B(x0,r)| r^σ0

|x−x₀|^d+σ0dx

≤ CkfkBM O,

where we have used the following classical inequality Z

|x−x0|≥2r

|f(x)−f_B(x0,r)| r^σ0

|x−x₀|^d+σ0dx≤Ckfk_{BM O},

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 funcfionsg and G are defined by

g(f)(x) =





∞

Z

0

|t∂_tT_t(f)(x)|²dt t





1/2

and

G(f)(x) =







∞

Z

0

Z

|x−y|<t

|t∂_tT_t(f)(y)|²dydt t^d+1







1/2

.

Proposition 4.5. The L-square function g is in the class K_L. Proposition 4.6. The L-square function G is in the class K_L.

Here we just give 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 (4.1) |t∂_tT_t(x, y+h)−t∂_tT_t(x, y)| ≤C|h|

√t δ

t^−d/2e^−c4|x−y|

2 t ,