ENDPOINT ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS RELATED TO SCHRODINGER OPERATORS

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