Let
b
b
be a
B
M
O
BMO
-function. It is well known that the linear commutator
[
b
,
T
]
[b, T]
of a Calderón-Zygmund operator
T
T
does not, in general, map continuously
H
1
(
R
n
)
H^1(\mathbb R^n)
into
L
1
(
R
n
)
L^1(\mathbb R^n)
. However, Pérez showed that if
H
1
(
R
n
)
H^1(\mathbb R^n)
is replaced by a suitable atomic subspace
H
b
1
(
R
n
)
\mathcal H^1_b(\mathbb R^n)
, then the commutator is continuous from
H
b
1
(
R
n
)
\mathcal H^1_b(\mathbb R^n)
into
L
1
(
R
n
)
L^1(\mathbb R^n)
. In this paper, we find the largest subspace
H
b
1
(
R
n
)
H^1_b(\mathbb R^n)
such that all commutators of Calderón-Zygmund operators are continuous from
H
b
1
(
R
n
)
H^1_b(\mathbb R^n)
into
L
1
(
R
n
)
L^1(\mathbb R^n)
. Some equivalent characterizations of
H
b
1
(
R
n
)
H^1_b(\mathbb R^n)
are also given. We also study the commutators
[
b
,
T
]
[b,T]
for
T
T
in a class
K
\mathcal K
of sublinear operators containing almost all important operators in harmonic analysis. When
T
T
is linear, we prove that there exists a bilinear operator
R
=
R
T
\mathfrak R= \mathfrak R_T
mapping continuously
H
1
(
R
n
)
×
B
M
O
(
R
n
)
H^1(\mathbb R^n)\times BMO(\mathbb R^n)
into
L
1
(
R
n
)
L^1(\mathbb R^n)
such that for all
(
f
,
b
)
∈
H
1
(
R
n
)
×
B
M
O
(
R
n
)
(f,b)\in H^1(\mathbb R^n)\times BMO(\mathbb R^n)
we have
[
b
,
T
]
(
f
)
=
R
(
f
,
b
)
+
T
(
S
(
f
,
b
)
)
,
\begin{equation}[b,T](f)= \mathfrak R(f,b) + T(\mathfrak S(f,b)), \end{equation}
where
S
\mathfrak S
is a bounded bilinear operator from
H
1
(
R
n
)
×
B
M
O
(
R
n
)
H^1(\mathbb R^n)\times BMO(\mathbb R^n)
into
L
1
(
R
n
)
L^1(\mathbb R^n)
which does not depend on
T
T
. In the particular case of
T
T
a Calderón-Zygmund operator satisfying
T
1
=
T
∗
1
=
0
T1=T^*1=0
and
b
b
in
B
M
O
log
(
R
n
)
BMO^\textrm {log}(\mathbb R^n)
, the generalized
B
M
O
BMO
type space that has been introduced by Nakai and Yabuta to characterize multipliers of
B
M
O
(
R
n
)
BMO(\mathbb {R}^n)
, we prove that the commutator
[
b
,
T
]
[b,T]
maps continuously
H
b
1
(
R
n
)
H^1_b(\mathbb R^n)
into
h
1
(
R
n
)
h^1(\mathbb R^n)
. Also, if
b
b
is in
B
M
O
(
R
n
)
BMO(\mathbb R^n)
and
T
∗
1
=
T
∗
b
=
0
T^*1 = T^*b = 0
, then the commutator
[
b
,
T
]
[b, T]
maps continuously
H
b
1
(
R
n
)
H^1_b (\mathbb R^n)
into
H
1
(
R
n
)
H^1(\mathbb R^n)
. When
T
T
is sublinear, we prove that there exists a bounded subbilinear operator
R
=
R
T
:
H
1
(
R
n
)
×
B
M
O
(
R
n
)
→
L
1
(
R
n
)
\mathfrak R= \mathfrak R_T: H^1(\mathbb R^n)\times BMO(\mathbb R^n)\to L^1(\mathbb R^n)
such that for all
(
f
,
b
)
∈
H
1
(
R
n
)
×
B
M
O
(
R
n
)
(f,b)\in H^1(\mathbb R^n)\times BMO(\mathbb R^n)
we have
|
T
(
S
(
f
,
b
)
)
|
−
R
(
f
,
b
)
≤
|
[
b
,
T
]
(
f
)
|
≤
R
(
f
,
b
)
+
|
T
(
S
(
f
,
b
)
)
|
.
\begin{equation}|T(\mathfrak S(f,b))|- \mathfrak R(f,b)\leq |[b,T](f)|\leq \mathfrak R(f,b) + |T(\mathfrak S(f,b))|. \end{equation}
The bilinear decomposition (1) and the subbilinear decomposition (2) allow us to give a general overview of all known weak and strong
L
1
L^1
-estimates.