Bilinear decompositions and commutators of singular integral operators
Authors: Ky Luong 
Published: November 2012

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis f comma b right-parenthesis element-of upper H Superscript 1 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis times upper B upper M upper O left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis">

(
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket b comma upper T right-bracket left-parenthesis f right-parenthesis equals German upper R left-parenthesis f comma b right-parenthesis plus upper T left-parenthesis German upper S left-parenthesis f comma b right-parenthesis right-parenthesis comma">

[
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R equals German upper R Subscript upper T Baseline colon upper H Superscript 1 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis times upper B upper M upper O left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis right-arrow upper L Superscript 1 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis f comma b right-parenthesis element-of upper H Superscript 1 Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis times upper B upper M upper O left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis">

(
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper T left-parenthesis German upper S left-parenthesis f comma b right-parenthesis right-parenthesis EndAbsoluteValue minus German upper R left-parenthesis f comma b right-parenthesis less-than-or-equal-to StartAbsoluteValue left-bracket b comma upper T right-bracket left-parenthesis f right-parenthesis EndAbsoluteValue less-than-or-equal-to German upper R left-parenthesis f comma b right-parenthesis plus StartAbsoluteValue upper T left-parenthesis German upper S left-parenthesis f comma b right-parenthesis right-parenthesis EndAbsoluteValue period">

|

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.