Abstract
Let T be a bilinear Calderón–Zygmund singular integral operator and let
T
*
{T^{*}}
be its corresponding truncated maximal operator. For any
b
∈
BMO
(
ℝ
n
)
{b\in\operatorname{BMO}({\mathbb{R}^{n}})}
and
b
→
=
(
b
1
,
b
2
)
∈
BMO
(
ℝ
n
)
×
BMO
(
ℝ
n
)
{\vec{b}=(b_{1},b_{2})\in\operatorname{BMO}({\mathbb{R}^{n}})\times%
\operatorname{BMO}({\mathbb{R}^{n}})}
, let
T
b
,
j
*
{T^{*}_{b,j}}
(
j
=
1
,
2
{j=1,2}
) and
T
b
→
*
{T^{*}_{\vec{b}}}
be the commutators in the j-th entry and the iterated commutators of
T
*
{T^{*}}
, respectively.
In this paper, for all
1
<
p
1
,
p
2
<
∞
{1<p_{1},p_{2}<\infty}
,
1
p
=
1
p
1
+
1
p
2
{\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}}
, we show that
T
b
,
j
*
{T^{*}_{b,j}}
and
T
b
→
*
{T^{*}_{\vec{b}}}
are compact operators from
L
p
1
(
w
1
)
×
L
p
2
(
w
2
)
{L^{p_{1}}(w_{1})\times L^{p_{2}}(w_{2})}
to
L
p
(
v
w
→
)
{L^{p}(v_{\vec{w}})}
if
b
,
b
1
,
b
2
∈
CMO
(
ℝ
n
)
{b,b_{1},b_{2}\in\operatorname{CMO}(\mathbb{R}^{n})}
and
w
→
=
(
w
1
,
w
2
)
∈
A
p
→
{\vec{w}=(w_{1},w_{2})\in A_{\vec{p}}}
,
v
w
→
=
w
1
p
/
p
1
w
2
p
/
p
2
{v_{\vec{w}}=w_{1}^{p/p_{1}}w_{2}^{p/p_{2}}}
. Here
CMO
(
ℝ
n
)
{\operatorname{CMO}(\mathbb{R}^{n})}
denotes the closure of
𝒞
c
∞
(
ℝ
n
)
{\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n})}
in the
BMO
(
ℝ
n
)
{\operatorname{BMO}(\mathbb{R}^{n})}
topology and
A
p
→
{A_{\vec{p}}}
is the multiple weights class.