Ann. Funct. Anal. 1 (2010), no. 2, 12–27
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL: www.emis.de/journals/AFA/
LIPSCHITZ ESTIMATES FOR MULTILINEAR COMMUTATOR
OF PSEUDO-DIFFERENTIAL OPERATORS
ZHIWEI WANG1 AND LANZHE LIU2∗
Communicated by J. Esterle
Abstract. In this paper, we prove the boundedness for some multilinear commutators generated by the pseudo-differential operator and Lipschitz functions.
1. Introduction
As the development of singular integral operators, their commutators and
multilinear operators have been well studied (see [4, 5, 6, 7, 8, 9, 10]). In
[4, 5, 6, 7, 8, 9, 10], the authors prove that the commutators and multilinear
operators generated by the singular integral operators and BM O functions are
bounded on Lp (Rn ) for 1 < p < ∞; Chanillo (see [2]) proves a similar result when
singular integral operators are replaced by the fractional integral operators. In
[4, 11, 12, 13, 16], the boundedness for the commutators and multilinear operators
generated by the singular integral operators and Lipschitz functions on Triebel–
Lizorkin and Lp (Rn )(1 < p < ∞) spaces are obtained. In [14], the weighted
boundedness for the commutators generated by the singular integral operators
and BM O and Lipschitz functions on Lp (Rn )(1 < p < ∞) spaces are obtained.
The purpose of this paper is to prove the weighted boundedness on Lebesgue
spaces for some multilinear operators associated to the pseudo-differential operators and the weighted Lipschitz functions. To do this, we first prove a sharp
function estimate for the multilinear operators. Our results are new, even in the
unweighted cases.
Date: Received: 25 July 2010; Accepted: 15 December 2010.
∗
Corresponding author.
2010 Mathematics Subject Classification. Primary 42B20; Secondary 42B25.
Key words and phrases. Multilinear commutator, pseudo-differential operator, Lipschitz
space, Lebesgue spaces, Triebel–Lizorkin space.
12
MULTILINEAR COMMUTATOR OF PSEUDO-DIFFERENTIAL OPERATORS
13
The purpose of this paper is to prove the boundedness on Lebesgue spaces for
some multilinear commutators generated by the pseudo-differential operator and
Lipschitz functions.
2. Notations and Theorems
In order to state our results, we begin by introducing the relevant notions and
definitions.
Throughout this paper, Q will denote a cube of Rn with sides parallel to the
axes.
R
1
The A1 weight is defined by A1 = {0 < ω ∈ L1loc : supQ3x |Q|
ω(y)dy ≤
Q
cω(x), a.e.}(see [5]).
The Lipschitz space Lipβ (Rn ) with 0 < β < 1 is the space of functions f such
that
|f (x) − f (y)|
||f ||Lipβ = sup
< ∞.
|x − y|β
x,y∈Rn
x6=y
For 0 < β < 1, 1 < p < ∞, m > 0, the Triebel–Lizorkin space F̃pmβ,∞ (Rn ) (see
[14]) is the space of functions f such that
Z
mβ
||f ||F̃pmβ,∞ = sup |Q|−1− n
|f (x) − fQ |dx < ∞.
Q3x̃
Q
Lp
Given some function bj ∈ Lipβ (Rn ), 1 ≤ j ≤ m, 0 < β < 1, we denote by Cjm
the family of all finite subsets γ = {γ(1), · · ·, γ(j)} of j different elements in
{1, · · ·, m}, where γ(i) < γ(j) when i < j.
For γ ∈ Cjm , we denote γ c = {1, · · ·, m}\γ.
For ~b = {b1 , · · ·, bm } and γ = {γ(1), · · ·, γ(j)} ∈ Cjm , we denote b~γ = {bγ(1) , · ·
·, bγ(j) }, and bγ = bγ(1) · · · bγ(j) , and ||b~γ ||Lipβ = ||bγ(1) ||Lipβ · · · ||bγ(j) ||Lipβ .
m
We say δ(x, ξ) ∈ S,σ
, if for x, ξ ∈ Rn ,
α β
∂ ∂
≤ Cα,β (1 + |ξ|)m−|β|+σ|α| ,
δ(x,
ξ)
∂xα ∂ξ β
where α, β are multiindex consisting of n nonnegative integers.
m
The pseudo-differential operators ψ · d · o· with symbols δ(x, ξ) ∈ S,σ
is given
by
Z
T (f )(x) =
e2πi(x,ξ) δ(x, ξ)fˆ(ξ)dξ,
Rn
where f is a Schwartz function and fˆ denotes the Fourier transform of f .
The pseudo-differential operators ψ · d · o· also have another expression
Z
T (f )(x) =
K(x, x − y)f (y)dy,
Rn
where K(x, x − y) =
R
Rn
δ(x, ξ)e2πi(x−y,ξ) dξ.
14
Z. WANG, L. LIU
Let bj , 1 ≤ j ≤ m be the fixed locally integrable functions on Rn . The multilinear commutator associated to the pseudo-differential operator is defined by
Z
m
Y
T~b (f )(x) =
K(x, x − y) (bj (x) − bj (y))f (y)dy.
Rn
j=1
Now, we state the main results as follows.
−na/2
Theorem 2.1. Let T be a ψ ·d·o· with symbol δ(x, ξ) ∈ S1−a,σ , 0 ≤ σ < 1−a, 0 <
1
a < 1, and bj ∈ Lipβ (Rn ) for 1 ≤ j ≤ m, 0 < β < m(1−a)
, and 1 < p < ∞. Then
T~b is bounded from Lp (Rn ) to F̃pmβ,∞ (Rn ).
−na/2
Theorem 2.2. Let T be a ψ · d · o· with symbol δ(x, ξ) ∈ S1−a,σ , 0 ≤ σ <
1
1 − a, 0 < a < 1, and bj ∈ Lipβ (Rn ) for 1 ≤ j ≤ m, 0 < β < m(1−a)
, and
mβ
1 < p < ∞, 1/q = 1/p − n . Then T~b is bounded from Lp (Rn ) to Lq (Rn ).
3. Preliminary Lemmas
Lemma 3.1. (see [14]) For 0 < β < 1, 1 ≤ p ≤ ∞,
1/p
Z
Z
β
β
−1− n
−n
−1
p
||b||Lipβ ≈ sup |Q|
|b(x)−bQ |dx ≈ sup |Q|
|Q|
|b(x) − bQ | dx
,
Q
Q
Q
−1
where bQ = |Q|
R
Q
Q
b(x)dx.
Lemma 3.2. (see [14]) Let Q1 ⊂ Q2 . Then |bQ1 − bQ2 | ≤ C||b||Lipβ |Q2 |β/n .
−na/2
Lemma 3.3. (see [1]) Let δ(x, ξ) ∈ S1−a,σ , 0 ≤ σ < 1 − a, 0 < a < 1, and
K(x, w) denote the inverse Fourier transformations in the ξ-variable
and in the
R
distribution sense of δ(x, ξ), that is informally K(x, w) = Rn δ(x, ξ)e2πi(w,ξ) dξ.
Then for |x − x0 | ≤ d ≤ 1/2 and N ≥ 0,
Z
1/2
2
|K(x, x − y) − K(x0 , x0 − y)| dy
(2N d)1−a ≤|y−x0 |≤(2N +1 d)1−a
≤ C|x − x0 |(1−a)(h−n/2) (2N +1 d)−h(1−a)
where h is an integer such that n/2 < h < n/2 + 1/(1 − a).
0
Lemma 3.4. (see [1]) Let δ(x, ξ) ∈ S,σ
, 0 < < 1, and, as usual, K(x, w) =
R
2πi(w,ξ)
δ(x,
ξ)e
dξ.
Then
for
|w|
≥
1/4
and arbitrarily large M, |K(x, w)| ≤
Rn
−2M
CM |w|
.
Lemma 3.5. (see [3]) For 0 < β < 1, 1 < p < ∞, m > 0, we have
Z
mβ
−1−
n
||f ||F̃pmβ,∞ ≈ sup |Q|
|f (x) − fQ |dx
Q3x̃
Q
Lp
Z
mβ
−1−
n
≈ sup inf |Q|
|f (x) − c|dx ,
Q3x̃ c∈CQ
R
where fQ = |Q|−1 Q f (x)dx.
Q
Lp
MULTILINEAR COMMUTATOR OF PSEUDO-DIFFERENTIAL OPERATORS
15
Lemma 3.6. (see [15, Section 1.9 of Chapter 5]) χQ ∈ A1 for any cube Q.
−na/2
Lemma 3.7. (see [1]) Let δ(x, ξ) ∈ S1−a,σ , 0 ≤ σ < 1 − a, 0 < a < 1, T (f )(x) =
R
e2πi(x,ξ) δ(x, ξ)fˆ(ξ)dξ, and ω ∈ Ap/2 with 2 ≤ p < ∞, ||T (f )||p,ω
Rn
1
R
= Rn |T (f )(x)|p ω(x)dx p . Then ||T (f )||p,ω ≤ Cp ||f ||p,ω for f ∈ C0∞ (Rn ).
Lemma 3.8. (see [15, Section 1.1 of Chapter 2]) Suppose 1 < p < ∞, 1 < r < ∞,
R
r1
1
r
Mr (f )(x) = supQ3x |Q|
|f
(y)|
dy
. Then
Q
Z
1/p
p
Z
≤C
Mr (f )(x) dx
Rn
p
|f (x)| dx
1/p
.
Rn
1/r
rmβ R
Lemma 3.9. (see [2]) Let Mr,mβ (f )(x) = supx∈Q |Q|−1+ n Q |f (y)|r dy
for
1 < r < ∞, β > 0, m > 0, and r < p < mβ/n, 1/q = 1/p − mβ/n. Then
||Mr,mβ (f )(x)||Lq ≤ C||f ||Lp .
4. Proof of Theorem 1
Proof. Fix a cube Q = Q(x̃, d) with the sidelength d. we first prove that there
exists some constant C and 1 < r < ∞ such that, for f ∈ Lp (Rn ),
Z
−1− mβ
n
sup inf |Q|
|T~b (x) − c|dx ≤ C||~b||Lipβ (Mr (T (f ))(x̃) + Mr (f )(x̃)) .
Q3x̃ c∈CQ
Q
We consider the case m = 1 and d ≤ 1.
Let f (x) = f1 (x)
R + f2 (x) with f1 (x) = f (x)χJ (x) and f2 (x) = f (x)χJ c (x), and
let (b1 )J = |J|−1 J b1 (y)dy, where J is a cube concentric with Q of side-length
d1−a .
Z
K(x, x − y)((b1 (x) − (b1 )J ) − (b1 (y) − (b1 )J ))f (y)dy
Tb~1 (f )(x) =
Rn
Z
Z
= (b1 (x) − (b1 )J )
K(x, x − y)f (y)dy −
K(x, x − y)(b1 (y) − (b1 )J )f1 (y)dy
Rn
Rn
Z
−
K(x, x − y)(b1 (y) − (b1 )J )f2 (y)dy
Rn
= (b1 (x) − (b1 )J )T (f )(x) − T ((b1 − (b1 )J )f1 )(x) − T ((b1 − (b1 )J )f2 )(x),
thus
β
−1− n
Z
|Q|
|Tb~1 (f )(x) − T (((b1 )J − b1 )f2 )(x̃)|dx
Z
Z
β
−1− n
|b1 (x) − (b1 )J ||T (f )(x)|dx + |Q|
|T ((b1 − (b1 )J )f1 )(x)|dx
Q
Q
Z
β
−1− n
|T ((b1 − (b1 )J )f2 )(x) − T ((b1 − (b1 )J )f2 )(x̃)|dx
+|Q|
Q
β
≤ |Q|−1− n
Q
=
A1,1 + A1,2 + A1,3 .
16
Z. WANG, L. LIU
For A1,1 , by Hölder’s inequality with exponent 1/r + 1/r0 = 1 and Lemma 1, we
have
β
−1− n
A1,1 ≤ C|Q|
−1
Z
|Q| |Q|
r0
1/r0 |b1 (x) − (b1 )J | dx
Z
−1
|Q|
≤ C|J|
Q
Z
−1
1/r0 r0
|J|
1/r
|T (f )(x)| dx
Q
β
−n
r
|b1 (x) − (b1 )J | dx
−1
Z
r
|Q|
1/r
|T (f )(x)| dx
J
Q
≤ C||b1 ||Lipβ Mr (T (f ))(x̃).
For A1,2 , by Hölder’s inequality with exponent 1/s + 1/s0 = 1 and 1/t + 1/t0 = 1,
such that st = r, and Lemma 7, 1, we have
β
−1− n
Z
s
A1,2 ≤ C|Q|
1/s
|T ((b1 − (b1 )J )f1 )(x)| dx
|Q|1/s
0
Rn
β
−1− n
Z
s
≤ C|Q|
s
1/s
|b1 (x) − (b1 )J | |f1 (x)| dx
|Q|1/s
0
Rn
β
−1− n
Z
s
s
1/s
|b1 (x) − (b1 )J | |f (x)| dx
= C|Q|
|Q|1/s
0
J
β
−1− n
≤ C|Q|
−1
1/s
|Q|
Z
−1
|J|
|f (x)|st dx
× |Q|
Z
st0
1/st0
|b1 (x) − (b1 )J | dx
J
1/st
|Q|1/s
0
Q
β
−n
≤ C|J|
−1
Z
|J|
st0
|b1 (x) − (b1 )J | dx
J
1/st0 −1
Z
|Q|
st
1/st
|f (x)| dx
Q
≤ C||b1 ||Lipβ Mr (f )(x̃).
For A1,3 , choose h such that n/2 < h < n/2 + 1/(1 − a) and n/2 − h + β < 0.
By Hölder’s inequality with exponent 1/r + 1/s + 1/2 = 1 and Lemma 2,3,1, we
have
|T ((b1 − (b1 )J )f2 )(x) − T ((b1 − (b1 )J )f2 )(x̃)|
Z
=
(K(x, x − y) − K(x̃, x̃ − y))(b1 (y) − (b1 )J )f2 (y)dy n
ZR
≤
|K(x, x − y) − K(x̃, x̃ − y)||b1 (y) − (b1 )J ||f (y)|dy
=
|y−x0 |>d1−a
∞
XZ
N =0
(2N d)1−a ≤|y−x̃|≤(2N +1 d)1−a
|K(x, x − y) − K(x̃, x̃ − y)||b1 (y) − (b1 )J ||f (y)|dy
MULTILINEAR COMMUTATOR OF PSEUDO-DIFFERENTIAL OPERATORS
=
∞ Z
X
N =0
17
|K(x, x − y) − K(x̃, x̃ − y)|
(2N d)1−a ≤|y−x̃|≤(2N +1 d)1−a
× |b1 (y) − (b1 )2(N +1)(1−a) J ||f (y)|dy
∞ Z
X
+
|K(x, x − y) − K(x̃, x̃ − y)|
(2N d)1−a ≤|y−x̃|≤(2N +1 d)1−a
N =0
× |(b1 )2(N +1)(1−a) J − (b1 )J ||f (y)|dy
1/2
∞ Z
X
2
≤C
|K(x, x − y) − K(x̃, x̃ − y)| dy
(2N d)1−a ≤|y−x̃|≤(2N +1 d)1−a
N =0
Z
s
×
1/s Z
r
|b1 (y) − (b1 )2(N +1)(1−a) J | dy
|y−x̃|≤(2N +1 d)1−a
∞ Z
X
+C
1/r
|f (y)| dy
|y−x̃|≤(2N +1 d)1−a
2
1/2
|K(x, x − y) − K(x̃, x̃ − y)| dy
(2N d)1−a ≤|y−x̃|≤(2N +1 d)1−a
N =0
× ||b~1 ||Lipβ (2
N +1
β(1−a)
d)
N +1
(2
n(1−a)/s
Z
r
1/r
|f (y)| dy
d)
|y−x̃|≤(2N +1 d)1−a
∞
X
≤C
|x − x̃|(1−a)(h−n/2) (2N +1 d)−h(1−a)+n(1−a)/2 (2N +1 d)β(1−a)
N =0
N +1
× (2
−β(1−a)
d)
N +1
(2
Z
−n(1−a)
s
|b1 (y) − (b1 )2(N +1)(1−a) J | dy
d)
|y−x̃|≤(2N +1 d)1−a
N +1
× (2
−n(1−a)
Z
1/r
r
|f (y)| dy
d)
|y−x̃|≤(2N +1 d)1−a
+C
∞
X
|x − x̃|(1−a)(h−n/2) (2N +1 d)−h(1−a)+n(1−a)/2
N =0
N +1
× ||b1 ||Lipβ (2
β(1−a)
d)
N +1
(2
−n(1−a)
Z
r
1/r
|f (y)| dy
d)
|y−x̃|≤(2N +1 d)1−a
≤C
∞
X
d(1−a)(h−n/2) (2N +1 d)(1−a)(n/2−h) (2N +1 d)β(1−a) ||b1 ||Lipβ Mr (f )(x̃)
N =0
+C
≤C
∞
X
N =0
∞
X
d(1−a)(h−n/2) (2N +1 d)(1−a)(n/2−h) (2N +1 d)β(1−a) ||b1 ||Lipβ Mr (f )(x̃)
2(N +1)(1−a)(n/2−h+β) dβ(1−a) ||b1 ||Lipβ Mr (f )(x̃)
N =0
β
≤ C|J| n ||b1 ||Lipβ Mr (f )(x̃)
thus
A1,3 ≤ C||b1 ||Lipβ Mr (f )(x̃).
1/s
18
Z. WANG, L. LIU
Combining all the estimates, we finish the case m = 1 and d ≤ 1.
In case m = 1 and d > 1, we proceeds the case as follows.
Let f (x) = f1 (x) + f2R(x) with f1 (x) = f (x)χ2J (x) and f2 (x) = f (x)χ(2Q)c (x),
and let (b1 )2Q = |2Q|−1 2Q b1 (y)dy. We have
Z
K(x, x − y)((b1 (x) − (b1 )2Q ) − (b1 (y) − (b1 )2Q ))f (y)dy
Z
= (b1 (x) − (b1 )2Q )
K(x, x − y)f (y)dy
Rn
Z
−
K(x, x − y)(b1 (y) − (b1 )2Q )f1 (y)dy
n
ZR
−
K(x, x − y)(b1 (y) − (b1 )2Q )f2 (y)dy
Tb~1 (f )(x) =
Rn
Rn
= (b1 (x) − (b1 )2Q )T (f )(x) − T ((b1 − (b1 )2Q )f1 )(x) − T ((b1 − (b1 )2Q )f2 )(x),
thus
β
−1− n
Z
|Q|
Q
β
−1− n
|Tb~1 (f )(x)|dx
Z
≤ |Q|
|b1 (x) − (b1 )2Q ||T (f )(x)|dx
Q
β
−1− n
Z
+ |Q|
|T ((b1 − (b1 )2Q )f1 )(x)|dx
Q
β
−1− n
Z
+ |Q|
|T ((b1 − (b1 )2Q )f2 )(x)|dx
Q
= A2,1 + A2,2 + A2,3 .
Similar to A1,1 , A2,1 ≤ C||b1 ||Lipβ Mr (T (f ))(x̃).
Similar to A1,2 , A2,1 ≤ C||b1 ||Lipβ Mr (f )(x̃).
For A2,3 , by Hölder’s inequality with exponent 1/r + 1/r0 = 1 and Lemma 4,
2, 1, we have
Z
|T ((b1 − (b1 )2Q )f2 )(x)| = K(x, x − y)(b1 (y) − (b1 )2Q )f2 (y)dy Rn
Z
≤
|K(x, x − y)||b1 (y) − (b1 )2Q ||f (y)|dy
|y−x̃|>2d
Z
≤C
|x − y|−2n |b1 (y) − (b1 )2Q ||f (y)|dy
|y−x̃|>2d
MULTILINEAR COMMUTATOR OF PSEUDO-DIFFERENTIAL OPERATORS
≤C
∞ Z
X
≤C
|x − y|−2n |b1 (y) − (b1 )2N +1 Q ||f (y)|dy
2N d≤|y−x̃|≤2N +1 d
N =1
+C
∞ Z
X
|x − y|−2n |(b1 )2N +1 Q − (b1 )2Q ||f (y)|dy
2N d≤|y−x̃|≤2N +1 d
N =1
∞
X
N +1
2
Z
−2n+n N +1 β N +1 −β
N +1 −n
d
(2
d) (2
d)
(2
d)
N +1
× (2
r
1/r
|f (y)| dy
|y−x̃|≤2N +1 d
N =1
19
−n
Z
r0
1/r0
|b1 (y) − (b1 )2N +1 Q | dy
d)
|y−x̃|≤2N +1 d
+C
≤C
∞
X
N =1
∞
X
N +1
2
Z
−2n+n N
+1
−n
(b1 )2N +1 Q − (b1 )2Q (2
d
d)
r
1/r
|f (y)| dy
|y−x̃|≤2N +1 d
−n N +1 β
2N +1 d
(2
d) ||b1 ||Lipβ Mr (f )(x̃)
N =1
+C
≤C
≤C
∞
X
N =1
∞
X
N =1
∞
X
−n N +1 β
2N +1 d
(2
d) ||b1 ||Lipβ Mr (f )(x̃)
2(N +1)(−n+β) d−n+β ||b1 ||Lipβ Mr (f )(x̃)
2(N +1)(−n+β) dβ ||b1 ||Lipβ Mr (f )(x̃)
N =1
β
≤ C|Q| n ||b1 ||Lipβ Mr (f )(x̃),
thus
A2,3 ≤ C||b1 ||Lipβ Mr (f )(x̃).
Combining all the estimates, we finish the case m = 1 and d > 1.
Now, we consider the case m ≥ 2 and d ≤ 1.
Let f (x) = f1 (x)
R + f2 (x) with f1 (x) = f (x)χJ (x) and f2 (x) = f (x)χJ c (x), and
let (bj )J = |J|−1 J bj (y)dy for 1 ≤ j ≤ m, where J is a cube concentric with Q
of side-length d1−a .
Z
T~b (f )(x) =
=
m
Y
K(x, x − y) ((bj (x) − (bj )J ) − (bj (y) − (bj )J ))f (y)dy
Rn
j=1
m
X
X
j=0
γ∈Cjm
m−j
(−1)
Z
(b(x) − bJ )γ
K(x, x − y)(b(y) − bJ )γ c f (y)dy
Rn
20
Z. WANG, L. LIU
Z
m
Y
=
(bj (x) − (bj )J )
j=1
+
K(x, x − y)f (y)dy
Rn
m−1
X
X
m−j
(−1)
Z
(b(x) − bJ )γ
m
m
Y
K(x, x − y) (bj (y) − (bj )J )f1 (y)dy
Z
+(−1)
Rn
+(−1)m
K(x, x − y)(b(y) − bJ )γ c f (y)dy
Rn
j=1 γ∈Cjm
Z
K(x, x − y)
Rn
j=1
m
Y
(bj (y) − (bj )J )f2 (y)dy
j=1
m
Y
=
(bj (x) − (bj )J )T (f )(x)
j=1
+
m−1
X
X
j=1
γ∈Cjm
(b(x) − bJ )γ T ((b − bJ )γ c f )(x)
m
Y
+(−1)m T ( (bj − (bj )J )f1 )(x)
j=1
m
Y
+(−1)m T (
(bj − (bj )J )f2 )(x),
j=1
thus
−1− mβ
n
m
Y
|T~b (f )(x) − T ( ((bj )J − bj )f2 )(x̃)|dx
Z
|Q|
Q
≤ |Q|−1−
mβ
n
j=1
Z Y
m
|bj (x) − (bj )J ||T (f )(x)|dx
Q j=1
+
m−1
X
X
Z
−1− mβ
n
|Q|
|(b(x) − bJ )γ ||T ((b − bJ )γ c f )(x)|dx
Q
j=1 γ∈Cjm
−1− mβ
n
Z
+|Q|
Q
−1− mβ
n
Z
m
Y
|T ( (bj − (bj )J )f1 )(x)|dx
j=1
m
Y
m
Y
(bj − (bj )J )f2 )(x) − T ( (bj − (bj )J )f2 )(x̃)|dx
|T (
+|Q|
Q
j=1
= A3,1 + A3,2 + A3,3 + A3,4 .
j=1
MULTILINEAR COMMUTATOR OF PSEUDO-DIFFERENTIAL OPERATORS
21
For A3,1 , by Hölder’s inequality with exponent 1/r1 + · · · + 1/rm + 1/r = 1 and
Lemma 1, we have
− mβ
n
A3,1 ≤ C|Q|
m Y
Z
−1
|J|
|bj (x) − (bj )J | dx
≤C
≤C
j=1
m
Y
−β/n
|J|
−1
Z
|Q|
rj
1/rj
|bj (x) − (bj )J | dx)
(|J|
1/r
Q
Z
−1
r
|T (f )(x)| dx
J
j=1
m Y
1/rj rj
1/r
Z
−1
r
|Q|
|T (f )(x)| dx
J
Q
||bj ||Lipβ Mr (T (f ))(x̃)
j=1
≤ C||~b||Lipβ Mr (T (f ))(x̃).
For A3,2 , by Hölder’s inequality with exponent 1/s + 1/s0 = 1 and 1/t + 1/t0 = 1,
such that st = r, and Lemma 1, 6, 7, we have
A3,2 ≤ C
m−1
X
− mβ
n
X
|Q|
−1
Z
|J|
|(b(x) − bJ )γ | dx
J
j=1 γ∈Cjm
−1
1/s0
s0
Z
1/s
s
× |Q|
|T ((b − bJ )γ c f )(x)| χQ (x)dx
Rn
≤ C
m−1
X
− mβ
n
X
|Q|
−1
Z
|J|
|(b(x) − bJ )γ | dx
J
j=1 γ∈Cjm
−1
1/s0
s0
Z
s
× |Q|
1/s
s
|(b(x) − bJ )γ c | |f (x)| χQ (x)dx
Rn
≤ C
m−1
X
X
j=1
γ∈Cjm
−1
− mβ
n
|Q|
−1
Z
s0
|J|
1/s0
|(b(x) − bJ )γ | dx
J
Z
s
× |Q|
s
1/s
|(b(x) − bJ )γ c | |f (x)| dx
Q
≤ C
m−1
X
X
− jβ
n
|J|
−1
|J|
×|J|
(m−j)β
n
s0
|(b(x) − bJ )γ | dx
−1
Z
|J|
st0
|(b(x) − bJ )γ c | dx
J
≤ C
m−1
X
X
1/s0
J
j=1 γ∈Cjm
−
Z
||b~γ ||Lipβ ||b~γ c ||Lipβ Mr (f )(x̃)
j=1 γ∈Cjm
≤ C||~b||Lipβ Mr (f )(x̃).
1/st0 −1
Z
|Q|
st
|f (x)| dx
Q
1/st
22
Z. WANG, L. LIU
For A3,3 , by Hölder’s inequality with exponent 1/s + 1/s0 = 1 and 1/t1 + · · · +
1/tm + 1/t = 1, such that st = r, and Lemma 1, 7, we have
!1/s
Z
m
Y
0
−1− mβ
s
n
A3,3 ≤ C|Q|
|T ( (bj − (bj )J )f1 )(x)| dx
|Q|1/s
Rn
!1/s
m
Y
Z
−1− mβ
n
j=1
≤ C|Q|
|bj (x) − (bj )J |s |f1 (x)|s dx
|Q|1/s
0
Rn j=1
Z Y
m
−1− mβ
n
≤ C|Q|
!1/s
|bj (x) − (bj )J |s |f (x)|s dx
|Q|1/s
0
J j=1
−1− mβ
n
1/s
≤ C|Q|
|Q|
m Y
−1
|J|
−1
Z
× |Q|
stj
1/stj
|bj (x) − (bj )J | dx
J
j=1
Z
1/st
st
|f (x)| dx
|Q|1/s
0
Q
≤ C
m Y
β
−n
|J|
−1
Z
1/stj
|bj (x) − (bj )J | dx)
(|J|
J
j=1
stj
−1
Z
× |Q|
st
1/st
|f (x)| dx
Q
≤ C
m
Y
||bj ||Lipβ Mr (f )(x̃)
j=1
≤ C||~b||Lipβ Mr (f )(x̃).
For A3,4 , choose h such that n/2 < h < n/2 + 1/(1 − a) and n/2 − h + mβ < 0.
By Hölder’s inequality with exponent 1/s + 1/r + 1/2 = 1 and Lemma 1, 2, 3,
we have
m
m
Y
Y
T ( (bj − (bj )J )f2 )(x) − T ( (bj − (bj )J )f2 )(x̃)
j=1
j=1
Z
m
Y
= (K(x, x − y) − K(x̃, x̃ − y)) (bj (y) − (bj )J )f2 (y)dy Rn
j=1
Z
m
Y
≤
|K(x, x − y) − K(x̃, x̃ − y)|
|bj (y) − (bj )J ||f (y)|dy
|y−x̃|>d1−a
≤
∞
X
j=1
Z
|K(x, x − y) − K(x̃, x̃ − y)|
N 1−a ≤|y−x̃|≤(2N +1 d)1−a
N =0 (2 d)
m
Y
×
|bj (y) − (bj )J ||f (y)|dy
j=1
MULTILINEAR COMMUTATOR OF PSEUDO-DIFFERENTIAL OPERATORS
≤
∞ Z
X
|K(x, x − y) − K(x̃, x̃ − y)|
(2N d)1−a ≤|y−x̃|≤(2N +1 d)1−a
N =0
m
Y
×
(|bj (y) − (bj )2(N +1)(1−a) J | + |(bj )2(N +1)(1−a) J − (bj )J |) |f (y)|dy
j=1
≤
∞ X
m X
X
N =0 j=0
|(b2(N +1)(1−a) J − bJ )γ |
γ∈Cjm
Z
|K(x, x − y) − K(x̃, x̃ − y)|
×
(2N d)1−a ≤|y−x̃|≤(2N +1 d)1−a
×|(b(y) − b2(N +1)(1−a) J )γ c ||f (y)|dy
∞ X
m X
X
≤ C
||b~γ ||Lipβ (2N +1 d)j(1−a)β
N =0 j=0 γ∈Cjm
Z
1/2
2
×
|K(x, x − y) − K(x̃, x̃ − y)| dy
(2N d)1−a ≤|y−x̃|≤(2N +1 d)1−a
Z
1/s
s
×
|(b(y) − b2(N +1)(1−a) J )γ c | dy
|y−x̃|≤(2N +1 d)1−a
Z
r
|f (y)| dy
×
≤ C
1/r
|y−x̃|≤(2N +1 d)1−a
∞ X
m X
X
||b~γ ||Lipβ (2N +1 d)j(1−a)β
N =0 j=0 γ∈Cjm
×|x − x̃|(1−a)(h−n/2) (2N +1 d)−h(1−a)+n(1−a)/2 (2N +1 d)(m−j)(1−a)β
×(2N +1 d)−(m−j)(1−a)β
Z
N +1 −n(1−a)
× (2
d)
s
1/s
|(b(y) − b2(N +1)(1−a) J )γ c | dy
|y−x̃|≤(2N +1 d)1−a
N +1
× (2
−n(1−a)
Z
r
1/r
|f (y)| dy
d)
|y−x̃|≤(2N +1 d)1−a
≤ C
∞ X
m X
X
||b~γ ||Lipβ (2N +1 d)j(1−a)β d(1−a)(h−n/2) (2N +1 d)(1−a)(n/2−h)
N =0 j=0 γ∈Cjm
×(2N +1 d)(m−j)(1−a)β ||b~γ c ||Lipβ Mr (f )(x̃)
∞ X
m X
X
≤ C
2(N +1)(1−a)(n/2−h+mβ) dm(1−a)β ||~b||Lipβ Mr (f )(x̃)
N =0 j=0 γ∈Cjm
≤ C|J|
mβ
n
||~b||Lipβ Mr (f )(x̃),
thus
A3,4 ≤ C||~b||Lipβ Mr (f )(x̃).
23
24
Z. WANG, L. LIU
Combining all the estimates, we finish the case m ≥ 2 and d ≤ 1.
In case m ≥ 2 and d > 1, we proceeds the case as follows.
Let f (x) = f1 (x) + f2R(x), with f1 (x) = f (x)χ2Q (x) and f2 (x) = f (x)χ(2Q)c (x),
and let (bj )2Q = |2Q|−1 2Q bj (y)dy for 1 ≤ j ≤ m.
m
Y
(bj (x) − (bj )2Q )T (f )(x)
T~b (f )(x) =
j=1
+
m−1
X
X
(b(x) − b2Q )γ T ((b − b2Q )γ c f )(x)
j=1 γ∈Cjm
m
Y
+(−1) T ( (bj − (bj )2Q )f1 )(x)
m
j=1
m
Y
+(−1)m T (
(bj − (bj )2Q )f2 )(x),
j=1
thus
−1− mβ
n
Z
|Q|
|T~b (f )(x)|dx
Q
−1− mβ
n
≤ |Q|
Z Y
m
|bj (x) − (bj )2Q ||T (f )(x)|dx
Q j=1
+
m−1
X
X
j=1
γ∈Cjm
−1− mβ
n
|Q|
Z
Q
mβ
n
|(b(x) − b2Q )γ ||T ((b − b2Q )γ c f )(x)|dx
Q
+|Q|
+|Q|−1−
Z
−1− mβ
n
Z
m
Y
|T ( (bj − (bj )2Q )f1 )(x)|dx
j=1
m
Y
|T (
Q
(bj − (bj )2Q )f2 )(x)|dx
j=1
= A4,1 + A4,2 + A4,3 + A4,4 .
Similar to A3,1 , A4,1 ≤ C||~b||Lipβ Mr (T (f ))(x̃).
Similar to A3,2 , A4,2 ≤ C||~b||Lipβ Mr (f )(x̃).
Similar to A3,3 , A4,3 ≤ C||~b||Lipβ Mr (f )(x̃).
For A4,4 , by Hölder’s inequality with exponent 1/r + 1/r0 = 1 and Lemma 1,
2, 4, we have
m
Y
T ( (bj − (bj )2Q )f2 )(x)
j=1
MULTILINEAR COMMUTATOR OF PSEUDO-DIFFERENTIAL OPERATORS
Z
m
Y
= K(x, x − y) (bj (y) − (bj )2Q )f2 (y)dy Rn
j=1
Z
m
Y
≤
|K(x, x − y)|
|bj (y) − (bj )2Q ||f (y)|dy
|y−x̃|>2d
Z
j=1
m
Y
|x − y|−2n
≤ C
|bj (y) − (bj )2Q ||f (y)|dy
|y−x̃|>2d
≤ C
≤ C
j=1
∞ Z
X
N =1
∞
X
−2n
|x − y|
2N d≤|y−x̃|≤2N +1 d
m
Y
|bj (y) − (bj )2Q ||f (y)|dy
j=1
(2N +1 d)−2n
N =1
m
Y
(|bj (y) − (bj )2N +1 Q | + |(bj )2N +1 Q − (bj )2Q |)|f (y)|dy
Z
×
≤ C
2N d≤|y−x̃|≤2N +1 d j=1
∞ X
m X
X
N +1 −2n
(2
d)
|(b2N +1 Q − b2Q )γ |
γ∈Cjm
N =1 j=0
Z
|(b(y) − b2N +1 Q )γ c ||f (y)|dy
×
≤ C
|y−x̃|≤2N +1 d
∞ X
m X
X
(2N +1 d)−2n+n ||b~γ ||Lipβ (2N +1 d)jβ (2N +1 d)(m−j)β
N =1 j=0 γ∈Cjm
N +1
×(2
−(m−j)β
d)
N +1
(2
−n
Z
r0
|(b(y) − b2N +1 Q )γ c | dy
d)
|y−x̃|≤2N +1 d
N +1
× (2
−n
Z
r
1/r
|f (x)| dy
d)
|y−x̃|≤2N +1 d
≤ C
∞ X
m X
X
(2N +1 d)−n ||b~γ ||Lipβ (2N +1 d)mβ ||b~γ c ||Lipβ Mr (f )(x̃)
N =1 j=0 γ∈Cjm
≤ C
∞ X
m X
X
2(N +1)(−n+mβ) d−n+mβ ||~b||Lipβ Mr (f )(x̃)
N =1 j=0 γ∈Cjm
≤ C
∞ X
m X
X
2(N +1)(−n+mβ) dmβ ||~b||Lipβ Mr (f )(x̃)
N =1 j=0 γ∈Cjm
≤ C
∞ X
m X
X
2(N +1)(−n+mβ) |Q|
N =1 j=0 γ∈Cjm
≤ C|Q|
mβ
n
||~b||Lipβ Mr (f )(x̃),
mβ
n
||~b||Lipβ Mr (f )(x̃)
1/r0
25
26
Z. WANG, L. LIU
thus
A4,4 ≤ C||~b||Lipβ Mr (f )(x̃).
Combining all the estimates, we finish the case m ≥ 2 and d > 1.
So, for any cube Q and c ∈ CQ and 1 < r < ∞ and f ∈ Lp ,
Z
−1− mβ
n
|Q|
|T~b (x) − c|dx ≤ C||~b||Lipβ (Mr (T (f ))(x̃) + Mr (f )(x̃)) ,
Q
thus
−1− mβ
n
Z
|T~b (x) − c|dx ≤ C||~b||Lipβ (Mr (T (f ))(x̃) + Mr (f )(x̃)) .
sup inf |Q|
Q3x̃ c∈CQ
Q
By Minkowski’s inequality and Lemma 7, 8, we have
Z
−1− mβ
n
||T~b (f )||F̃pmβ,∞ ≈ sup inf |Q|
|T~b (f )(x) − c|dx
Q3x̃ c∈CQ
Q
Lp
≤ C||~b||Lipβ (||Mr (T (f ))(x̃)||Lp + C||Mr (f )(x̃)||Lp )
≤ C||~b||Lip (||T (f )||Lp + ||f ||Lp )
β
≤ C||~b||Lipβ ||f ||Lp .
This completes the proof of Theorem 1.
5. Proof of Theorem 2
Proof. Similar to Theorem 1, we can prove that there exists some constant c and
1 < r < ∞, such that for f ∈ Lp ,
Z
−1
|Q|
|T~b (x) − c|dx ≤ C||~b||Lipβ (Mr,mβ (T (f ))(x̃) + Mr,mβ (f )(x̃)) .
Q
Further, we have
(T~b (f ))# (x̃) ≤ C||~b||Lipβ (Mr,mβ (T (f ))(x̃) + Mr,mβ (f )(x̃)).
By Minkowski’s inequality and Lemma 7, 9, we have
||T~b (f )||Lq ≤ C||M (T~b (f ))(x̃)||Lq
≤ C||(T~b (f ))# (x̃)||Lq
≤ C||~b||Lipβ (||Mr,mβ (T (f ))(x̃)||Lq + ||Mr,mβ (f )(x̃)||Lq )
≤ C||~b||Lip (||T (f )||Lp + ||f ||Lp )
β
≤ C||~b||Lipβ ||f ||Lp .
This completes the proof of Theorem 2.
MULTILINEAR COMMUTATOR OF PSEUDO-DIFFERENTIAL OPERATORS
27
References
1. S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), 7–16.
2. S. Chanillo and A. Torchinsky, Sharp function and weighted Lp estimates for a class of
pseudo-differential operators, Ark. Math. 24 (1986), 1–25.
3. W.G. Chen, Besov estimates for a class of multilinear singular integrals, Acta Math. Sinica
16 (2000), 613–626.
4. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several
variables, Ann. of Math. 103 (1976), 611–635.
5. J. Garcia-Cuerva and M.J.L. Herrero, A theory of Hardy spaces associated to Herz spaces,
Proc. London Math. Soc. 69 (1994), 605–628.
6. S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Math. 16
(1978), 263–270.
7. L.Z. Liu, Boundedness of multilinear operator on Triebel–Lizorkin spaces, Inter J. Math.
Math. Sci. 5 (2004), 259–271.
8. L.Z. Liu, The continuity of commutators on Triebel–Lizorkin spaces, Integral Equations
Operator Theory 49 (2004), 65–76.
9. L.Z. Liu, Boundedness for multilinear Littlewood-Paley operators on Hardy and Herz-Hardy
spaces, Extracta Math. 19 (2004), 243–255.
10. S.Z. Lu, Four lectures on real H p spaces, World Scientific, River Edge, NI, 1995.
11. S.Z. Lu, Q. Wu and D.C. Yang, Boundedness of commutators on Hardy type spaces, Sci.
China Ser. A 45 (2002), 984–997.
12. S.Z. Lu and D.C. Yang, The decomposition of the weighted Herz spaces and its applications,
Sci. China Ser. A 38 (1995), 147–158.
13. S.Z. Lu and D.C. Yang, The weighted Herz type Hardy spaces and its applications, Sci.
China Ser. A 38 (1995), 662–673.
14. M. Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rochbeg and Weiss, Indiana Univ. Math. J. 44 (1995), 1–17.
15. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory
Integrals, Princeton Univ Press, Princeton, 1993.
16. A. Torchinsky and S. Wang, A note on the Marcinkiewicz integral, Colloq. Math.
60/61(1990), 235–243.
1,2
College of Mathematics, Changsha University of Science and Technology,
Changsha, 410077, P. R. of China.
E-mail address: wangzhiwei402@163.com
E-mail address: lanzheliu@163.com