Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
28 (2012), 153–166
www.emis.de/journals
ISSN 1786-0091
CONTINUITY FOR MULTILINEAR COMMUTATOR OF
LITTLEWOOD-PALEY OPERATOR ON BESOV SPACES
TAN LU, CHUANGXIA HUANG AND LIU LANZHE
Abstract. In this paper, we prove the the continuity for the multilinear commutator associated to the Littlewood-Paley operator on the Besov
spaces.
1. Introduction
As the development of the singular integral operators, their commutators
have been well studied (see [1, 2, 3, 14]). From [2, 3, 9, 13], we know that the
commutators and multilinear operators generated by the singular integral operators and the Lipschitz functions are bounded on the Triebel-Lizorkin and
Lebesgue spaces. The purpose of this paper is to introduce the multilinear
commutator associated to the Littlewood-Paley operator and prove the continuity properties for the multilinear commutator on the Besov spaces.
2. Preliminaries and Theorem
First, let us introduce some notations. Throughout this paper, Q will denote
a cube of Rn with sides parallel to the axes. For a locally integrable function
f , the sharp function of f is defined by
Z
1
#
f (x) = sup
|f (y) − fQ |dy,
Q3x |Q| Q
R
where, and in what follows, fQ = |Q|−1 Q f (x)dx. It is well-known that(see
[15, 16])
Z
1
#
f (x) ≈ sup inf
|f (y) − c|dy.
Q3x c∈C |Q| Q
2010 Mathematics Subject Classification. 42B20, 42B25.
Key words and phrases. Multilinear operator; Littlewood-Paley Operator; Besov space.
153
154
TAN LU, CHUANGXIA HUANG AND LIU LANZHE
For β ≥ 0, the Besov space ∧˙ β (Rn ) is the space of functions f such that
[β]+1
||f ||∧˙ β =
sup
∆
f
(x)
h
/|h|β < ∞,
x, h ∈ Rn
h 6= 0
where ∆kh denotes the k-th difference operator(see [13]).
For bj ∈ ∧˙ β (Rn ) (j = 1, · · · , m), set
||~b||∧˙ β =
m
Y
||bj ||∧˙ β .
j=1
Given some functions bj (j = 1, . . . , m) and a positive integer m and 1 ≤
j ≤ m, we denote by Cjm the family of all finite subsets σ = {σ(1), · · ·, σ(j)}
of {1, · · ·, m} of j different elements. For σ ∈ Cjm , set σ c = {1, · · ·, m} \ σ.
For ~b = (b1 , · · ·, bm ) and σ = {σ(1), · · ·, σ(j)} ∈ Cjm , set ~bσ = (bσ(1) , · · ·, bσ(j) ),
bσ = bσ(1) · · · bσ(j) and ||~bσ ||∧˙ β = ||bσ(1) ||∧˙ β · · · ||bσ(j) ||∧˙ β .
Definition 1. Let 0 < p, q ≤ ∞, α ∈ R. For k ∈ Z, set Bk = {x ∈ Rn : |x| ≤
2k } and Ck = Bk \ Bk−1 . Denote by χk the characteristic function of Ck and
χ0 the characteristic function of B0 .
(1) The homogeneous Herz space is defined by
K̇qα, p (Rn ) = {f ∈ Lqloc (Rn \ {0}) : ||f ||K̇qα,
where
"
||f ||K̇qα,
∞
X
=
p
p
< ∞},
#1/p
2kαp ||f χk ||pLq
;
k=−∞
(2) The nonhomogeneous Herz space is defined by
Kqα, p (Rn ) = {f ∈ Lqloc (Rn ) : ||f ||Kqα,
where
"
||f ||Kqα,
p
=
∞
X
p
< ∞},
#1/p
2kαp ||f χk ||pLq + ||f χB0 ||pLq
;
k=1
And the usual modification is made when p = q = ∞.
Definition 2. Let 1 ≤ q < ∞, α ∈ R. The central Campanato space is
defined by (see [17])
CLα, q (Rn ) = {f ∈ Lqloc (Rn ) : ||f ||CLα, q < ∞},
where
−α
||f ||CLα, q = sup |B(0, r)|
r>0
1
|B(0, r)|
Z
1/q
|f (x) − fB(0,r) | dx
.
q
B(0,r)
CONTINUITY FOR MULTILINEAR COMMUTATOR . . .
155
Definition 3. Fix δ > 0. Let ψ be a fixed function which satisfies the following
properties:
R
(1) Rn ψ(x)dx = 0;
(2) |ψ(x)| ≤ C(1 + |x|)−(n+1−δ) ;
(3) |ψ(x + y) − ψ(x)| ≤ C|y| (1 + |x|)−(n+−δ) when 2|y| < |x|.
n+1
We denote that Γ(x) = {(y, t) ∈ R+
: |x − y| < t} and the characteristic
function of Γ(x) by χΓ(x) . The Littlewood-Paley multilinear commutator is
defined by
Z Z
1/2
~b
~b
2 dydt
Sδ (f )(x) =
|Ft (f )(x, y)| n+1
,
t
Γ(x)
where
Z Y
m
~b
(bj (x) − bj (z))ψt (y − z)f (z)dz,
Ft (f )(x, y) =
Rn j=1
and ψt (x) = t−n+δ ψ(x/t) for t > 0. We also define that
Z Z
1/2
2 dydt
|f ∗ ψt (y)| n+1
Sδ (f )(x) =
,
t
Γ(x)
which is the Littlewood-Paley operator(see [16]).
RR
Let H be the space H = h : ||h|| = ( Rn+1 |h(y, t)|2 t(dydt
)1/2 < ∞, then,
n+1)
+
~
for each fixed x ∈ Rn , Ftb (f )(x, y) may be viewed as a mapping from [0, +∞)
to H, and it is clear that
Sδ (f )(x) = ||χΓ(x) Ft (f )(x)||
and
~
~
Sδb (f )(x) = ||χΓ(x) Ftb (f )(x, y)||.
~
Note that when b1 = · · · = bm , Sδb is just the commutator of order m. It is
well known that commutators are of great interest in harmonic analysis and
have been widely studied by many authors (see [2, 3, 4, 7, 6, 5, 8, 9, 10, 13, 14]).
Our main purpose is to study the boundedness properties for the multilinear
commutator on Besov spaces.
Now, we state our theorems as following.
Theorem 1. Let 0 < δ < n, 1 < r < n/δ, 1/s = 1/r − δ/n, 0 < β < 1/2m
~
and bj ∈ ∧˙ β (Rn ) for j = 1, · · · , m. Then Sδb is bounded from Lp (Rn ) to
∧˙ (δ+mβ)−n/p (Rn ) for any n/(δ + mβ) ≤ p ≤ n/δ.
Theorem 2. Let 0 < δ < n, 0 < β < 1/2m, 1 < q1 < n/(δ + mβ), 1/q2 =
1/q1 − (δ + mβ)/n, −n/q2 − 1/2 < α ≤ −n/q2 and bj ∈ ∧˙ β (Rn ) for j =
~
1, · · · , m. Then Sδb is bounded from K̇qα,∞
(Rn ) to CL−α/n−1/q2 ,q2 (Rn ).
1
Remark 1. Theorem 2 also hold for the nonhomogeneous Herz type Hardy
space.
156
TAN LU, CHUANGXIA HUANG AND LIU LANZHE
3. Proofs of Theorems
To prove the theorems, we need the following lemmas.
Lemma 1 (see [13]). For 0 < β < 1, 1 ≤ p ≤ ∞, we have
Z
1
||b||∧˙ β ≈ sup
|b(x) − bQ |dx
1+β/n
Q |Q|
Q
1/p
Z
1
1
p
|b(x) − bQ | dx
≈ sup
β/n
|Q| Q
Q |Q|
Z
1
≈ sup inf
|b(x) − c|dx
c |Q|1+β/n Q
Q
1/p
Z
1
1
p
≈ sup inf
|b(x) − c| dx
.
c |Q|β/n
|Q| Q
Q
Lemma 2 (see [12]). For α < 0, 0 < q < ∞, we have
||f ||K̇qα,
∞
≈ sup 2µα ||f χBµ ||Lq .
µ∈Z
Lemma 3. Let 0 < η < n, 1 < p < n/η. Suppose b ∈ ∧˙ β (Rn ), then
|b2k+1 B − bB | ≤ C||b||∧˙ β k|2k+1 B|β/n for k ≥ 1.
Proof.
|b2k+1 B − bB | ≤
k
X
|b2j+1 B − b2j B |
j=0
Z
1
|b(y) − b2j+1 B |dy
≤
|2j B| 2j B
j=0
1/p
Z
k X
1
p
|b(y) − b2j+1 B | dy
≤C
j+1 B|
|2
j+1 B
2
j=0
k
X
≤ C||b||∧˙ β
k
X
|2j+1 B|β/n ≤ C||b||∧˙ β k|2k+1 B|β/n .
j=0
Lemma 4 (see [16]). Let 0 < δ < n, 1 < p < n/δ and 1/q = 1/p − δ/n. Then
Sδ is bounded from Lp (Rn ) to Lq (Rn ).
Lemma 5 (see [7]). Let 0 ≤ η < n, 1 < r < n/η, 1/r − 1/s = η/n and
~
bj ∈ ∧˙ β (Rn ) for j = 1, · · · , m. Then Sδb is bounded from Lr (Rn ) to Ls (Rn ).
Proof of Theorem 1. It is only to prove that there exists a constant C0 such
that
Z
1
~
|Sδb (f )(x) − C0 |dx ≤ C||f ||Lp .
1+(δ+mβ)/n−1/p
|Q|
Q
CONTINUITY FOR MULTILINEAR COMMUTATOR . . .
157
Fix a cube Q, Q = Q(x0 , d), we decompose f into f = f1 + f2 with f1 =
f χQ , f2 = f χ(Rn \Q) .
When m = 1, for C0 = Sδ (((b1 )Q − b1 )f2 )(x0 ), we have
Sδb1 (f )(x) = (b1 (x)−(b1 )Q )Sδ (f )(x)−Sδ ((b1 −(b1 )Q )f1 )(x)−Sδ ((b1 −(b1 )Q )f2 )(x).
Then
|Sδb1 (f )(x) − Sδ (((b1 )Q − b1 )f2 )(x0 )| =
= ||χΓ(x) Ftb1 (f )(x, y)|| − ||χΓ(x0 ) Ft (((b1 )Q − b1 )f2 )(y)||
≤ ||χΓ(x) Ftb1 (f )(x, y) − χΓ(x0 ) Ft (((b1 )Q − b1 )f2 )(y)||
≤ ||χΓ(x) (b1 (x) − (b1 )Q )Ft (f )(y)|| + ||χΓ(x) Ft ((b1 − (b1 )Q )f1 )(y)||
+||χΓ(x) Ft ((b1 − (b1 )Q )f2 )(y) − χΓ(x0 ) Ft ((b1 − (b1 )Q )f2 )(y)||
= A(x) + B(x) + C(x).
For A(x), for 1 < p < q < n/δ, 1/q = 1/p − δ/n, by the boundness of Sδ from
Lp (Rn ) to Lq (Rn ), by Hölder’s inequality with exponent1/q + 1/q 0 = 1 and
Lemma 1, we have
Z
1
|A(x)|dx ≤
|Q|1+(δ+β)/n−1/p Q
1/q
1/q0 Z
Z
1
q
q0
|Sδ (f )(x)| dx
|(b1 (x) − (b1 )Q )| dx
≤C
|Q|1+(δ+β)/n−1/p
Q
Q
1/p
1/q0 Z
Z
0
|Q|β/n+1/q
1
1
p
q0
≤C
|f (x)| dx
|(b1 (x) − (b1 )Q )| dx
|Q|1+(δ+β)/n−1/p |Q|β/n |Q| Q
Q
|Q|1+(δ+β)/n−1/p
||b1 ||∧˙ β ||f ||Lp
|Q|1+(δ+β)/n−1/p
≤ C||b1 ||∧˙ β ||f ||Lp .
≤C
For B(x), denoting p = rt, 1 < r < s < n/δ, 1/s = 1/r − δ/n, by the
boundness of Sδ from Lr (Rn ) to Ls (Rn ), by Hölder’s inequality with exponent
1/t + 1/t0 = 1 and Lemma 1, we have
Z
1
|B(x)|dx ≤
|Q|1+(δ+β)/n−1/p Q
1/s
Z
1
1
s
≤C
|Sδ (b1 (x) − (b1 )Q )f χQ (x)| dx
|Q|(δ+β)/n−1/p |Q| Rn
Z
1/r
1
r
≤C
|(b1 (x) − (b1 )Q )f (x)| dx
|Q|(δ+β)/n−1/p+1/s
Q
Z
1/rt0 Z
1/rt
1
rt0
rt
≤C
|b1 (x) − (b1 )Q | dx
|f (x)| dx
|Q|(δ+β)/n−1/p+1/s
Q
Q
158
TAN LU, CHUANGXIA HUANG AND LIU LANZHE
0
|Q|β/n+1/rt
1
≤C
(δ+β)/n−1/p+1/s
|Q|
|Q|β/n
1/rt0 Z
1/rt
Z
1
rt0
rt
|b1 (x) − (b1 )Q | dx
|f (x)| dx
|Q| Q
Q
|Q|(δ+β)/n−1/p+1/s
||b1 ||∧˙ β ||f ||Lp
|Q|(δ+β)/n−1/p+1/s
≤ C||b1 ||∧˙ β ||f ||Lp .
≤C
For C(x), by the Minkowski’s inequality, we have
C(x) ≤
"Z Z
≤
Z
Z
n+1
R+
≤C
2
|χΓ(x) − χΓ(x0 ) ||b1 (z) − (b1 )Q ||ψt (y − z)||f (z)|
Qc
dydt
tn+1
#1/2
|b1 (z) − (b1 )Q ||f (z)|
Qc
Z Z
1/2
Z Z
1−n
1−n
t
dydt
t
dydt
dz
× −
2n+2−2δ
2n+2−2δ |x0 −y|≤t (t + |y − z|)
|x−y|≤t (t + |y − z|)
Z
≤
|b1 (z) − (b1 )Q ||f (z)| ×
Qc
1/2
ZZ dydt 
1
1
(t + |x + y − z|)2n+2−2δ − (t + |x0 + y − z|)2n+2−2δ tn−1  dz


×
|y|≤t,|x+y−z|≤t
Z
≤
Qc


|b1 (z) − (b1 )2Q ||f (z)| 
Z
Z
1/2
|x − x0 |t

dydt dz,
(t + |x + y − z|)2n+3−2δ
1−n
|y|≤t,|x+y−z|≤t
by 2t + |x + y − z| ≥ 2t + |x − z| − |y| ≥ t + |x − z| , for |y| ≤ t and
Z ∞
tdt
= C|x − z|−2n−1+2δ .
2n+3−2δ
(t
+
|x
−
z|)
0
Thus, for x ∈ Q, by Hölder’s inequality with exponent 1/p + 1/p0 = 1 and
Lemma 1, 3, we have
C(x) ≤
Z
≤C
1/2
22n+3 |x0 − x|t1−n dydt
dz
|b1 (z) − (b1 )Q ||f (z)|
2n+3−2δ
Qc
|y|≤t (2t + 2|x + y − z|)
1/2
Z Z
Z
t1−n dydt
1/2
≤C
dz
|b1 (z) − (b1 )Q ||f (z)||x − x0 |
2n+3−2δ
Qc
|y|≤t (2t + |x + y − z|)
Z Z
1/2
Z
t1−n dydt
1/2
≤C
|b1 (z) − (b1 )Q ||f (z)||x − x0 |
dz
2n+3−2δ
Qc
|y|≤t (t + |x − z|)
Z Z
CONTINUITY FOR MULTILINEAR COMMUTATOR . . .
Z
Z
≤C
≤C
≤C
|b1 (z) − (b1 )Q ||f (z)||x − x0 |
Qc
∞
XZ
k=0
∞
X
2
k=0
Z
×
≤C
∞
0
tdt
(t + |x − z|)2n+3−2δ
1/2
dz
|x0 − x|1/2 |x0 − z|−(n+1/2−2δ) |b1 (z) − (b1 )Q ||f (z)|dz
2k+1 Q\2k Q
−k/2
1/2
159
1
k+1
|2 Q|1−δ/n
Z
|f (z)| dz
Z
×
2k+1 Q
p0
|b1 (z) − (b1 )2Q |
2k+1 Q
∞
X
1
2−k/2 k+1 1−δ/n
|2 Q|
k=0
1/p
p
1/p0
dz
Z
1/p
×
|f (z)| dz
p
2k+1 Q
p0
1/p0
p0
(|b1 (z) − (b1 )2k+1 Q | + |(b1 )2k+1 Q − (b1 )2Q | )dz
×
≤C
2k+1 Q
∞
X
−k/2
2
k=0
1
|2k+1 Q|1−δ/n
1/p
Z
|f (z)| dz
p
×
2k+1 Q
0
× |2k+1 Q|β/n+1/p
1/p0
Z
1
1
p0
|(b1 (z) − (b1 )2k+1 Q )| dz
|2k+1 Q|β/n |2k+1 Q| 2k+1 Q
k+1
1/p0
+|(b1 )2k+1 Q − (b1 )Q ||2 Q|
≤C
≤C
≤C
∞
X
k=0
∞
X
k=0
∞
X
0
−k/2
2
0
|2k+1 Q|β/n+1/p ||b1 ||∧˙ β + k|2k+1 Q|β/n+1/p ||b1 ||∧˙ β
||f ||Lp
|2k+1 Q|1−δ/n
0
k2−k/2 |2k+1 Q|β/n+1/p −1+δ/n ||b1 ||∧˙ β ||f ||Lp
0
k2k(−1/2+δ+β−n/p) |Q|β/n+1/p −1+δ/n ||b1 ||∧˙ β ||f ||Lp
k=0
(δ+β)/n−1/p
≤ C|Q|
|||b1 ||∧˙ β ||f ||Lp .
Thus,
1
|Q|1+(δ+β)/n−1/p
Z
||b1 ||∧˙ β ||f ||Lp
|C(x)|dx ≤ C
|Q|1+(δ+β)/n−1/p
Q
≤ C||b1 ||∧˙ β ||f ||Lp .
Z
|Q|(δ+β)/n−1/p dx
Q
This completes the case m = 1.
Now, we consider the Case m ≥ 2. we have, for b = (b1 , · · · , bm ),
~
Ftb (f )(x, y) =
160
TAN LU, CHUANGXIA HUANG AND LIU LANZHE
Z
m
Y
=
=
[(bj (x) − (bj )Q ) − (bj (z) − (bj )Q )] ψt (y − z)f (z)dz
Rn j=1
m X
X
Z
m−j
(−1)
(b(x) − (b)Q )σ
=
m
Y
(b(z) − (b)Q )σc ψt (y − z)f (z)dz
Rn
j=0 σ∈Cjm
m
Y
(bj (x) − (bj )Q )Ft (f )(y) + (−1) Ft ( (bj − (bj )Q ))f )(y)
m
j=1
+
=
m−1
X
X
j=1
σ∈Cjm
m
Y
Z
(−1)m−j (b(x) − (b)Q )σ
(b(z) − b(x))σc ψt (y − z)f (z)dz
Rn
m
Y
(bj (x) − (bj )Q )Ft (f )(y) + (−1) Ft ( (bj − (bj )Q ))f )(y)
m
j=1
+
j=1
m−1
X
j=1
X
~
(b(x) − (b)Q )σ Ftbσc (f )(x, y).
j=1 σ∈Cjm
m
Y
Thus, set C0 = Sδ ( (bj − (bj )2Q )f2 )(x0 ),
j=1
~
|Sδb (f )(x)
m
Y
− Sδ ( (bj − (bj )2Q )f2 )(x0 )| ≤
j=1
m
Y
≤ ||χΓ(x) (bj (x) − (bj )2Q )Ft (f )(x)||
j=1
+
m−1
X
X
~
||χΓ(x) (b(x) − (b)2Q )σ Ftbσc (f )(x)||
j=1 σ∈Cjm
m
Y
+ ||χΓ(x) Ft ( (bj − (bj )2Q )f1 )(x)||
+ ||χΓ(x) Ft (
j=1
m
Y
m
Y
(bj − (bj )2Q )f2 )(x) − χΓ(x0 ) Ft ( (bj − (bj )2Q )f2 )(x0 )||
j=1
j=1
= I1 (x) + I2 (x) + I3 (x) + I4 (x).
For I1 (x), for 1 < p < q < n/δ, 1/q = 1/p − δ/n, by the boundness of Sδ from
0
+1/q =
Lp (Rn ) to Lq (Rn ), by Hölder’s inequality with exponent1/q10 +· · ·+1/qm
1 and Lemma 1, we have
Z
1
|I1 (x)|dx =
|Q|1+(δ+mβ)/n−1/p Q
CONTINUITY FOR MULTILINEAR COMMUTATOR . . .
=
Z
1
|
|Q|1+(δ+mβ)/n−1/p
≤C
Q
m
Y
(bj (x) − (bj )Q )Sδ (f )(x)|dx
j=1
m Z
Y
1
|Q|1+(δ+mβ)/n−1/p
j=1
161
qj0
|(bj (x) − (bj )Q )| dx
1/qj0 Z
Q
1/q
|Sδ (f )(x)| dx
q
Q
0
0
m
|Q|mβ/n+1/q1 +···qm Y
1
1
≤C
|Q|1+(δ+mβ)/n−1/p j=1 |Q|mβ/n |Q|
Z
1/p
p
×
|f (x)| dx
Z
qj0
|(b1 (x) − (b1 )Q )| dx
1/qj0
×
Q
Q
|Q|1+(δ+mβ)/n−1/p ~
||b||∧˙ β ||f ||Lp
|Q|1+(δ+mβ)/n−1/p
≤ C||~b||∧˙ ||f ||Lp .
≤C
β
For I2 (x), denoting p = rt, 1 < r < s < n/δ, 1/s = 1/r−δ/n, by the boundness
of Sδ from Lr (Rn ) to Ls (Rn ), by Hölder’s inequality with 1/s0 + 1/s = 1,
1/t0 + 1/t = 1 and Lemma 1, we have
Z
1
|Q|1+(δ+mβ)/n−1/p
=
m−1
X
X
j=1
σ∈Cjm
≤C
m−1
X
|I2 (x)|dx =
Q
Z
1
|Q|1+(δ+mβ)/n−1/p
X
j=1 σ∈Cjm
|(b(x) − bQ )σ Sδ ((b − bQ )σc f )(x)|dx
Q
1
1/s0
|(b(x) − bQ )σ | dx
×
Z
|Q|1+(δ+mβ)/n−1/p
s0
Q
Z
1/s
s
Sδ ((b − bQ )σc f )(x)| dx
×
Q
≤C
m−1
X
X
j=1 σ∈Cjm
1/s0
Z
0
1
1
|Q||σ|β/n+1/s
s0
|(b(x) − bQ )σ | dx
|Q|1+(δ+mβ)/n−1/p |Q||σ|β/n |Q| Q
Z
1/r
r
×
|(b − bQ )σc f (x)| dx
Q
≤C
m−1
X
j=1
1/s0
Z
X |Q||σ|β/n+1/s0
1
1
s0
|(b(x) − bQ )σ | dx
1+mβ/n−1/p |Q||σ|β/n |Q|
|Q|
Q
m
σ∈C
j
Z
1/rt0 Z
1/rt
rt0
rt
×
|(b(x) − bQ )σc | dx
|f (x)| dx
Q
Q
162
TAN LU, CHUANGXIA HUANG AND LIU LANZHE
≤C
m−1
X
X
j=1 σ∈Cjm
1/s0
Z
0
1
1
|Q||σ|β/n+1/s
s0
|(b(x) − bQ )σ | dx
|Q|1+(δ+mβ)/n−1/p |Q||σ|β/n |Q| Q
|σ c |β/n+1/rt0
×|Q|
≤C
m−1
X
1
|Q||σc |β/n
1
|Q|
1/rt0
|(b(x) − bQ )σc | dx
||f ||Lp
Z
rt0
Q
X |Q|
||~bσ ||∧˙ β ||~bσc ||∧˙ β ||f ||Lp
1+(δ+mβ)/n−1/p
|Q|
m
σ∈C
mβ/n+1/s0 +1/rt0
j=1
j
≤ C||~b||∧˙ β ||f ||Lp .
For I3 (x), for 1 < r < s < n/δ, 1/s = 1/r − δ/n, by the boundness of Sδ from
Lr (Rn ) to Ls (Rn ), taking 1 < r < p < ∞, p = rt, 1/t1 + · · · + 1/tm + 1/t = 1,
by Hölder’s inequality and Lemma 1, we have
Z
1
|I3 (x)|dx
|Q|1+(δ+mβ)/n−1/p Q
1/s
Z
m
Y
1
1
s
≤
|Sδ ( (bj − (bj )Q ))f χB (x)| dx
|Q|(δ+mβ)/n−1/p |Q| Rn
j=1
1/r
Z Y
m
1
r
≤
|( (bj − (bj )Q )f (x)| dx
|Q|(δ+mβ)/n−1/p+1/s
Q j=1
1/rtj Z
1/rt
m Z
Y
1
rtj
rt
≤
|bj (x) − (bj )Q | dx
|f (x)| dx
|Q|(δ+mβ)/n−1/p+1/s j=1
Q
Q
1/rtj
Z
m
|Q|mβ/n+1/rt1 +···1/rtm Y
1
1
rtj
≤
|bj (x) − (bj )Q | dx
||f ||Lp
|Q|(δ+mβ)/n−1/p+1/s j=1 |Q|mβ/n |Q| Q
≤ C||~b||∧˙ β ||f ||Lp .
For I4 (x), similar to the proof of C(x) in the case m = 1, we get
Z
m
Y
1/2
−(n+1/2−δ)
I4 (x) ≤ C
|x0 − x| |x0 − z|
| (bj (z) − (bj )Q )||f (z)|dz.
(2Q)c
j=1
Thus, we choose 1 < pj < ∞, j = 1, · · · , m, 1/p1 + · · · + 1/pm + 1/p = 1, by
Hölder’s inequality and Lemma 1, 3, we have
I4 (x) ≤
∞ Z
X
≤C
k=0
≤C
∞
X
k=0
|x0 − x|
1/2
−(n+1/2−δ)
|x0 − z|
2k+1 Q\2k Q
2−k/2
1
|2k+1 Q|1−δ/n
j=1
Z
1/p
|f (z)|p dz
2k+1 Q
m
Y
| (bj (z) − (bj )Q )||f (z)|dz
×
CONTINUITY FOR MULTILINEAR COMMUTATOR . . .
×
m Z
Y
p0j
|(bj (z) − (bj )Q )| dz
1/p0j
2k+1 Q
j=1
≤C
∞
X
−k/2
2
k=0
×
163
m Z
Y
1
k+1
|2 Q|1−δ/n
Z
1/p
|f (z)| dz
p
2k+1 Q
p0j
p0j
(|bj (z) − (bj )2k+1 Q | + |(bj )2k+1 Q − (bj )2Q | )dz
1/p0j
2k+1 Q
j=1
≤C
∞
X
−k/2
2
k=0
1
|2k+1 Q|1−δ/n
Z
1/p
|f (z)| dz
p
2k+1 Q
m Y
0
×
|2k+1 Q|mβ/n+1/p
1/p0
Z
1
1
p0
|(bj (z) − (bj )2k+1 Q )| dz
|2k+1 Q|mβ/n |2k+1 Q| 2k+1 Q
j=1
k+1
1/p0
+|(bj )2k+1 Q − (bj )Q ||2 Q|
≤C
∞
X
k=0
2−k/2
1
|2k+1 Q|1−δ/n
×
m Y
k+1
mβ/n+1/p0
k+1
mβ/n+1/p0
×
|2 Q|
||bj ||∧˙ β + k|2 Q|
||bj ||∧˙ β ||f ||Lp
j=1
≤C
∞
X
0
k2k(−1/2+δ+mβ−n/p) |Q|mβ/n+1/p −1+δ/n ||~b||∧˙ β ||f ||Lp
k=0
≤ C|Q|(δ+mβ)/n−1/p |||~b||∧˙ β ||f ||Lp .
Thus,
1
Z
|Q|1+(δ+mβ)/n−1/p
≤ C||~b||∧˙ β ||f ||Lp
|I4 (x)|dx ≤
Q
Z
1
|Q|1+(δ+mβ)/n−1/p
|Q|(δ+mβ)/n−1/p dx ≤ C||~b||∧˙ β ||f ||Lp .
Q
Proof of Theorem 2. Fix a ball B = B(0, l), there exists 0 ∈ Z such that
20 −1 ≤ l < 20 . We choose x0 such that 2l < |x0 | < 3l. It is only to prove that
1/q2
Z
1
~b
~b
0 (α+n/q2 )
q2
2
|S (f )(x) − Sδ (f2 )(x0 )| dx
≤ C||f ||K̇qα,∞ .
1
20 n |x|<20 δ
We write, for f1 = f χ4B0 and f2 = f χRn \4B0 , then
~
~
~
~
~
|Sδb (f )(x) − Sδb (f2 )(x0 )| ≤ |Sδb (f1 )(x)| + |Sδb (f2 )(x) − Sδb (f2 )(x0 )|.
164
TAN LU, CHUANGXIA HUANG AND LIU LANZHE
So
1
0 (α+n/q2 )
2
Z
20 n
≤2
0 (α+n/q2 )
|x|<20
1
~
|Sδb (f )(x)
Z
−
~
Sδb (f2 )(x0 )|q2 dx
1/q2
1/q2
~
|Sδb (f1 )(x)|q2 dx
20 n |x|<20
1/q2
Z
1
~b
~b
q2
0 (α+n/q2 )
+2
|S (f2 )(x) − Sδ (f2 )(x0 )| dx
20 n |x|<20 δ
= J1 + J2 .
~
For J1 , by the (Lq1 , Lq2 )-boundedness of Sδb (see Lemma 4) and Lemma 2, we
get
Z
1/q1
0 (α+n/q2 ) −0 n/q2
q1
J1
≤ C2
2
|f1 (x)| dx
Rn
≤ C2
0 α
||f χB0 ||Lq1
≤ C||f ||K̇qα,∞ .
1
For J2 , similar to the estimates of Theorem 1, set 1/v1 + · · · + 1/vm + 1/q1 = 1,
by Hölder’s inequality and recall that −1/q2 < α, 1/q2 = 1/q1 − (δ + mβ)/n,
we obtain
~
~
|Sδb (f2 )(x) − Sδb (f2 )(x0 )| ≤
m
m
Y
Y
≤ |Sδ ( (bj − (bj )B )(f2 )(x) − Sδ ( (bj − (bj )B )(f2 )(x0 )|
+|
j=1
m
Y
j=1
(bj (x) − (bj )B )||Sδ (f2 )(x) − Sδ (f2 )(x0 )| = W1 (x) + W2 (x).
j=1
For W1 (x), similar to the proof of C(x) in Theorem 1, by the Minkowski’s
inequality, we have
m
∞ Z
Y
X
1/2
−(n+1/2−δ)
|x0 − x| |x0 − z|
| (bj (z) − (bj )B )||f (z)|dz
W1 (x)
≤C
≤C
k=1 B0 +k
∞ Z
X
j=1
20 /2
m
Y
| (bj (z) − (bj )B )||f (z)|dz
(0 +k)(n+1/2−δ)
2
j=1
k=1
1/vj
Z
∞
m
/2
X
Y
20
vj
|bj (z) − (bj )B | dy
×
≤C
(0 +k)(n+1/2−δ)
2
B
+k
j=1
0
k=1
Z
1/q1
×
|f (z)q1 dz
B0 +k
B0 +k
CONTINUITY FOR MULTILINEAR COMMUTATOR . . .
≤C
∞
X
20 /2
2(0 +k)(n+1/2−δ)
k=1
165
2(0 +k)(mβ+n/v1 +···n/vm )
1/vj
Z
1
1
vj
×
||f χ0 +k ||Lq1
|bj (z) − (bj )B | dz
mβ/n |B
|B
|
|
+k
+k
B
0
0
0 +k
j=1
m
Y
≤C
≤C
∞
X
2k(mβ+n−n/q1 −n+δ−1/2) 20 (mβ+n−n/q1 −n+δ) ||~b||∧˙ β ||f χ0 +k ||Lq1
k=1
∞
X
2k(δ+mβ−n/q1 −α−1/2) 20 (δ+mβ−n/q1 −α) ||~b||∧˙ β 2(0 +k)α ||f χ0 +k ||Lq1
k=1
≤ C20 (−n/q2 −α) ||~b||∧˙ β ||f ||K̇qα,∞ .
1
For W2 (x), by Hölder’s inequality with 1/q10 + 1/q1 = 1, we have,
m
∞ Z
Y
X
||x − z|−n+δ − |x0 − z|−n+δ ||f (z)|dz
W2 (x)
≤ C| (bj (x) − (bj )B )|
≤ C|
j=1
m
Y
k=1 B0 +k
∞ Z
X
j=1
k=1
(bj (x) − (bj )B )|
m
∞
Y
X
≤ C| (bj (x) − (bj )B )|
≤ C|
×2
j=1
m
Y
k=1
∞
X
j=1
k=1
(bj (x) − (bj )B )|
|x − z|−n+δ−1 |x − x0 ||f (z)|dz
B0 +k
20
2(0 +k)(n+1−δ)
!1/q1
Z
2(0 +k)(n−n/q1 )
|f (z)|q1 dz
B0 +k
2k(δ−n−1−α+n−n/q1 ) 20 (δ−n−α+n−n/q1 ) ×
||f χ0 +k ||Lq1
m
Y
0 (δ−α−n/q1 )
≤ C2
| (bj (x) − (bj )B )||f ||K̇qα,∞ .
(0 +k)α
1
j=1
Thus, by Hölder’s inequality with 1/v1 + · · · + 1/vm = 1, we have
1/q2
Z
1
q2
0 (α+n/q2 )
|W2 (x)| dx
2
20 n |x|<20
1/q2
Z
m
Y
1
q2
0 (α+n/q2 ) 0 (δ−α−n/q1 )
| (bj (x) − (bj )B )| dx
||f ||K̇qα,∞
≤ C2
2
1
20 n |x|<20 j=1
≤ C20 (α+n/q2 ) 20 (δ−α−n/q1 ) 20 (mβ+n/q2 (1/v1 +···+1/vm ))
1/q2 vj Z
m Y
1
1
1
q2 v j
× 0 n
|(bj (x) − (bj )B )| dx
||f ||K̇qα,∞
1
2 |B|mβ/n j=1 |B| |x|<20
≤ C||~b||∧˙ β ||f ||K̇qα,∞ .
1
166
TAN LU, CHUANGXIA HUANG AND LIU LANZHE
Thus,
J2 ≤ C||~b||∧˙ β ||f ||K̇qα,∞ .
1
References
[1] S. Chanillo. A note on commutators. Indiana Univ. Math. J., 31(1):7–16, 1982.
[2] W. Chen. A Besov estimate for multilinear singular integrals. Acta Math. Sin. (Engl.
Ser.), 16(4):613–626, 2000.
[3] S. Janson. Mean oscillation and commutators of singular integral operators. Ark. Mat.,
16(2):263–270, 1978.
[4] L. Lanzhe. Triebel-Lizorkin space estimates for multilinear operators of sublinear operators. Proc. Indian Acad. Sci. Math. Sci., 113(4):379–393, 2003.
[5] L. Liu. Boundedness for multilinear Littlewood-Paley operators on Hardy and HerzHardy spaces. Extracta Math., 19(2):243–255, 2004.
[6] L. Liu. Boundedness of multilinear operators on Triebel-Lizorkin spaces. Int. J. Math.
Math. Sci., (5-8):259–271, 2004.
[7] L. Liu. The continuity of commutators on Triebel-Lizorkin spaces. Integral Equations
Operator Theory, 49(1):65–75, 2004.
[8] L.-z. Liu, S.-z. Lu, and J.-s. Xu. Boundedness for commutators of Littlewood-Paley
operators. Adv. Math. (China), 32(4):473–480, 2003.
[9] S. Lu, Y. Meng, and Q. Wu. Lipschitz estimates for multilinear singular integrals. II.
Acta Math. Sci. Ser. B Engl. Ed., 24(2):291–300, 2004.
[10] S. Lu, Q. Wu, and D. Yang. Boundedness of commutators on Hardy type spaces. Sci.
China Ser. A, 45(8):984–997, 2002.
[11] S. Z. Lu. Four lectures on real H p spaces. World Scientific Publishing Co. Inc., River
Edge, NJ, 1995.
[12] S. Z. Lu and D. C. Yang. The weighted Herz-type Hardy space and its applications.
Sci. China Ser. A, 38(6):662–673, 1995.
[13] M. Paluszyński. Characterization of the Besov spaces via the commutator operator of
Coifman, Rochberg and Weiss. Indiana Univ. Math. J., 44(1):1–17, 1995.
[14] C. Pérez and R. Trujillo-González. Sharp weighted estimates for multilinear commutators. J. London Math. Soc. (2), 65(3):672–692, 2002.
[15] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory
integrals, volume 43 of Princeton Mathematical Series. Princeton University Press,
Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.
[16] A. Torchinsky. Real-variable methods in harmonic analysis, volume 123 of Pure and
Applied Mathematics. Academic Press Inc., Orlando, FL, 1986.
[17] D. C. Yang. The central Campanato spaces and its application. Approx. Theory Appl.
(N.S.), 10(4):85–99, 1994.
Received February 13, 2011.
College of Mathematics,
Changsha University of Science and Technology,
Changsha 410077, P. R. of China
E-mail address: lanzheliu@163.com