paper 16-18-2009. 128KB Jun 04 2011 12:03:07 AM
ACTA UNIVERSITATIS APULENSIS
No 18/2009
SHARP FUNCTION ESTIMATE FOR MULTILINEAR
COMMUTATOR OF SINGULAR INTEGRAL WITH VARIABLE
CALDERÓN-ZYGMUND KERNEL
Zhiqiang Wang and Lanzhe Liu
Abstract: In this paper, we prove the sharp function inequality for the
multilinear commutator related to the singular integral operator with variable
Calderón-Zygmund kernel. By using the sharp inequality, we obtain the Lp norm inequality for the multilinear commutator.
2000 Mathematics Subject Classification: 42B20, 42B25.
1. Introduction
As the development of singular integral operators, their commutators have
been well studied (see [1-4]). Let T be the Calderón-Zygmund singular integral operator, a classical result of Coifman, Rocherberg and Weiss (see [3])
states that commutator [b, T ](f ) = T (bf ) − bT (f )(where b ∈ BM O(Rn )) is
bounded on Lp (Rn ) for 1 < p < ∞. In [6-8], the sharp estimates for some
multilinear commutators of the Calderón-Zygmund singular integral operators
are obtained. The main purpose of this paper is to prove the sharp function
inequality for the multilinear commutator related to the singular integral operator with variable Calderón-Zygmund kernel. By using the sharp inequality,
we obtain the Lp -norm inequality for the multilinear commutator.
2. Notations and Results
First let us introduce some notations (see [4][8][9]). In this paper, Q will
denote a cube
of Rn with sides parallel to the axes, and for a cube Q let
R
fQ = |Q|−1 Q f (x)dx and the sharp function of f is defined by
1 Z
f (x) = sup
|f (y) − fQ |dy.
Q∋x |Q| Q
#
169
Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
It is well-known that (see [4])
1 Z
f (x) ≈ sup inf
|f (y) − C|dy.
Q∋x c∈C |Q| Q
#
We say that b belongs to BM O(Rn ) if b# belongs to L∞ (Rn ) and define
||b||BM O = ||b# ||L∞ . It has been known that (see [9])
||b − b2k Q ||BM O ≤ Ck||b||BM O .
Let M be the Hardy-Littlewood maximal operator, that is that
−1
M (f )(x) = sup |Q|
x∈Q
Z
Q
|f (y)|dy;
we write that Mp (f ) = (M (|f |p ))1/p for 0 < p < ∞.
For bj ∈ BM O(Rn )(j = 1, · · · , m), set
||~b||BM O =
m
Y
||bj ||BM O .
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)} ∈ C m , set ~bσ = (bσ(1) , · · ·, bσ(j) ),
j
~
bσ = bσ(1) · · · bσ(j) and ||bσ ||BM O = ||bσ(1) ||BM O · · · ||bσ(j) ||BM O .
In this paper, we will study some multilinear commutators as follows.
Definition 1. Let K(x) = Ω(x)/|x|n : Rn \ {0} → R. K is said to be a
Calderón-Zygmund kernel if
(a) Ω ∈ C ∞ (Rn \ {0});
(b) Ω
is homogeneous of degree zero;
R
(c) Σ Ω(x)xα dσ(x) = 0 for all multi-indices α ∈ (N ∪ {0})n with |α| = N ,
where Σ = {x ∈ Rn : |x| = 1} is the unit sphere of Rn .
Definition 2. Let K(x, y) = Ω(x, y)/|y|n : Rn × (Rn \ {0}) → R. K is
said to be a variable Calderón-Zygmund kernel if
(d) K(x, ·) is a Calderón-Zygmund
kernel for a.e. x ∈ Rn ;
∂ | γ|
= M < ∞.
(e) max|γ|≤2n ∂ γ y Ω(x, y) ∞ n
L (R ×Σ)
170
Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
Suppose bj (j = 1, · · · , m) are the fixed locally integrable functions on Rn .
Let T be the singular integral operator with variable Calderón-Zygmund kernel
as
Z
K(x, x − y)f (y)d(y),
T (f )(x) =
Rn
Ω(x, x − y)
and that Ω(x, y)/|y|n is a variable Calderón|x − y|n
Zygmund kernel. The multilinear commutator of singular integral with variable
Calderón-Zygmund kernel is defined by
where K(x, x − y) =
T~b (f )(x) =
Z
m
Y
Rn j=1
(bj (x) − bj (y))K(x, x − y)f (y)dy.
Note that when b1 = ··· = bm , T~b is just the m order commutator (see [1][5]).
It is well known that commutators are of great interest in harmonic analysis
and have been widely studied by many authors (see [1-3][5-8]). Our main
purpose is to establish the sharp inequality for the multilinear commutator.
Now we state our theorems as following.
Theorem 1. Let bj ∈ BM O(Rn ) for j = 1, · · · , m. Then for any 1 <
r < ∞, there exists a constant C > 0 such that for any f ∈ C0∞ (Rn ) and any
x ∈ Rn ,
(T~b (f ))# (x) ≤ C||~b||BM O Mr (f )(x) +
m X
X
j=1 σ∈Cjm
Mr (T~bσc (f ))(x) .
Theorem 2. Let bj ∈ BM O(Rn ) for j = 1, · · · , m. Then T~b is bounded on
L (Rn ) for 1 < p < ∞.
p
3. Proof of Theorem
To prove the theorems, we need the following lemmas.
Lemma 1. (see[10]) Let 1 < p < ∞ and T be the singular integral operator
with variable Calderón-Zygmund kernel. Then T is bounded on Lp (Rn ).
Lemma 2. Let 1 < r < ∞, bj ∈ BM O(Rn ) for j = 1, · · · , k. Then
k
k
Y
1 Z Y
||bj ||BM O
|bj (y) − (bj )Q |dy ≤ C
|Q| Q j=1
j=1
171
Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
and
1/r
k
1 Z Y
|bj (y) − (bj )Q |r dy
|Q| Q j=1
≤C
k
Y
||bj ||BM O .
j=1
Proof. Choose 1 < pj < ∞ j = 1, · · · , k such that 1/p1 + · · · + 1/pk = 1, we
obtain, by Hölder’s inequality,
k
k
Y
1 Z Y
1 Z
|bj (y)−(bj )Q |dy ≤
|bj (y) − (bj )Q |pj dy
|Q| Q j=1
|Q|
Q
j=1
!1/pj
≤C
k
Y
||bj ||BM O
j=1
and
1
|Q|
Z
k
Y
Q j=1
r
1/r
|bj (y) − (bj )Q | dy
C
≤
k
Y
k
Y
j=1
1 Z
|bj (y) − (bj )Q |pj r dy
|Q| Q
!1/pj r
≤
||bj ||BM O .
j=1
The lemma follows.
Proof of Theorem 1. It suffices to prove for f ∈ C0∞ (Rn ) and some constant
C0 , the following inequality holds:
1
|Q|
Z
Q
|T~b (f )(x) − C0 |dx ≤ C||~b||BM O Mr (f )(x̃) +
m
X
X
j=1 σ∈Cjm
Mr (T~bσc (f )(x̃)) .
Fix a cube Q = Q(x0 , r) and x̃ ∈ Q.
We first consider the Case m=1 . Write, for f1 = f χ2Q and f2 = f χ(2Q)c ,
Tb1 (f )(x) = (b1 (x)−(b1 )2Q )T (f )(x)−T ((b1 −(b1 )2Q )f1 )(x)−T ((b1 −(b1 )2Q )f2 )(x).
Let C0 = T (((b1 )2Q − b1 )f2 )(x0 ), then
|Tb1 (f )(x) − C0 |
≤ |(b1 (x) − (b1 )2Q )T (f )(x) + T ((b1 )2Q − (b1 )f1 )(x)
+T (((b1 )2Q − b1 )f2 )(x) − T (((b1 )2Q − b1 )f2 )(x0 )|
≤ |(b1 (x) − (b1 )2Q )T (f )(x)| + |T ((b1 )2Q − (b1 )f1 )(x)|
+|T (((b1 )2Q − b1 )f2 )(x) − T (((b1 )2Q − b1 )f2 )(x0 )|
= A(x) + B(x) + C(x).
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Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
For A(x), by Hölder’s inequality with exponent 1/r + 1/r′ = 1, we get
1 Z
A(x)dx
|Q| Q
1 Z
|b1 (x) − (b1 )2Q ||T (f )(x)|dx
=
|Q| Q
1 Z
′
≤ C
|b1 (x) − (b1 )2Q |r dx
|2Q| 2Q
≤ C||b1 ||BM O Mr (T (f ))(x̃).
!1/r′
1 Z
|T (f )(x)|r dx
|Q| Q
!1/r
For B(x), choose p such that 1 < p < r, and 1 < q < ∞, pq = r, by the
boundedness of T on Lp (Rn ) and the Hölder’s inequality, we obtain
1 Z
B(x)dx
|Q| Q
1 Z
T ((b1 − (b1 )2Q )f1 )(x)dx
=
|Q| Q
≤
1 Z
[T ((b1 − (b1 )2Q )f χ2Q )(x)]p dx
|Q| Rn
!1/p
!1/p
1 Z
(|b1 (x) − (b1 )2Q ||f (x)χ2Q (x)|)p dx
≤ C
|Q| Rn
1/pq
1/pq′ Z
Z
1
pq
1/pq ′
|f (x)| dx
|b1 (x) − (b1 )2Q |
dx
≤ C
|Q|1/p 2Q
2Q
!1/r
1 Z
≤ C
|f (x)|r dx
|2Q| 2Q
≤ C||b1 ||BM O Mr (f )(x̃).
1 Z
′
|b1 (x) − (b1 )2Q |pq dx
|2Q| 2Q
!1/pq′
For C(x), by [11], we know that
T~b (f )(x) =
gk
∞ X
X
k=1 h=1
ahk (x)
Z
Rn
m
Yhk (x − y) Y
(bj (x) − bj (y))f (y)dy
|x − y|n+m j=1
where gk ≤ Ck n−2 , ||ahk ||L∞ ≤ Ck −2n , |Yhk (x − y)| ≤ Ck n/2−1 and
Y (x − y)
Yhk (x0 − y)
hk
≤ Ck n/2 |x − x0 |/|x0 − y|n+1
−
|x − y|n
|x0 − y|n
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Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
for |x − y| > 2|x0 − x| > 0. So we get, by Minkowski’s inequality and Hölder’s
inequality,
Z
C(x) =
Rn
(K(x, x − y) − K(x0 , x0 − y))((b1 )2Q
− b1 (y))f2 (y)dy
Ω(x, x − y)
Ω(x0 , x0 − y)
−
|(b1 )2Q − b1 (y)||f (y)|dy
≤C
|x0 − y|n
(2Q)c |x − y|n
Z
gk
∞ X
∞ Z
X
X
Yhk (x0 − y)
Yhk (x − y)
|(b1 )2Q −b1 (y)|·
−
|ahk (x)|
≤C
|x0 − y|n
Rn |x − y|n
2l+1 Q\2l Q
Z
k=1 h=1
l=1
·|f (y)|dy ≤ C
∞
X
k
−2n
·k
n/2
l=1
k=1
≤C
∞
X
k −3n/2
≤C
1
−l
2
|2l+1 Q|
l=1
≤C
∞
X
l=1
k=1
∞
X
∞ Z
X
∞
X
Z
2l+1 Q\2l Q
|x − x0 |
|b1 (y) − (b1 )2Q ||f (y)|dy
|x0 − y|n+1
Z
r
|b1 (y) − (b1 )2Q ||f (y)|dy
(2l r)n+1 2l+1 Q
r′
2l+1 Q
|b1 (y) − (b1 )2Q | dy
!1/r′
1
|2l+1 Q|
Z
2l+1 Q
r
|f (y)| dy
!1/r
l2−l ||b1 ||BM O Mr (f )(x̃) ≤ C||b1 ||BM O Mr (f )(x̃),
l=1
thus
1 Z
C(x)dx ≤ C||b1 ||BM O Mr (f )(x̃).
|Q| Q
Now, we consider the Case m ≥ 2, we have known that, for b = (b1 , · · · , bm ),
T~b (f )(x) =
=
=
Z
m
Y
Rn j=1
m X
X
Z
Rn
m
Y
j=1
(bj (x) − bj (y)) K(x, x − y)f (y)dy
[(bj (x) − (bj )2Q ) − (bj (y) − (bj )2Q )] K(x, x − y)f (y)dy
j=0 σ∈Cjm
m−j
(−1)
(b(x) − (b)2Q )σ
Z
Rn
(b(y) − (b)2Q )σ K(x, x − y)f (y)dy
= (b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )T (f )(x)
+(−1)m T ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f )(x)
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Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
+
m−1
X
j=1
m−j
X
(−1)
(b(x) − (b)2Q )σ
σ∈Cjm
Z
Rn
(b(y) − b(x))σc K(x, x − y)f (y)dy
= (b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )T (f )(x)
+(−1)m T ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f )(x)
+
m−1
X
j=1
X
σ∈Cjm
(−1)m−j (b(x) − (b)2Q )σ T~bσc (f )(x),
thus,
|T~b (f )(x) − T ((b1 − (b1 )2B ) · · · (bm − (bm )2B )f2 )(x0 )|
≤ |(b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )T (f )(x)|
+
m−1
X
X
j=1 σ∈Cjm
|(b(x) − (b)2Q )σ T~bσc (f )(x)|
+|T ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f1 )(x)|+
|T ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f2 )(x) − T ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f2 )(x0 )|
= I1 (x) + I2 (x) + I3 (x) + I4 (x).
For I1 (x), by Hölder’s inequality with exponent 1/p1 +· · ·+1/pm +1/r = 1,
where 1 < pj < ∞, j = 1, · · · , m, we get
1 Z
I1 (x)dx
|Q| Q
1 Z
|b1 (x) − (b1 )2Q | · · · |bm (x) − (bm )2Q ||T (f )(x)|dx
≤
|Q| Q
≤
m
Y
j=1
1 Z
|bj (x) − (bj )2Q |pj dx
|Q| Q
!1/pj
≤ C||~b||BM O Mr (T (f ))(x̃).
For I2 (x), by the Minkowski’s and Hölder’s inequality, we get
1 Z
I2 (x)dx
|Q| Q
≤
m−1
X
X
j=1 σ∈Cjm
1 Z
|(b(x) − (b)2Q )σ ||T~bσc (f )(x)|dx
|Q| Q
175
!1/r
1 Z
|T (f )(x)|r dx
|Q| Q
Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
≤C
m−1
X
!1/r′
1 Z
′
|(b(x) − (b)2Q )σ |r dµ(x)
|2Q| 2Q
X
j=1 σ∈Cjm
≤C
m−1
X
j=1
X
σ∈Cjm
1 Z
|T~ (f )(x)|r dx
|Q| Q bσc
!1/r
||~bσ ||BM O Mr (T~bσc (f ))(x̃).
For I3 (x), choose 1 < p < r, 1 < qj < ∞, j = 1, · · · , m such that
1/q1 + · · · + 1/qm + p/r = 1, by the boundedness of T on Lp (Rn ) and Hölder’s
inequality, we get
1 Z
I3 (x)dx
|Q| Q
1 Z
|T ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f χ2Q )(x)|p dx
|Q| Rn
≤
!1/p
1 Z
≤ C
|b1 (x) − (b1 )2Q |p · · · |bm (x) − (bm )2Q |p |f (x)χ2Q (x)|p dx
|Q| Rn
1 Z
|f (x)|r dx
≤ C
|2Q| 2Q
!1/r
≤ C||~b||BM O Mr (f )(x̃).
m
Y
j=1
!1/p
!1/pqj
1 Z
|bj (x) − (bj )2B |pqj dx
|2Q| 2Q
For I4 (x), choose 1 < pj < ∞ j = 1, · · · , m such that 1/p1 +· · ·+1/pm +1/r = 1,
by Minkowski’s inequality and Hölder’s inequality, we obtain
I4 (x) ≤ C
≤C
Z
(2Q)c
∞ Z
X
l=1
m
Ω(x, x − y)
Ω(x0 , x0 − y) Y
(bj (y) − (bj )2Q )||f (y)|dy
|
−
|x − y|n
|x0 − y|n j=1
2l+1 Q\2l Q
"
gk
∞ X
X
#
Y (x − y)
Yhk (x0 − y)
hk
|ahk (x)|
−
|x0 − y|n
Rn |x − y|n
k=1 h=1
m
Y
×|
Z
(bj (y) − (bj )2Q )||f (y)|dy
j=1
≤C
∞
X
k
−2n
∞ Z
X
l=1
k=1
≤C
·k
n/2
∞
X
k=1
k −3n/2
∞
X
l=1
2l+1 Q\2l Q
m
|x − x0 | Y
| (bj (y) − (bj )2Q )||f (y)|dy
|x0 − y|n+1 j=1
Z
m
Y
r
(bj (y) − (bj )2Q )||f (y)|dy
|
(2l r)n+1 2l+1 Q j=1
176
Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
≤C
∞
X
2
l=1
·
m
Y
j=1
≤C
∞
X
l=1
thus
lm 2−l
1
−l
1
|2l+1 Q|
m
Y
|2l+1 Q|
Z
Z
2l+1 Q
r
|f (y)| dy
pj
2l+1 Q
!1/r
|bj (y) − (bj )2Q | dy
·
!1/pj
||bj ||BM O Mr (f )(x̃) ≤ C||~b||BM O Mr (f )(x̃),
j=1
1 Z
I4 (x)dx ≤ C||~b||BM O Mr (f )(x̃).
|Q| Q
This completes the proof of the theorem.
Proof of Theorem 2. Choose 1 < r < p in Theorem 1. We first consider the
case m=1, we have
||Tb1 (f )||Lp ≤
≤
≤
≤
≤
||M (Tb1 (f ))||Lp ≤ Ck(Tb1 (f ))# kLp
CkMr (T (f ))kLp + CkMr (f )kLp
CkT (f )kLp + CkMr (f )kLp
Ckf kLp + Ckf kLp
Ckf kLp .
When m ≥ 2, we may get the conclusion of Theorem 2 by induction. This
finishes the proof.
References
[1] A. P. Calderón and A. Zygmund, On singular integrals with variable
kernels, Appl. Anal., 7, (1978), 221-238.
[2] F. Chiarenza, M. Frasca and P. Longo, Interior W 2,p -estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat., 40,
(1991), 149-168.
[3] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for
Hardy spaces in several variables, Ann. of Math., 103, (1976), 611-635.
[4] G. Di Fazio and M. A. Ragusa, Interior estimates in Morrey spaces
for strong solutions to nondivergence form equations with discontinuous coefficients, J. Func. Anal., 112, (1993), 241-256.
177
Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
[5] J. Garcia-Cuerva and J.L.Rubio de Francia, Weighted norm inequalities
and related topics, North-Holland Math., 16, Amsterdam, 1985.
[6] L. Z. Liu, The continuity for multilinear singular integral operators with
variable Calderón-Zygmund kernel on Hardy and Herz spaces, Siberia Electronic Math. Reports, 2, (2005), 156-166.
[7] L. Z. Liu, Good λ estimate for multilinear singular integral operators
with variable Calderón-Zygmund kernel, Kragujevac J. of Math., 27, (2005),
19-30.
[8] L. Z. Liu, Weighted estimates of multilinear singular integral operators
with variable Calderón-Zygmund kernel for the extreme cases, Vietnam J. of
Math., 34, (2006), 51-61.
[9] S. Z. Lu, D. C. Yang and Z. S. Zhou, Oscillatory singular integral operators with Calderón-Zygmund kernels, Southeast Asian Bull. of Math., 23,
(1999), 457-470.
[10] C. Pérez, Endpoint estimate for commutators of singular integral operators, J. Func. Anal., 128, (1995), 163-185.
[11] C. Pérez and G. Pradolini, Sharp weighted endpoint estimates for commutators of singular integral operators, Michigan Math. J., 49, (2001), 23-37.
[12] C. Pérez and R.Trujillo-Gonzalez, Sharp Weighted estimates for multilinear commutators, J. London Math. Soc, 65, (2002), 672-692.
[13] E. M. Stein, Harmonic Analysis: real variable methods, orthogonality
and oscillatory integrals, Princeton Univ. Press, Princeton NJ, 1993.
[14] A. Torchinsky, Real variable methods in harmonic analysis, Pure and
Applied Math., 123, Academic Press, New York, 1986.
[15] H. Xu and L. Z. Liu., Weighted boundedness for multilinear singular
integral operator with variable Calderón-Zygmund kernel, African Diaspora J.
of Math., 6, (2008), 1-12.
Authors:
Zhiqiang Wang and Lanzhe Liu
College of Mathematics
Changsha University of Science and Technology
Changsha, 410077, P.R.of China
E-mail: [email protected]
178
No 18/2009
SHARP FUNCTION ESTIMATE FOR MULTILINEAR
COMMUTATOR OF SINGULAR INTEGRAL WITH VARIABLE
CALDERÓN-ZYGMUND KERNEL
Zhiqiang Wang and Lanzhe Liu
Abstract: In this paper, we prove the sharp function inequality for the
multilinear commutator related to the singular integral operator with variable
Calderón-Zygmund kernel. By using the sharp inequality, we obtain the Lp norm inequality for the multilinear commutator.
2000 Mathematics Subject Classification: 42B20, 42B25.
1. Introduction
As the development of singular integral operators, their commutators have
been well studied (see [1-4]). Let T be the Calderón-Zygmund singular integral operator, a classical result of Coifman, Rocherberg and Weiss (see [3])
states that commutator [b, T ](f ) = T (bf ) − bT (f )(where b ∈ BM O(Rn )) is
bounded on Lp (Rn ) for 1 < p < ∞. In [6-8], the sharp estimates for some
multilinear commutators of the Calderón-Zygmund singular integral operators
are obtained. The main purpose of this paper is to prove the sharp function
inequality for the multilinear commutator related to the singular integral operator with variable Calderón-Zygmund kernel. By using the sharp inequality,
we obtain the Lp -norm inequality for the multilinear commutator.
2. Notations and Results
First let us introduce some notations (see [4][8][9]). In this paper, Q will
denote a cube
of Rn with sides parallel to the axes, and for a cube Q let
R
fQ = |Q|−1 Q f (x)dx and the sharp function of f is defined by
1 Z
f (x) = sup
|f (y) − fQ |dy.
Q∋x |Q| Q
#
169
Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
It is well-known that (see [4])
1 Z
f (x) ≈ sup inf
|f (y) − C|dy.
Q∋x c∈C |Q| Q
#
We say that b belongs to BM O(Rn ) if b# belongs to L∞ (Rn ) and define
||b||BM O = ||b# ||L∞ . It has been known that (see [9])
||b − b2k Q ||BM O ≤ Ck||b||BM O .
Let M be the Hardy-Littlewood maximal operator, that is that
−1
M (f )(x) = sup |Q|
x∈Q
Z
Q
|f (y)|dy;
we write that Mp (f ) = (M (|f |p ))1/p for 0 < p < ∞.
For bj ∈ BM O(Rn )(j = 1, · · · , m), set
||~b||BM O =
m
Y
||bj ||BM O .
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)} ∈ C m , set ~bσ = (bσ(1) , · · ·, bσ(j) ),
j
~
bσ = bσ(1) · · · bσ(j) and ||bσ ||BM O = ||bσ(1) ||BM O · · · ||bσ(j) ||BM O .
In this paper, we will study some multilinear commutators as follows.
Definition 1. Let K(x) = Ω(x)/|x|n : Rn \ {0} → R. K is said to be a
Calderón-Zygmund kernel if
(a) Ω ∈ C ∞ (Rn \ {0});
(b) Ω
is homogeneous of degree zero;
R
(c) Σ Ω(x)xα dσ(x) = 0 for all multi-indices α ∈ (N ∪ {0})n with |α| = N ,
where Σ = {x ∈ Rn : |x| = 1} is the unit sphere of Rn .
Definition 2. Let K(x, y) = Ω(x, y)/|y|n : Rn × (Rn \ {0}) → R. K is
said to be a variable Calderón-Zygmund kernel if
(d) K(x, ·) is a Calderón-Zygmund
kernel for a.e. x ∈ Rn ;
∂ | γ|
= M < ∞.
(e) max|γ|≤2n ∂ γ y Ω(x, y) ∞ n
L (R ×Σ)
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Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
Suppose bj (j = 1, · · · , m) are the fixed locally integrable functions on Rn .
Let T be the singular integral operator with variable Calderón-Zygmund kernel
as
Z
K(x, x − y)f (y)d(y),
T (f )(x) =
Rn
Ω(x, x − y)
and that Ω(x, y)/|y|n is a variable Calderón|x − y|n
Zygmund kernel. The multilinear commutator of singular integral with variable
Calderón-Zygmund kernel is defined by
where K(x, x − y) =
T~b (f )(x) =
Z
m
Y
Rn j=1
(bj (x) − bj (y))K(x, x − y)f (y)dy.
Note that when b1 = ··· = bm , T~b is just the m order commutator (see [1][5]).
It is well known that commutators are of great interest in harmonic analysis
and have been widely studied by many authors (see [1-3][5-8]). Our main
purpose is to establish the sharp inequality for the multilinear commutator.
Now we state our theorems as following.
Theorem 1. Let bj ∈ BM O(Rn ) for j = 1, · · · , m. Then for any 1 <
r < ∞, there exists a constant C > 0 such that for any f ∈ C0∞ (Rn ) and any
x ∈ Rn ,
(T~b (f ))# (x) ≤ C||~b||BM O Mr (f )(x) +
m X
X
j=1 σ∈Cjm
Mr (T~bσc (f ))(x) .
Theorem 2. Let bj ∈ BM O(Rn ) for j = 1, · · · , m. Then T~b is bounded on
L (Rn ) for 1 < p < ∞.
p
3. Proof of Theorem
To prove the theorems, we need the following lemmas.
Lemma 1. (see[10]) Let 1 < p < ∞ and T be the singular integral operator
with variable Calderón-Zygmund kernel. Then T is bounded on Lp (Rn ).
Lemma 2. Let 1 < r < ∞, bj ∈ BM O(Rn ) for j = 1, · · · , k. Then
k
k
Y
1 Z Y
||bj ||BM O
|bj (y) − (bj )Q |dy ≤ C
|Q| Q j=1
j=1
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Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
and
1/r
k
1 Z Y
|bj (y) − (bj )Q |r dy
|Q| Q j=1
≤C
k
Y
||bj ||BM O .
j=1
Proof. Choose 1 < pj < ∞ j = 1, · · · , k such that 1/p1 + · · · + 1/pk = 1, we
obtain, by Hölder’s inequality,
k
k
Y
1 Z Y
1 Z
|bj (y)−(bj )Q |dy ≤
|bj (y) − (bj )Q |pj dy
|Q| Q j=1
|Q|
Q
j=1
!1/pj
≤C
k
Y
||bj ||BM O
j=1
and
1
|Q|
Z
k
Y
Q j=1
r
1/r
|bj (y) − (bj )Q | dy
C
≤
k
Y
k
Y
j=1
1 Z
|bj (y) − (bj )Q |pj r dy
|Q| Q
!1/pj r
≤
||bj ||BM O .
j=1
The lemma follows.
Proof of Theorem 1. It suffices to prove for f ∈ C0∞ (Rn ) and some constant
C0 , the following inequality holds:
1
|Q|
Z
Q
|T~b (f )(x) − C0 |dx ≤ C||~b||BM O Mr (f )(x̃) +
m
X
X
j=1 σ∈Cjm
Mr (T~bσc (f )(x̃)) .
Fix a cube Q = Q(x0 , r) and x̃ ∈ Q.
We first consider the Case m=1 . Write, for f1 = f χ2Q and f2 = f χ(2Q)c ,
Tb1 (f )(x) = (b1 (x)−(b1 )2Q )T (f )(x)−T ((b1 −(b1 )2Q )f1 )(x)−T ((b1 −(b1 )2Q )f2 )(x).
Let C0 = T (((b1 )2Q − b1 )f2 )(x0 ), then
|Tb1 (f )(x) − C0 |
≤ |(b1 (x) − (b1 )2Q )T (f )(x) + T ((b1 )2Q − (b1 )f1 )(x)
+T (((b1 )2Q − b1 )f2 )(x) − T (((b1 )2Q − b1 )f2 )(x0 )|
≤ |(b1 (x) − (b1 )2Q )T (f )(x)| + |T ((b1 )2Q − (b1 )f1 )(x)|
+|T (((b1 )2Q − b1 )f2 )(x) − T (((b1 )2Q − b1 )f2 )(x0 )|
= A(x) + B(x) + C(x).
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Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
For A(x), by Hölder’s inequality with exponent 1/r + 1/r′ = 1, we get
1 Z
A(x)dx
|Q| Q
1 Z
|b1 (x) − (b1 )2Q ||T (f )(x)|dx
=
|Q| Q
1 Z
′
≤ C
|b1 (x) − (b1 )2Q |r dx
|2Q| 2Q
≤ C||b1 ||BM O Mr (T (f ))(x̃).
!1/r′
1 Z
|T (f )(x)|r dx
|Q| Q
!1/r
For B(x), choose p such that 1 < p < r, and 1 < q < ∞, pq = r, by the
boundedness of T on Lp (Rn ) and the Hölder’s inequality, we obtain
1 Z
B(x)dx
|Q| Q
1 Z
T ((b1 − (b1 )2Q )f1 )(x)dx
=
|Q| Q
≤
1 Z
[T ((b1 − (b1 )2Q )f χ2Q )(x)]p dx
|Q| Rn
!1/p
!1/p
1 Z
(|b1 (x) − (b1 )2Q ||f (x)χ2Q (x)|)p dx
≤ C
|Q| Rn
1/pq
1/pq′ Z
Z
1
pq
1/pq ′
|f (x)| dx
|b1 (x) − (b1 )2Q |
dx
≤ C
|Q|1/p 2Q
2Q
!1/r
1 Z
≤ C
|f (x)|r dx
|2Q| 2Q
≤ C||b1 ||BM O Mr (f )(x̃).
1 Z
′
|b1 (x) − (b1 )2Q |pq dx
|2Q| 2Q
!1/pq′
For C(x), by [11], we know that
T~b (f )(x) =
gk
∞ X
X
k=1 h=1
ahk (x)
Z
Rn
m
Yhk (x − y) Y
(bj (x) − bj (y))f (y)dy
|x − y|n+m j=1
where gk ≤ Ck n−2 , ||ahk ||L∞ ≤ Ck −2n , |Yhk (x − y)| ≤ Ck n/2−1 and
Y (x − y)
Yhk (x0 − y)
hk
≤ Ck n/2 |x − x0 |/|x0 − y|n+1
−
|x − y|n
|x0 − y|n
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Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
for |x − y| > 2|x0 − x| > 0. So we get, by Minkowski’s inequality and Hölder’s
inequality,
Z
C(x) =
Rn
(K(x, x − y) − K(x0 , x0 − y))((b1 )2Q
− b1 (y))f2 (y)dy
Ω(x, x − y)
Ω(x0 , x0 − y)
−
|(b1 )2Q − b1 (y)||f (y)|dy
≤C
|x0 − y|n
(2Q)c |x − y|n
Z
gk
∞ X
∞ Z
X
X
Yhk (x0 − y)
Yhk (x − y)
|(b1 )2Q −b1 (y)|·
−
|ahk (x)|
≤C
|x0 − y|n
Rn |x − y|n
2l+1 Q\2l Q
Z
k=1 h=1
l=1
·|f (y)|dy ≤ C
∞
X
k
−2n
·k
n/2
l=1
k=1
≤C
∞
X
k −3n/2
≤C
1
−l
2
|2l+1 Q|
l=1
≤C
∞
X
l=1
k=1
∞
X
∞ Z
X
∞
X
Z
2l+1 Q\2l Q
|x − x0 |
|b1 (y) − (b1 )2Q ||f (y)|dy
|x0 − y|n+1
Z
r
|b1 (y) − (b1 )2Q ||f (y)|dy
(2l r)n+1 2l+1 Q
r′
2l+1 Q
|b1 (y) − (b1 )2Q | dy
!1/r′
1
|2l+1 Q|
Z
2l+1 Q
r
|f (y)| dy
!1/r
l2−l ||b1 ||BM O Mr (f )(x̃) ≤ C||b1 ||BM O Mr (f )(x̃),
l=1
thus
1 Z
C(x)dx ≤ C||b1 ||BM O Mr (f )(x̃).
|Q| Q
Now, we consider the Case m ≥ 2, we have known that, for b = (b1 , · · · , bm ),
T~b (f )(x) =
=
=
Z
m
Y
Rn j=1
m X
X
Z
Rn
m
Y
j=1
(bj (x) − bj (y)) K(x, x − y)f (y)dy
[(bj (x) − (bj )2Q ) − (bj (y) − (bj )2Q )] K(x, x − y)f (y)dy
j=0 σ∈Cjm
m−j
(−1)
(b(x) − (b)2Q )σ
Z
Rn
(b(y) − (b)2Q )σ K(x, x − y)f (y)dy
= (b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )T (f )(x)
+(−1)m T ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f )(x)
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Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
+
m−1
X
j=1
m−j
X
(−1)
(b(x) − (b)2Q )σ
σ∈Cjm
Z
Rn
(b(y) − b(x))σc K(x, x − y)f (y)dy
= (b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )T (f )(x)
+(−1)m T ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f )(x)
+
m−1
X
j=1
X
σ∈Cjm
(−1)m−j (b(x) − (b)2Q )σ T~bσc (f )(x),
thus,
|T~b (f )(x) − T ((b1 − (b1 )2B ) · · · (bm − (bm )2B )f2 )(x0 )|
≤ |(b1 (x) − (b1 )2Q ) · · · (bm (x) − (bm )2Q )T (f )(x)|
+
m−1
X
X
j=1 σ∈Cjm
|(b(x) − (b)2Q )σ T~bσc (f )(x)|
+|T ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f1 )(x)|+
|T ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f2 )(x) − T ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f2 )(x0 )|
= I1 (x) + I2 (x) + I3 (x) + I4 (x).
For I1 (x), by Hölder’s inequality with exponent 1/p1 +· · ·+1/pm +1/r = 1,
where 1 < pj < ∞, j = 1, · · · , m, we get
1 Z
I1 (x)dx
|Q| Q
1 Z
|b1 (x) − (b1 )2Q | · · · |bm (x) − (bm )2Q ||T (f )(x)|dx
≤
|Q| Q
≤
m
Y
j=1
1 Z
|bj (x) − (bj )2Q |pj dx
|Q| Q
!1/pj
≤ C||~b||BM O Mr (T (f ))(x̃).
For I2 (x), by the Minkowski’s and Hölder’s inequality, we get
1 Z
I2 (x)dx
|Q| Q
≤
m−1
X
X
j=1 σ∈Cjm
1 Z
|(b(x) − (b)2Q )σ ||T~bσc (f )(x)|dx
|Q| Q
175
!1/r
1 Z
|T (f )(x)|r dx
|Q| Q
Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
≤C
m−1
X
!1/r′
1 Z
′
|(b(x) − (b)2Q )σ |r dµ(x)
|2Q| 2Q
X
j=1 σ∈Cjm
≤C
m−1
X
j=1
X
σ∈Cjm
1 Z
|T~ (f )(x)|r dx
|Q| Q bσc
!1/r
||~bσ ||BM O Mr (T~bσc (f ))(x̃).
For I3 (x), choose 1 < p < r, 1 < qj < ∞, j = 1, · · · , m such that
1/q1 + · · · + 1/qm + p/r = 1, by the boundedness of T on Lp (Rn ) and Hölder’s
inequality, we get
1 Z
I3 (x)dx
|Q| Q
1 Z
|T ((b1 − (b1 )2Q ) · · · (bm − (bm )2Q )f χ2Q )(x)|p dx
|Q| Rn
≤
!1/p
1 Z
≤ C
|b1 (x) − (b1 )2Q |p · · · |bm (x) − (bm )2Q |p |f (x)χ2Q (x)|p dx
|Q| Rn
1 Z
|f (x)|r dx
≤ C
|2Q| 2Q
!1/r
≤ C||~b||BM O Mr (f )(x̃).
m
Y
j=1
!1/p
!1/pqj
1 Z
|bj (x) − (bj )2B |pqj dx
|2Q| 2Q
For I4 (x), choose 1 < pj < ∞ j = 1, · · · , m such that 1/p1 +· · ·+1/pm +1/r = 1,
by Minkowski’s inequality and Hölder’s inequality, we obtain
I4 (x) ≤ C
≤C
Z
(2Q)c
∞ Z
X
l=1
m
Ω(x, x − y)
Ω(x0 , x0 − y) Y
(bj (y) − (bj )2Q )||f (y)|dy
|
−
|x − y|n
|x0 − y|n j=1
2l+1 Q\2l Q
"
gk
∞ X
X
#
Y (x − y)
Yhk (x0 − y)
hk
|ahk (x)|
−
|x0 − y|n
Rn |x − y|n
k=1 h=1
m
Y
×|
Z
(bj (y) − (bj )2Q )||f (y)|dy
j=1
≤C
∞
X
k
−2n
∞ Z
X
l=1
k=1
≤C
·k
n/2
∞
X
k=1
k −3n/2
∞
X
l=1
2l+1 Q\2l Q
m
|x − x0 | Y
| (bj (y) − (bj )2Q )||f (y)|dy
|x0 − y|n+1 j=1
Z
m
Y
r
(bj (y) − (bj )2Q )||f (y)|dy
|
(2l r)n+1 2l+1 Q j=1
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Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
≤C
∞
X
2
l=1
·
m
Y
j=1
≤C
∞
X
l=1
thus
lm 2−l
1
−l
1
|2l+1 Q|
m
Y
|2l+1 Q|
Z
Z
2l+1 Q
r
|f (y)| dy
pj
2l+1 Q
!1/r
|bj (y) − (bj )2Q | dy
·
!1/pj
||bj ||BM O Mr (f )(x̃) ≤ C||~b||BM O Mr (f )(x̃),
j=1
1 Z
I4 (x)dx ≤ C||~b||BM O Mr (f )(x̃).
|Q| Q
This completes the proof of the theorem.
Proof of Theorem 2. Choose 1 < r < p in Theorem 1. We first consider the
case m=1, we have
||Tb1 (f )||Lp ≤
≤
≤
≤
≤
||M (Tb1 (f ))||Lp ≤ Ck(Tb1 (f ))# kLp
CkMr (T (f ))kLp + CkMr (f )kLp
CkT (f )kLp + CkMr (f )kLp
Ckf kLp + Ckf kLp
Ckf kLp .
When m ≥ 2, we may get the conclusion of Theorem 2 by induction. This
finishes the proof.
References
[1] A. P. Calderón and A. Zygmund, On singular integrals with variable
kernels, Appl. Anal., 7, (1978), 221-238.
[2] F. Chiarenza, M. Frasca and P. Longo, Interior W 2,p -estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat., 40,
(1991), 149-168.
[3] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for
Hardy spaces in several variables, Ann. of Math., 103, (1976), 611-635.
[4] G. Di Fazio and M. A. Ragusa, Interior estimates in Morrey spaces
for strong solutions to nondivergence form equations with discontinuous coefficients, J. Func. Anal., 112, (1993), 241-256.
177
Z. Wang, L. Liu - Sharp Function Estimate for Multilinear Commutator of...
[5] J. Garcia-Cuerva and J.L.Rubio de Francia, Weighted norm inequalities
and related topics, North-Holland Math., 16, Amsterdam, 1985.
[6] L. Z. Liu, The continuity for multilinear singular integral operators with
variable Calderón-Zygmund kernel on Hardy and Herz spaces, Siberia Electronic Math. Reports, 2, (2005), 156-166.
[7] L. Z. Liu, Good λ estimate for multilinear singular integral operators
with variable Calderón-Zygmund kernel, Kragujevac J. of Math., 27, (2005),
19-30.
[8] L. Z. Liu, Weighted estimates of multilinear singular integral operators
with variable Calderón-Zygmund kernel for the extreme cases, Vietnam J. of
Math., 34, (2006), 51-61.
[9] S. Z. Lu, D. C. Yang and Z. S. Zhou, Oscillatory singular integral operators with Calderón-Zygmund kernels, Southeast Asian Bull. of Math., 23,
(1999), 457-470.
[10] C. Pérez, Endpoint estimate for commutators of singular integral operators, J. Func. Anal., 128, (1995), 163-185.
[11] C. Pérez and G. Pradolini, Sharp weighted endpoint estimates for commutators of singular integral operators, Michigan Math. J., 49, (2001), 23-37.
[12] C. Pérez and R.Trujillo-Gonzalez, Sharp Weighted estimates for multilinear commutators, J. London Math. Soc, 65, (2002), 672-692.
[13] E. M. Stein, Harmonic Analysis: real variable methods, orthogonality
and oscillatory integrals, Princeton Univ. Press, Princeton NJ, 1993.
[14] A. Torchinsky, Real variable methods in harmonic analysis, Pure and
Applied Math., 123, Academic Press, New York, 1986.
[15] H. Xu and L. Z. Liu., Weighted boundedness for multilinear singular
integral operator with variable Calderón-Zygmund kernel, African Diaspora J.
of Math., 6, (2008), 1-12.
Authors:
Zhiqiang Wang and Lanzhe Liu
College of Mathematics
Changsha University of Science and Technology
Changsha, 410077, P.R.of China
E-mail: [email protected]
178