PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE
Nouvelle série, tome 95 (109) (2014), 201–214
DOI: 10.2298/PIM1409201L
ESTIMATES OF MULTILINEAR
SINGULAR INTEGRAL OPERATORS
AND MEAN OSCILLATION
Lanzhe Liu
Communicated by Stevan Pilipović
Abstract. We prove the boundedness properties for some multilinear operators related to certain integral operators from Lebesgue spaces to Orlicz
spaces. The operators include Calderón–Zygmund singular integral operator,
Littlewood–Paley operator and Marcinkiewicz operator.
1. Introduction
We are going to consider some integral operators as follows. Let m be a positive
integer and A be a function on Rn . We denote
X 1
Rm+1 (A; x, y) = A(x) −
Dα A(y)(x − y)α .
α!
|α|6m
Definition 1.1. Let T : S → S ′ be a linear operator and there exists a locally
integrable function K(x, y) on Rn × Rn r {x = y} such that
Z
K(x, y)f (y) dy
T f (x) =
Rn
for every bounded and compactly supported function f and x ∈
/ supp f , where K
satisfies: |K(x, y)| 6 C|x − y|−n and for fixed 0 < ε 6 1,
|K(y, x) − K(z, x)| 6 C|y − z|ε |x − z|−n−ε
if 2|y − z| 6 |x − z|. The multilinear operator related to the integral operator T is
defined by
Z
Rm+1 (A; x, y)
T A (f )(x) =
K(x, y)f (y) dy.
|x − y|m
2010 Mathematics Subject Classification: 42B20; 42B25.
Supported by the Scientific Research Fund of Hunan Provincial Education Departments
(13K013).
201
202
LIU
Definition 1.2. Let Ft (x, y) be defined on Rn × Rn × [0, +∞); we denote
Z
Ft (f )(x) =
Ft (x, y)f (y) dy
Rn
for every bounded and compactly supported function f and
Z
Rm+1 (A; x, y)
A
Ft (f )(x) =
Ft (x, y)f (y) dy.
|x − y|m
Rn
Let H = {h : ||h|| < ∞} be a Banach space. For each fixed x ∈ Rn , we consider
Ft (f )(x) and FtA (f )(x) as mappings from [0, +∞) to H. Then, the multilinear
operators related to Ft are defined by S A (f )(x) = kFtA (f )(x)k, where kFt (x, y)k 6
C|x − y|−n and for fixed ε > 0
kFt (y, x) − Ft (z, x)k 6 C|y − z|ε |x − z|−n−ε
if 2|y − z| 6 |x − z|. We also define S(f )(x) = kFt (f )(x)k.
Note that when m = 0, T A and S A are just the commutators of T and S
with A (see [4, 7–9, 12]). While when m > 0, they are nontrivial generalizations
of the commutators. Let T be the Calderón–Zygmund singular integral operator.
A classical result of Coifman, Rochberg and Weiss [4] states that the commutator
[b, T ] = T (bf ) − bT f (where b ∈ BMO(Rn )) is bounded on Lp (Rn ) for 1 < p < ∞.
Chanillo [1] proves a similar result when T is replaced by a fractional integral
operator. In [8], the boundedness properties for the commutators related to the
Calderón–Zygmund singular integral operators from Lebesgue spaces to Orlicz ones
are obtained. It is well known that multilinear operators are of great interest in
harmonic analysis and have been widely studied by many authors (see [2,3,5]). Our
main purpose is to prove the boundedness properties for the multilinear operators
T A and S A from Lebesgue spaces to Orlicz ones.
Let us introduce some notations. Throughout the paper, Q will denote a cube
of Rn with sides parallel to the axes. For any locally integrable function f , the
sharp function of f is defined by
Z
1
f # (x) = sup
|f (y) − fQ | dy,
Q∋x |Q| Q
R
where, and in what follows, fQ = |Q|−1 Q f (x) dx. It is well known [6] that
Z
1
f # (x) ≈ sup inf
|f (y) − c| dy.
Q∋x c∈C |Q| Q
Let M be the Hardy–Littlewood maximal operator defined by
Z
−1
M (f )(x) = sup |Q|
|f (y)| dy.
Q∋x
p
1/p
We write Mp f = (M (f ))
Q
. For 1 6 r < ∞ and 0 6 β < n, let
1/r
Z
1
r
Mβ,r (f )(x) = sup
|f (y)| dy
.
1−rβ/n
Q∋x |Q|
Q
ESTIMATES OF MULTILINEAR SINGULAR INTEGRAL OPERATORS
203
We say that f belongs to BMO(Rn ) if f # belongs to L∞ (Rn ) and kf kBMO =
kf # kL∞ . More generally, let ϕ be a nondecreasing positive function and define
BMOϕ (Rn ) as the space of all the functions f satisfying
Z
1
|f (y) − fQ | dy 6 Cϕ(r).
|Q(x, r)| Q(x,r)
For β > 0, the Lipschitz space Lipβ (Rn ) is the space of functions f satisfying
kf kLipβ = sup |f (x) − f (y)|/|x − y|β < ∞.
x6=y
For an f , let mf (t) = |{x ∈ Rn : |f (x)| > t}| denote the distribution function of f .
Let ψ be a nondecreasing convex function on R+ with ψ(0) = 0 and let ψ −1
be its inverseR function. The Orlicz space Lψ (Rn ) is defined by the set of functions
f such that ψ(λ|f
(x)|) dx < ∞ for some λ > 0. The norm is given by kf kLψ =
R
inf λ>0 λ−1 (1 + ψ(λ|f (x)|) dx).
We shall prove the following theorems in Section 2.
Theorem 1.1. Let 1 < p < ∞ and ϕ, ψ be two nondecreasing positive functions
on R+ with ϕ(t) = tn/p ψ −1 (t−n ) (or equivalently ψ −1 (t) = t1/p ϕ(t−1/n ). Suppose
that ψ is convex, ψ(0) = 0, ψ(2t) 6 Cψ(t). Let T be the same as in Definition 1.1
and bounded on Lp (Rn ) for all 1 < p < ∞. Then T A : Lp (Rn ) → Lψ (Rn ) is
bounded if Dα A ∈ BMOϕ (Rn ) for all α with |α| = m.
Theorem 1.2. Let 1 < p < ∞ and ϕ, ψ be two non-decreasing positive functions on R+ with ϕ(t) = tn/p ψ −1 (t−n ) (or equivalently ψ −1 (t) = t1/p ϕ(t−1/n ).
Suppose that ψ is convex, ψ(0) = 0, ψ(2t) 6 Cψ(t). Let S be the same as in Definition 1.2 and bounded on Lp (Rn ) for all 1 < p < ∞. Then S A : Lp (Rn ) → Lψ (Rn )
is bounded if Dα A ∈ BMOϕ (Rn ) for all α with |α| = m.
2. Proofs of Theorems
We begin with the following preliminary lemmas.
Lemma 2.1. [7] Let ϕ be a nondecreasing positive function on R+
R and η be an
n
infinitely differentiable
function
on
R
with
compact
support
such
that
η(x) dx = 1.
R
Denote bt (x) = Rn b(x − ty)η(y) dy. Then kb − bt kBMO 6 Cϕ(t)kbkBMOϕ .
Lemma 2.2. [7] Let 0 < β < 1 and ϕ be a nondecreasing positive function on
R+ or β = 1. Then kbt kLipβ 6 Ct−β ϕ(t)kbkBMOϕ .
Lemma 2.3. [7] Suppose 1 6 p2 < p < p1 < ∞, ρ is a nonincreasing function
on R+ , B is a linear operator such that mB(f ) (t1/p1 ρ(t)) 6 Ct−1 if kf kLp1 6 1
R∞
and mB(f ) (t1/p2 ρ(t)) 6 Ct−1 if kf kLp2 6 1. Then 0 mB(f ) (t1/p ρ(t)) dt 6 C if
kf kLp 6 (p/p1 )1/p .
Lemma 2.4. [1] Suppose that 1 6 r < p < n/β and 1/q = 1/p − β/n. Then
kMβ,r (f )kLq 6 Ckf kLp .
204
LIU
Lemma 2.5. [9] Suppose that 1 6 r < ∞ and b ∈ Lipβ . Then
k(b − bQ )f χ2Q kLr 6 C|Q|1/r kbkLipβ Mβ,r (f )(x)
where χ2Q is the characteristic function of 2Q.
Lemma 2.6. [3] Let A be a function on Rn and Dα A ∈ Lq (Rn ) for all α with
|α| = m and some q > n. Then
1/q
Z
X 1
|Rm (A; x, y)| 6 C|x − y|m
|Dα A(z)|q dz
,
|
Q̃(x,
y)|
Q̃(x,y)
|α|=m
√
where Q̃ is the cube centered on x and having side length 5 n|x − y|.
To prove the theorems of the paper, we need the following
Key Lemma 1. Let 0 < β < n, T and S be the same as in Definitions 1.1 and
1.2. Suppose that Q = Q(x0 , d) is a cube with supp f ⊂ (2Q)c and x, x̃ ∈ Q.
(a) If Dα A ∈ BMO(Rn ) for all α with |α| = m, then
X
|T A (f )(x) − T A (f )(x0 )| 6 C
kDα AkBMO Mr (f )(x̃) for any r > 1;
|α|=m
(b) If Dα A ∈ Lipβ (Rn ) for all α with |α| = m, then
X
kDα AkLipβ Mβ,1 (f )(x̃);
|T A (f )(x) − T A (f )(x0 )| 6 C
|α|=m
(c) If Dα A ∈ BMO(Rn ) for all α with |α| = m, then
X
kFtA (f )(x) − FtA (f )(x0 )k 6 C
kDα AkBMO Mr (f )(x̃) for any r > 1;
|α|=m
α
n
(d) If D A ∈ Lipβ (R ) for all α with |α| = m, then
X
kDα AkLipβ Mβ,1 (f )(x̃).
kFtA (f )(x) − FtA (f )(x0 )k 6 C
|α|=m
Proof. Let Ã(x) = A(x) −
1
α
α
|α|=m α! (D A)Q x ,
P
then
Rm+1 (A; x, y) = Rm+1 (Ã; x, y) and Dα Ã = Dα A − (Dα A)Q for |α| = m.
Suppose that supp f ⊂ (2Q)c and x, x̃ ∈ Q = Q(x0 , d). Note that |x0 − y| ≈ |x − y|
for y ∈ (2Q)c . We write
Z K(x, y)
K(x0 , y)
T A (f )(x) − T A (f )(x0 ) =
−
Rm (Ã; x, y)f (y) dy
m
|x0 − y|m
Rn |x − y|
Z
K(x0 , y)f (y)
+
[Rm (Ã; x, y) − Rm (Ã; x0 , y)] dy
|x0 − y|m
Rn
X 1 Z K(x, y)(x − y)α
K(x0 , y)(x0 − y)α
−
−
Dα Ã(y)f (y) dy
α! Rn
|x − y|m
|x0 − y|m
|α|=m
:= I + II + III.
ESTIMATES OF MULTILINEAR SINGULAR INTEGRAL OPERATORS
205
(a) By Lemma 2.6 and the following inequality (see [10]), for b ∈ BMO(Rn ),
|bQ1 − bQ2 | 6 C log(|Q2 |/|Q1 |)kbkBMO for Q1 ⊂ Q2 ,
we know that, for x ∈ Q and y ∈ 2k+1 Q r 2k Q with k > 1,
X
|Rm (Ã; x, y)| 6 C|x − y|m
(kDα AkBMO + |(Dα A)Q − (Dα A)Q(x,y) |)
|α|=m
m
6 Ck|x − y|
X
|α|=m
kDα AkBMO ,
thus
|x − x0 |
|x − x0 |ε
+
|Rm (Ã; x, y)kf (y)| dy
|I| 6 C
|x0 − y|m+n+1
|x0 − y|m+n+ε
Rn r2Q
∞ Z
X
X
|x − x0 |
|x − x0 |ε
α
6C
kD AkBMO
k
+
|f (y)| dy
|x0 − y|n+1
|x0 − y|n+ε
2k+1 Qr2k Q
Z
k=1
|α|=m
6C
kDα AkBMO
X
kDα AkBMO M (f )(x̃).
|α|=m
6C
∞
X
X
|α|=m
k(2−k + 2−kε )
k=1
1
|2k+1 Q|
Z
2k+1 Q
|f (y)| dy
For II, by the formula (see [3]):
Rm (Ã; x, y) − Rm (Ã; x0 , y) =
X 1
Rm−|η| (Dη Ã; x, x0 )(x − y)η
η!
|η|<m
and Lemma 2.6, we get
Z
|II| 6 C
|Rm (Ã; x, y) − Rm (Ã; x0 , y)|
|f (y)| dy
|x0 − y|m+n
∞ Z
X
X
|x − x0 |
6C
kDα AkBMO
|f (y)| dy
n+1
k+1 Qr2k Q |x0 − y|
k=1 2
|α|=m
X
6C
kDα AkBMO M (f )(x̃).
Rn r2Q
|α|=m
For III, similar to the estimates of I, we obtain, for any r > 1 with 1/r + 1/r′ = 1,
Z
|x − x0 |
|x − x0 |ε
|III| 6 C
+
|Dα A(y) − (Dα A)Q kf (y)| dy
|x0 − y|n+1
|x0 − y|n+ε
Rn r2Q
1/r
Z
∞
X X
1
−k
−kε
r
6C
(2 + 2 )
|f (y)| dy
|2k+1 Q| 2k+1 Q
|α|=m k=1
×
1
|2k+1 Q|
Z
2k+1 Q
1/r′
|D A(x) − (D A)Q | dx
α
α
r′
206
LIU
6C
X
|α|=m
kDα AkBMO Mr (f )(x̃).
Thus
|T A (f )(x) − T A (f )(x0 )| 6 C
X
|α|=m
kDα AkBMO Mr (f )(x̃).
(b) By Lemma 2.6 and the following inequality, for b ∈ Lipβ ,
Z
1
kbkLipβ |x − y|β dy 6 kbkLipβ (|x − x0 | + d)β ,
|b(x) − bQ | 6
|Q| Q
we get
|Rm (Ã; x, y)| 6
X
|α|=m
kDα AkLipβ (|x − y| + d)m+β ,
then
|x − x0 |
|x − x0 |ε
|I| 6 C
+
|Rm (Ã; x, y)kf (y)| dy
|x0 − y|m+n+1
|x0 − y|m+n+ε
Rn r2Q
∞ Z
X
X
|x − x0 |ε
|x − x0 |
6C
kDα AkLipβ
+
|f (y)| dy
n+1−β
|x0 −y|n+ε−β
2k+1 Qr2k Q |x0 −y|
Z
k=1
|α|=m
6C
kDα AkLipβ
X
kDα AkLipβ Mβ,1 (f )(x̃),
|α|=m
6C
∞
X
X
|α|=m
(2−k + 2−kε )
k=1
1
|2k+1 Q|1−β/n
Z
2k+1 Q
|f (y)| dy
|Rm (Ã; x, y) − Rm (Ã; x0 , y)|
|f (y)| dy
|x0 − y|m+n
∞ Z
X
X
|x − x0 |
6C
kDα AkLipβ
|f (y)| dy
n+1−β
k+1 Qr2k Q |x0 − y|
k=1 2
|α|=m
X
6C
kDα AkLipβ Mβ,1 (f )(x̃),
|II| 6 C
Z
Rn r2Q
|α|=m
|x − x0 |
|x − x0 |ε
|III| 6 C
+
|Dα A(y) − (Dα A)Q kf (y)| dy
|x0 − y|n+1−β
|x0 − y|n+ε−β
Rn r2Q
Z
∞
X
X
1
6C
kDα AkLipβ
(2−k + 2−kε ) k+1 1−β/n
|f (y)| dy
|2 Q|
k+1 Q
2
k=1
|α|=m
X
6C
kDα AkLipβ Mβ,1 (f )(x̃).
Z
|α|=m
Thus
|T A (f )(x) − T A (f )(x0 )| 6 C
X
|α|=m
kDα AkLipβ Mβ,1 (f )(x̃).
ESTIMATES OF MULTILINEAR SINGULAR INTEGRAL OPERATORS
207
The same argument as in the proof of (a) and (b) will give the proof of (c) and
(d), we omit the details.
Now we are in the position to prove our theorems.
Proof of Theorem 1.1. We prove it in several steps. First, we prove
X
(1)
(T A (f ))# 6 C
kDα AkBMO Mr (f )
|α|=m
for
P any 11 < rα < ∞.α Fix a cube Q = Q(x0 , d) and x̃ ∈ Q. Let Ã(x) = A(x) −
|α|=m α! (D A)Q x . We write, for f1 = f χ2Q and f2 = f χRn r2Q ,
Z
Rm+1 (A; x, y)
T A (f )(x) =
K(x, y)f (y) dy
|x − y|m
Rn
Z
Z
Rm+1 (A; x, y)
Rm (Ã; x, y)
K(x,
y)f
(y)
dy
+
K(x, y)f1 (y) dy
=
2
m
|x
−
y|
|x − y|m
n
n
R
R
X 1 Z K(x, y)(x − y)α
−
Dα Ã(y)f1 (y) dy;
α! Rn
|x − y|m
|α|=m
then
1
|Q|
Z A
T (f )(x) − T A (f2 )(x0 ) dx 6 1
T Rm (Ã; x, ·) f1 (x)dx
|Q| Q
|x − ·|m
Q
Z X
1
1 (x − ·)α α
+
T
D Ãf1 (x)dx
|Q|
α! |x − ·|m
Z
Q |α|=m
+
1
|Q|
Z
Q
A
T (f2 )(x) − T A (f2 )(x0 ) dx := I1 + I2 + I3 .
Now, for I1 , if x ∈ Q and y ∈ 2Q, using Lemma 2.6, we get
X
Rm (Ã; x, y) 6 C|x − y|m
kDα AkBMO ,
|α|=m
r
thus, by the L boundedness of T for any 1 < r < ∞ and Holder’ inequality, we
obtain
Z
X
X
1
I1 6 C
kDα AkBMO
|T (f1 )(x)| dx 6 C
kDα AkBMO kT (f1 )kLr |Q|−1/r
|Q| Q
|α|=m
|α|=m
X
X
α
−1/r
6C
kD AkBMO kf1 kLr |Q|
6C
kDα AkBMO Mr (f )(x̃).
|α|=m
|α|=m
For I2 , for any q > 1, l > 1 and denoting r = ql, by Lq -boundedness of T , we gain
Z X
C
α
α
I2 6
(D A − (D A)Q )f1 (x)dx
T
|Q|
Q
|α|=m
1/q
X 1 Z
α
α
q
6C
|T ((D A − (D A)Q )f1 )(x)| dx
|Q| Q
|α|=m
208
LIU
X
6 C|Q|−1/q
|α|=m
k(Dα A − (Dα A)Q )f1 kLq
1/(ql′ ) 1/(ql)
Z
X 1 Z
1
α
α
ql′
|D A(y) − (D A)Q | dy
|f (y)|ql dy
6C
|Q| Q
|Q| Q
|α|=m
X
6C
kDα AkBMO Mr (f )(x̃).
|α|=m
P
For I3 , by using Key Lemma, we have I3 6 C |α|=m kDα AkBMO Mr (f )(x̃). We
now put these estimates together, and
P taking the supremum over all Q such that
x̃ ∈ Q, we obtain (T A (f ))# (x̃) 6 C |α|=m kDα AkBMO Mr (f )(x̃); Thus, taking r
such that 1 < r < p, we obtain
X
(2)
kT A(f )kLp 6 Ck(T A (f ))# kLp 6 C
kDα AkBMO kMr (f )kLp
|α|=m
X
6C
|α|=m
α
kDα AkBMO kf kLp .
n
Secondly, we prove that, for D A ∈ Lipβ (R ) with |α| = m
X
(3)
(T A (f ))# 6 C
kDα AkLipβ (Mβ,r ((f )) + Mβ,1 (f ))
|α|=m
for any 1 6 r < n/β. In fact, by Lemma 2.6, we have, for x ∈ Q and y ∈ 2Q
X
sup |Dα A(z) − (Dα A)Q |
|Rm (Ã; x, y)| 6 C|x − y|m
|α|=m
z∈2Q
6 C|x − y|m |Q|β/n
X
|α|=m
kDα AkLipβ
and by Lemma 2.5, we have
k(Dα A − (Dα A)2Q )f χ2Q kLr 6 C|Q|1/r kDα AkLipβ Mβ,r (f )(x).
Similarly to the proof of (1), we obtain
Z
Z A
1
T (f )(x) − T A (f )(x0 ) dx 6 1
T Rm (Ã; x, ·) f1 (x) dx
m
|Q| Q
|Q| Q
|x − ·|
Z X
1
1 (x − ·)α α
+
T
D Ãf1 (x) dx
|Q|
α! |x − ·|m
Q |α|=m
+
6
1
|Q|
X
|α|=m
Z
Q
A
T (f2 )(x) − T A (f2 )(x0 ) dx
α
kD AkLipβ
C
|Q|1/r−β/n
Z
Q
1/r
|T (f1 )(x)| dx
1/r
X C Z
α
r
+
|T (D Ãf χ2Q )(x)| dx
|Q| Q
|α|=m
r
ESTIMATES OF MULTILINEAR SINGULAR INTEGRAL OPERATORS
+
6C
1
|Q|
X
Z
|α|=m
Q
209
A
T (f2 )(x) − T A (f2 )(x0 ) dx
kDα AkLipβ
1
kf1 kLr
|Q|1/r−β/n
1/r
X 1 Z
α
α
r
|(D A(x) − (D A)Q )f (x)χ2Q (x)| dx
+C
|Q| Rn
|α|=m
Z
A
1
T (f2 )(x) − T A (f2 )(x0 ) dx
+
|Q| Q
X
6C
kDα AkLipβ (Mβ,r (f )(x̃) + Mβ,1 (f )(x̃));
|α|=m
Thus, taking 1 6 r < p < n/β, 1/q = 1/p − β/n and by Lemma 2.4, we obtain
(4)
kT A(f )kLq 6 Ck(T A (f ))# kLq
X
6C
kDα AkLipβ (kMβ,r (f )kLq + kMβ,1(f )kLq )
|α|=m
6C
X
|α|=m
kDα AkLipβ kf kLp .
Now we verify that T A satisfies the conditions of Lemma 2.3. In fact, for any
1 < pi < n/β, 1/qi = 1/pi − β/n (i = 1, 2) and kf kLpi 6 1, note that T A (f )(x) =
T A−As (f )(x)+T As (f )(x) and Dα (As ) = (Dα A)s , by (2) and Lemma 2.1, we obtain
X
kT A−As (f )kLpi 6 C
kDα (A − As )kBMO kf kLpi
|α|=m
6C
X
|α|=m
kDα A − (Dα A)s kBMO 6 C
X
|α|=m
kDα AkBMOϕ ϕ(s),
and by (4) and Lemma 2.2, we obtain
X
X
kDα AkBMOϕ .
k(Dα A)s kLipβ kf kLpi 6 Cs−β ϕ(s)
kT As (f )kLqi 6 C
|α|=m
|α|=m
Thus, for s = t−1/n ,
mT A (f ) (t1/pi ϕ(t−1/n )) 6 mT A−As (f ) (t1/pi ϕ(t−1/n )/2) + mT As (f ) (t1/pi ϕ(t−1/n )/2)
pi −β
qi ϕ(s)
s ϕ(s)
6C
+
= Ct−1 .
t1/pi ϕ(s)
t1/pi ϕ(s)
Taking 1 < p2 < p < p1 < n/β and by Lemma 2.3, we get, for kf kLp 6 (p/p1 )1/p ,
Z
Z ∞
A
ψ(|T (f )(x)|) dx =
mT A (f ) (ψ −1 (t)) dt 6 C,
Rn
A
0
and thus, kT (f )kLψ 6 C. This completes the proof of Theorem 1.1.
210
LIU
Proof of Theorem 1.2. Let Q, Ã(x), f1 and f2 be the same as the proof of
Theorem 1.1, we write
Z
Rm+1 (Ã; x, y)
A
Ft (f )(x) =
Ft (x, y)f (y) dy
|x − y|m
n
R
Z
Z
Rm+1 (Ã; x, y)
Rm (Ã; x, y)
F
(x,
y)f
(y)
dy
+
Ft (x, y)f1 (y) dy
=
t
m
|x − y|
|x − y|m
Rn
Rn
Z
X 1
Ft (x, y)(x − y)α α
−
D Ã(y)f1 (y) dy,
α! Rn
|x − y|m
|α|=m
then
Z
Z
A
A
1
S (f )(x) − S A (f2 )(x0 ) dx = 1
|Ft (f )(x)k − kFtA (f2 )(x0 )k dx
|Q| Q
|Q| Q
Z 1
Ft Rm (Ã; x, ·) f1 (x) dx
6
|Q| Q |x − ·|m
Z
X 1 1
(x − ·)α α
+
Ft
D Ãf1 (x)
dx
m
α! |Q| Q
|x − ·|
|α|=m
Z
1
+
kF A (f2 )(x) − FtA (f2 )(x0 )k dx.
|Q| Q t
By using the same argument as in the proof of Theorem 1.1 will give the proof of
Theorem 1.2, so we omit the details.
3. Applications
In this section we shall apply Theorems 1.1 and 1.2 to some particular operators such as the Calderón–Zygmund singular integral operator, Littlewood–Paley
operator, and Marcinkiewicz operator.
3.1. Calderón–Zygmund singular integral operator. Let T be the Calderón–Zygmund operator [4, 6, 10], i.e., the multilinear operator related to T is
defined by
Z
Rm+1 (A; x, y)
T A (f )(x) =
K(x, y)f (y) dy.
|x − y|m
Then it is easily to verify that Key Lemma holds for T A , thus T satisfies the
conditions in Theorem 1.1. So, the conclusion of Theorem 1.1 holds for T A .
3.2. Littlewood–Paley operator. Let ε > 0 and ψ be a fixed functions
satisfying
(1) |ψ(x)| 6 C(1 + |x|)−(n+1) ,
(2) |ψ(x + y) − ψ(x)| 6 C|y|ε (1 + |x|)−(n+1+ε) when 2|y| < |x|;
The multilinear Littlewood-Paley operator is defined by
Z ∞
1/2
dt
gψA (f )(x) =
|FtA (f )(x)|2
,
t
0
ESTIMATES OF MULTILINEAR SINGULAR INTEGRAL OPERATORS
211
where
Rm+1 (A; x, y)
ψt (x − y)f (y) dy
|x − y|m
Rn
and ψt (x) = t−n ψ(x/t) for t > 0. We write that Ft (f ) = ψt ∗ f . We also define
1/2
Z ∞
2 dt
,
gψ (f )(x) =
|Ft (f )(x)|
t
0
FtA (f )(x) =
Z
which is the Littlewood–Paley g function [11];
R∞
1/2
Let H = h : khk = 0 |h(t)|2 dt/t
< ∞ be space. Then, for each
fixed x ∈ Rn , FtA (f )(x) may be regarded as a mapping from [0, +∞) to H, and
it is clear that gψ (f )(x) = kFt (f )(x)k and gψA (f )(x) = kFtA (f )(x)k. It has been
known that gψ is bounded on Lp (Rn ) for all 1 < p < ∞. Thus it is only to verify
that Key Lemma holds for gψA . In fact, we write, for a cube Q = Q(x0 , d) with
supp f ⊂ (2Q)c , x, x̃ ∈ Q = Q(x0 , d),
Z ψt (x − y) ψt (x0 − y)
FtA (f )(x) − FtA (f )(x0 ) =
−
Rm (Ã; x, y)f (y) dy
|x − y|m
|x0 − y|m
Rn
Z
ψt (x0 − y)
+
(Rm (Ã; x, y) − Rm (Ã; x0 , y))f (y) dy
m
Rn |x0 − y|
X 1 Z
(x − y)α ψt (x − y) (x0 − y)α ψt (x0 − y)
−
−
Dα Ã(y)f (y) dy
α! Rn
|x − y|m
|x0 − y|m
|α|=m
:= J1 + J2 + J3 ;
By the condition of ψ and Minkowski’s inequality, we obtain, for any r > 1,
Z
|Rm (Ã; x, y)kf (y)|
kJ1 k 6 C
|x0 − y|m
Rn
2 1/2
Z ∞ t|x − x0 |ε
dt
t|x − x0 |
×
+
dy
n+1
n+1+ε
|x
−
y|(t
+
|x
−
y|)
(t
+
|x
−
y|)
t
0
0
0
0
Z
|x − x0 |
|x − x0 |ε
6C
+
|Rm (Ã; x, y)kf (y)| dy
m+n+1
|x0 − y|
|x0 − y|m+n+ε
(2Q)c
X
6C
kDα AkBMO M (f )(x̃),
|α|=m
kJ2 k 6 C
X
|α|=m
α
kD AkBMO
∞ Z
X
k=1
2k+1 r2k Q
|x − x0 |
|f (y)| dy
|x0 − y|n+1
6 CkDα AkBMO M (f )(x̃),
∞ Z
X X
|x − x0 |ε
|x − x0 |
kJ3 k 6 C
+
|Dα Ã(y)kf (y)| dy
n+1
n+ε
|x
−
y|
|x
−
y|
k+1
k
0
0
r2 Q
|α|=m k=1 2
X
6C
kDα AkBMO Mr (f )(x̃).
|α|=m
212
LIU
From the above estimates, we see that Theorem 1.2 holds for gψA .
3.3. Marcinkiewicz
integral operator. Let Ω be homogeneous of degree
R
zero on Rn and S n−1 Ω(x′ ) dσ(x′ ) = 0. Assume that Ω ∈ Lipγ (S n−1 ) for 0 < γ 6 1,
i.e., there exists a constant M > 0 such that for any x, y ∈ S n−1 , |Ω(x) − Ω(y)| 6
M |x − y|γ . The multilinear Marcinkiewicz integral operator is defined by
Z ∞
1/2
A
2 dt
µA
(f
)(x)
=
|F
(f
)(x)|
,
Ω
t
t3
0
where
FtA (f )(x)
=
Z
|x−y|6t
and we write
Ft (f )(x) =
Ω(x − y) Rm+1 (A; x, y)
f (y) dy,
|x − y|n−1 |x − y|m
Z
|x−y|6t
We also define
µΩ (f )(x) =
Z
0
∞
Ω(x − y)
f (y) dy.
|x − y|n−1
dt
|Ft (f )(x)|2 3
t
1/2
,
which is the Marcinkiewicz integral [12].
R∞
1/2
Let H = h : khk = 0 |h(t)|2 dt/t3
< ∞ be a space. Then it is clear that
A
µΩ (f )(x) = kFt (f )(x)k and µA
Ω (f )(x) = kFt (f )(x)k. Now, it is only to verify that
A
Key Lemma holds for µΩ . In fact, for a cube Q = Q(x0 , d) with supp f ⊂ (2Q)c ,
x, x̃ ∈ Q = Q(x0 , d) and r > 1, we have
kFtA (f )(x) − FtA (f )(x0 )k
Z ∞Z
Ω(x − y)Rm (Ã; x, y)
6
f (y) dy
|x − y|m+n−1
0
|x−y|6t
2 1/2
Z
dt
Ω(x0 − y)Rm (Ã; x0 , y)
−
f (y) dy 3
m+n−1
|x0 − y|
t
|x0 −y|6t
α
X Z ∞ Z
Ω(x − y)(x − y)
+
|x − y|m+n−1
0
|x−y|6t
|α|=m
2 1/2
dt
Ω(x0 − y)(x0 − y)α
α
D
Ã(y)f
(y)
dy
t3
m+n−1
|x0 −y|6t |x0 − y|
2 1/2
Z ∞Z
|Ω(x − y)||Rm (Ã; x, y)|
dt
6
|f
(y)|
dy
m+n−1
|x
−
y|
t3
0
|x−y|6t, |x0 −y|>t
Z ∞Z
2 1/2
|Ω(x0 − y)||Rm (Ã; x0 , y)|
dt
+
|f
(y)|
dy
m+n−1
|x
−
y|
t3
0
0
|x−y|>t, |x0 −y|6t
Z ∞Z
Ω(x − y)R (Ã; x, y)
m
+
|x − y|m+n−1
0
|x−y|6t,|x0−y|6t
−
Z
ESTIMATES OF MULTILINEAR SINGULAR INTEGRAL OPERATORS
213
2 1/2
Ω(x0 − y)Rm (Ã; x0 , y) dt
−
|f (y)| dy
|x0 − y|m+n−1
t3
α
X Z ∞ Z
Ω(x − y)(x − y)
+
|x − y|m+n−1
0
|x−y|6t
|α|=m
2 1/2
dt
Ω(x0 − y)(x0 − y)α
α
D Ã(y)f (y) dy 3
−
m+n−1
|x
−
y|
t
0
|x0 −y|6t
:= L1 + L2 + L3 + L4
Z
and
L1 6 C
Z
|f (y)||Rm (Ã; x, y)|
|x − y|m+n−1
Z
Z
|f (y)||Rm (Ã; x, y)|
|x − y|m+n−1
Rn
6C
Rn
6C
Z
(2Q)c
6C
X
|α|=m
|x−y|6t<|x0−y|
dt
t3
1/2
1
1
−
|x − y|2
|x0 − y|2
dy
1/2
dy
|f (y)||Rm (Ã; x, y)| |x0 − x|1/2
dy
|x − y|m+n−1
|x − y|3/2
kDα AkBMO M (f )(x̃).
P
Similarly, we have L2 6 C |α|=m kDα AkBMO M (f )(x̃).
For L3 , by the following inequality [12]:
Ω(x − y)
Ω(x0 − y) |x − x0 |
|x − x0 |γ
|x − y|n−1 − |x0 − y|n−1 6 C |x0 − y|n + |x0 − y|n−1+γ ,
we gain
L3 6 C
X
|α|=m
α
kD AkBMO
Z
(2Q)c
×
6C
X
|α|=m
6C
X
|α|=m
kDα AkBMO
∞
X
|x − x0 |
|x − x0 |γ
+
|x0 − y|n
|x0 − y|n−1+γ
Z
|x0 −y|6t,|x−y|6t
dt
t3
1/2
|f (y)| dy
k(2−k + 2−γk )M (f )(x)
k=1
α
kD AkBMO M (f )(x̃);
Similarly to the cases of L1 , L2 and L3 , we obtain for L4
∞ Z
X X
|x − x0 |
|x − x0 |1/2
|x − x0 |γ
L4 6 C
+
+
|x0 − y|n+1
|x0 − y|n+γ
|x0 − y|n+1/2
2k+1 Qr2k Q
|α|=m k=1
× |Dα Ã(y)kf (y)| dy
214
LIU
6C
∞
X X
k(2−k + 2−k/2 + 2−γk )
|α|=m k=1
6C
X
|α|=m
1
|2k+1 Q|
Z
2k+1 Q
|Dα Ã(y)kf (y)| dy
kDα AkBMO Mr (f )(x̃).
Thus, Theorem 1.2 holds for µA
Ω.
References
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2. J. Cohen, J. Gosselin, On multilinear singular integral operators on Rn , Studia Math. 72
(1982), 199–223.
3.
, A BMO estimate for multilinear singular integral operators, Illinois J. Math. 30
(1986), 445–465.
4. R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611–635.
5. Y. Ding, S. Z. Lu, Weighted boundedness for a class rough multilinear operators, Acta Math.
Sinica 17 (2001), 517–526.
6. J. Garcia-Cuerva, J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics,
North-Holland Math. 116, Amsterdam, 1985.
7. S. Janson, Mean oscillation and commutators of singular integral operators, Ark. for Mat. 16
(1978), 263–270.
8. L. Z. Liu, Continuity for commutators of Littlewood-Paley operator on certain Hardy spaces,
J. Korean Math. Soc. 40 (2003), 41–60.
9. M. Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J. 44 (1995), 1–17.
10. E. M. Stein, Harmonic analysis: real variable methods, orthogonality and oscillatory integrals,
Princeton Univ. Press, Princeton NJ, 1993.
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12. A. Torchinsky, S. Wang, A note on the Marcinkiewicz integral, Colloq. Math. 60/61 (1990),
235–24.
Department of Mathematics
Hunan University
Changsha 410082
P. R. of China
lanzheliu@163.com
(Received 11 06 2011)