ARCHIVUM MATHEMATICUM (BRNO)
Tomus 50 (2014), 77–96
MEAN OSCILLATION AND BOUNDEDNESS
OF MULTILINEAR INTEGRAL OPERATORS
WITH GENERAL KERNELS
Liu Lanzhe
Abstract. In this paper, the boundedness properties for some multilinear
operators related to certain integral operators from Lebesgue spaces to Orlicz
spaces are proved. The integral operators include singular integral operator
with general kernel, Littlewood-Paley operator, Marcinkiewicz operator and
Bochner-Riesz operator.
1. Introduction and results
As the development of singular integral operators, their commutators and
multilinear operators have been well studied (see [3]–[7], [18]–[20]). Let T be
the Calderón-Zygmund singular integral operator and b ∈ BMO(Rn ), a classical
result of Coifman, Rochberg and Weiss (see [6]) stated that the commutator
[b, T ](f ) = T (bf ) − bT (f ) is bounded on Lp (Rn ) for 1 < p < ∞. The purpose of
this paper is to introduce some multilinear operator associated to certain integral
operators with general kernels (see [1, 10, 15]) and prove the boundedness properties
of the multilinear operators from Lebesgue spaces to Orlicz spaces.
In this paper, we are going to consider some integral operators as following (see
[1]).
Let l and mj be the positive integers (j = 1, . . . , l), m1 + · · · + ml = m and bj
be the functions on Rn (j = 1, . . . , l). Set, for 1 ≤ j ≤ l,
X 1
Rmj +1 (bj ; x, y) = bj (x) −
Dα bj (y)(x − y)α .
α!
|α|≤mj
Definition 1. Let T : S → S 0 be a linear operator such that T is bounded on
L2 (Rn ) and has a kernel K, that is there exists a locally integrable function K(x, y)
2010 Mathematics Subject Classification: primary 42B20; secondary 42B25.
Key words and phrases: multilinear operator, singular integral operator, BMO space, Orlicz
space, Littlewood-Paley operator, Marcinkiewicz operator, Bochner-Riesz operator.
Supported by the Scientific Research Fund of Hunan Provincial Education Departments
(13K013).
Received November 11, 2013, revised February 2014. Editor V. Müller.
DOI: 10.5817/AM2014-2-77
78
L. LANZHE
on Rn × Rn \ {(x, y) ∈ Rn × Rn : x = y} such that
Z
T (f )(x) =
K(x, y)f (y)dy
Rn
for every bounded and compactly supported function f , where K satisfies:
|K(x, y)| ≤ C|x − y|−n ,
Z
(|K(x, y) − K(x, z)| + |K(y, x) − K(z, x)|) dx ≤ C ,
2|y−z|<|x−y|
and there is a sequence of positive constant numbers {Ck } such that for any k ≥ 1,
Z
1/q
(|K(x, y) − K(x, z)| + |K(y, x) − K(z, x)|)q dy
2k |z−y|≤|x−y|<2k+1 |z−y|
0
≤ Ck (2k |z − y|)−n/q ,
where 1 < q 0 < 2 and 1/q + 1/q 0 = 1. The multilinear operator related to the
operator T is defined by
Z Ql
j=1 Rmj +1 (bj ; x, y)
b
K(x, y)f (y) dy .
T (f )(x) =
|x − y|m
n
R
Definition 2. Let F (x, y, t) define on Rn × Rn × [0, +∞), we denote that
Z
Ft (f )(x) =
F (x, y, t)f (y) dy
Rn
and
Ftb (f )(x)
Z
=
Rn
Ql
j=1
Rmj +1 (bj ; x, y)
|x − y|m
F (x, y, t)f (y) dy
for every bounded and compactly supported function f . Let H be the Banach
space H = {h : khk < ∞}. For each fixed x ∈ Rn , we view Ft (f )(x) and Ftb (f )(x)
as a mapping from [0, +∞) to H. Then, the multilinear operators related to Ft is
defined by
S b (f )(x) = kFtb (f )(x)k ,
where Ft satisfies:
kF (x, y, t)k ≤ C|x − y|−n ,
Z
(kF (x, y, t)−F (x, z, t)k+kF (y, x, t)−F (z, x, t)k) dx ≤ C ,
2|y−z|<|x−y|
and there is a sequence of positive constant numbers {Ck } such that for any k ≥ 1,
Z
1/q
(kF (x, y, t)−F (x, z, t)k+kF (y, x, t)−F (z, x, t)k)q dy
2k |z−y|≤|x−y|<2k+1 |z−y|
0
≤ Ck (2k |z − y|)−n/q ,
where 1 < q 0 < 2 and 1/q + 1/q 0 = 1. We also define that S(f )(x) = kFt (f )(x)k.
MULTILINEAR INTEGRAL OPERATORS
79
Note that the classical Calderón-Zygmund singular integral operator satisfies
Definition 1 (see [8, 19, 20, 22, 23]) and that T b and S b are just the commutators
of T and S with b if m = 0 (see [6, 9, 11, 19, 20]). While when m > 0, it is
non-trivial generalizations of the commutators. Let T be the Calderón-Zygmund
singular integral operator, a classical result of Coifman, Rochberg and Weiss (see
[6]) states that the commutator [b, T ] = T (bf ) − bT f (where b ∈ BMO(Rn )) is
bounded on Lp (Rn ) for 1 < p < ∞, Chanillo (see [2]) proves a similar result when
T is replaced by the fractional integral operator. In [9], Janson proved boundedness
properties for the commutators related to the Calderón-Zygmund singular integral
operators from Lebesgue spaces to Orlicz spaces. It is well known that multilinear
operators are of great interest in harmonic analysis and have been widely studied
by many authors (see [3]–[5], [7]). The main purpose of this paper is to prove the
boundedness properties for the multilinear operators T b and S b from Lebesgue
spaces to Orlicz spaces.
Let us introduce some notations. Throughout this 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 ,
Q3x |Q| Q
R
where, and in what follows, fQ = |Q|−1 Q f (x) dx. It is well-known that (see
[8, 22])
Z
1
f # (x) ≈ sup inf
|f (y) − c| dy .
Q3x c∈C |Q| Q
Let M be the Hardy-Littlewood maximal operator defined by
Z
1
M (f )(x) = sup
|f (y)| dy .
Q3x |Q| Q
We write that Mp f = (M (f p ))1/p for 0 < p < ∞. For 1 ≤ r < ∞ and 0 < β < n,
let
1/r
Z
1
r
Mβ,r (f )(x) = sup
|f
(y)|
dy
.
1−rβ/n
Q3x |Q|
Q
We say that f belongs to BMO(Rn ) if f # belongs to L∞ (Rn ) and kf kBMO =
kf kL∞ . More generally, let ρ be a non-decreasing positive function on [0, +∞)
and define BMOρ (Rn ) as the space of all functions f such that
Z
1
|f (y) − fQ | dy ≤ Cρ(r) .
|Q(x, r)| Q(x,r)
#
For β > 0, the Lipschitz space Lipβ (Rn ) is the space of functions f such that
kf kLipβ = sup |f (x) − f (y)|/|x − y|β < ∞ .
x6=y
For f , mf denotes the distribution function of f , that is mf (t) = |{x ∈ Rn :
|f (x)| > t}|.
80
L. LANZHE
Let ρ be a non-decreasing convex function on [0, +∞) with ρ(0) = 0. ρ−1 denotes
the inverse function
of ρ. The Orlicz space Lρ (Rn ) is defined by the set of functions
R
f such that Rn ρ(λ|f (x)|)dx < ∞ for some λ > 0. The Luxemburg norm is given
by (see [21])
Z
−1
kf kLρ = inf λ
1+
ρ(λ|f (x)|) dx .
λ>0
Rn
We shall prove the following theorems in Section 2.
Theorem 1. Let 0 < β ≤ 1, q 0 < p < n/lβ and ϕ, ψ be two non-decreasing positive
functions on [0, +∞) with (ψ l )−1 (t) = t1/p ϕl (t−1/n ). Suppose that ψ is convex,
ψ(0) = 0, ψ(2t) ≤ Cψ(t). Let T be the same as in Definition 1 and the sequence
{k l Ck } ∈ l1 . Then T b is bounded from Lp (Rn ) to Lψl (Rn ) if Dα bj ∈ BMO(Rn )
for all α with |α| = mj and j = 1, . . . , l.
Theorem 2. Let 0 < β ≤ 1, q 0 < p < n/mβ and ϕ, ψ be two non-decreasing
positive functions on [0, +∞) with (ψ l )−1 (t) = t1/p ϕl (t−1/n ). Suppose that ψ is
convex, ψ(0) = 0, ψ(2t) ≤ Cψ(t). Let S be the same as in Definition 2 and
the sequence {Ck } ∈ l1 . Then S b is bounded from Lp (Rn ) to Lψl (Rn ) if Dα bj ∈
BMO(Rn ) for all α with |α| = mj and j = 1, . . . , l.
Remark.
(a) If l = 1 and ψ −1 (t) = t1/p ϕ(t−1/n ), then T b and S b are all
bounded on from Lp (Rn ) to Lψ (Rn ) under the conditions of Theorems 1 and 2.
(b) If l = 1, ϕ(t) ≡ 1 and ψ(t) = tp for 1 < p < ∞, then T b and S b are all
bounded on Lp (Rn ) if Dα b ∈ BMOϕ (Rn ) for all α with |α| = m.
(c) If l = 1, ψ(t) = ts and ϕ(t) = tn(1/p−1/s) for 1 < p < s < ∞, then, by
BMOtβ (Rn ) = Lipβ (Rn ) (see [9, Lemma 4]), T b and S b are all bounded from
Lp (Rn ) to Ls (Rn ) if Dα b ∈ Lipn(1/p−1/s) (Rn ) for all α with |α| = m.
2. Proof of theorems
We begin with the following preliminary lemmas.
Lemma 1 (see [1]). Let T and S be the the same as Definitions 1 and 2, the
sequence {Ck } ∈ l1 . Then T and S are bounded on Lp (Rn ) for 1 < p < ∞.
Lemma 2 (see [9]). Let ρ be a non-decreasing positive function on [0, +∞) and
n
ηR be an infinitely differentiable function
that
R on R with compact support such
t
t
η(x)
dx
=
1.
Denote
that
b
(x)
=
b(x
−
ty)η(y)
dy.
Then
kb
−
b
k
≤
BMO
n
n
R
R
Cρ(t)kbkBMOρ .
Lemma 3 (see [1]). Let 0 < β < 1 or β = 1 and ρ be a non-decreasing positive
function on [0, +∞). Then kbt kLipβ ≤ Ct−β ρ(t)kbkBMOρ .
Lemma 4 (see [1]). Suppose 1 ≤ p2 < p < p1 < ∞, ρ is a non-increasing function on R+ , B is a linear or sublinear operator such that mB(f ) (t1/p1 ρ(t)) ≤ Ct−1 if
R∞
kf kLp1 ≤ 1 and mB(f ) (t1/p2 ρ(t)) ≤ Ct−1 if kf kLp2 ≤ 1. Then 0 mB(f ) (t1/p ρ(t)) dt ≤
C if kf kLp ≤ (p/p1 )1/p .
Lemma 5 (see [2]). Suppose that 0 < β < n, 1 ≤ r < p < n/β and 1/s = 1/p−β/n.
Then kMβ,r (f )kLs ≤ Ckf kLp .
MULTILINEAR INTEGRAL OPERATORS
81
Lemma 6 (see [5]). Let b be a function on Rn and Dα A ∈ Lq (Rn ) for all α with
|α| = m and some q > n. Then
|Rm (b; x, y)| ≤ C|x − y|
m
X |α|=m
1
|Q̃(x, y)|
Z
|Dα b(z)|q dz
1/q
,
Q̃(x,y)
√
where Q̃ is the cube centered at x and having side length 5 n|x − y|.
To prove the theorems of the paper, we need the following
Key Lemma. Let T and S be the same as in Definitions 1 and 2. Suppose that
Q = Q(x0 , d) is a cube with supp f ⊂ (2Q)c and x, x̃ ∈ Q.
(I) If the sequence {k l Ck } ∈ l1 and Dα bj ∈ BMO(Rn ) for all α with |α| = mj
and j = 1, . . . , l, then
l X
Y
|T b (f )(x) − T b (f )(x0 )| ≤ C
j=1
kDαj bj kBMO Mr (f )(x̃) for any r > q 0 ;
|αj |=mj
(II) If the sequence {Ck } ∈ l1 , 0 < β ≤ 1 and Dα bj ∈ Lipβ (Rn ) for all α with
|α| = mj and j = 1, . . . , l, then
|T b (f )(x)−T b (f )(x0 )| ≤ C
l X
Y
j=1
kDαj bj kLipβ Mlβ,r (f )(x̃) for any r > q 0 ;
|αj |=mj
(III) If the sequence {k l Ck } ∈ l1 and Dα bj ∈ BMO(Rn ) for all α with |α| = mj
and j = 1, . . . , l, then
kFtb (f )(x)−Ftb (f )(x0 )k ≤ C
l X
Y
j=1
kDαj bj kBMO Mr (f )(x̃) for any r > q 0 ;
|αj |=mj
(IV) If the sequence {k l Ck } ∈ l1 , 0 < β ≤ 1 and Dα bj ∈ Lipβ (Rn ) for all α
with |α| = mj and j = 1, . . . , l, then
kFtb (f )(x)−Ftb (f )(x0 )k ≤ C
l X
Y
j=1
kDαj bj kLipβ Mlβ,r (f )(x̃) for any r > q 0 .
|αj |=mj
√
Proof. Without P
loss of generality, we may assume l = 2. Let Q̃ = 5 nQ and
1
α
α
α
b̃j (x) = bj (x) −
α! (D bj )Q̃ x , then Rm (bj ; x, y) = Rm (b̃j ; x, y) and D b̃j =
|α|=m
82
L. LANZHE
Dα bj − (Dα bj )Q̃ for |α| = mj . We write, for supp f ⊂ (2Q)c and x, x̃ ∈ Q,
T b (f )(x) − T b (f )(x0 ) =
2
K(x, y)
K(x0 , y) Y
−
Rmj (b̃j ; x, y)f (y) dy
m
|x0 − y|m j=1
Rn |x − y|
Z
Z
Rm2 (b̃2 ; x, y)
K(x0 , y)f (y) dy
Rm1 (b̃1 ; x, y) − Rm1 (b̃1 ; x0 , y)
|x0 − y|m
Rn
Z
Rm1 (b̃1 ; x0 , y)
+
Rm2 (b̃2 ; x, y) − Rm2 (b̃2 ; x0 , y)
K(x0 , y)f (y) dy
|x0 − y|m
n
R
Z
i
h R (b̃ ; x, y)(x−y)α1
X 1
Rm2 (b̃2 ; x0 , y)(x0−y)α1
m2 2
−
K(x,
y)
−
K(x
,
y)
0
α1 ! R n
|x−y|m
|x0−y|m
+
|α1 |=m1
× Dα1 b̃1 (y)f (y) dy
i
X 1 Z h Rm (b̃1 ; x, y)(x−y)α2
Rm1 (b̃1 ; x0 , y)(x0−y)α2
1
−
K(x,
y)
−
K(x
,
y)
0
α2 ! R n
|x−y|m
|x0−y|m
|α2 |=m2
× Dα2 b̃2 (y)f (y) dy
Z h
i
X
(x − y)α1 +α2
(x0 − y)α1 +α2
1
K(x, y) −
K(x0 , y)
+
m
m
α1 !α2 ! Rn
|x − y|
|x0 − y|
|α1 |=m1 , |α2 |=m2
× Dα1 b̃1 (y)Dα2 b̃2 (y)f (y) dy
= I1 + I2 + I3 + I4 + I5 + I6 .
(I). By Lemma 6 and the following inequality (see [10]), for b ∈ BMO(Rn ),
|bQ1 − bQ2 | ≤ C log(|Q2 |/|Q1 |)kbkBMO for Q1 ⊂ Q2 ,
we know that, for x ∈ Q and y ∈ 2k+1 Q \ 2k Q with k ≥ 1,
X
|Rm (b̃; x, y)| ≤ C|x − y|m
(kDα bkBMO + |(Dα b)Q̃(x,y) − (Dα b)Q̃ |)
|α|=m
m
≤ Ck|x − y|
X
kDα bkBMO .
|α|=m
Note that |x − y| ∼ |x0 − y| for x ∈ Q and y ∈ Rn \ Q̃, by the conditions on K and
recalling r > q 0 , we obtain
Z
|I1 | ≤
Rn \2Q
2
Y
1
1
−
y)|
|Rmj (b̃j ; x, y)| |f (y)| dy
|K(x,
|x − y|m
|x0 − y|m
j=1
Z
+
Rn \2Q
|K(x, y) − K(x0 , y)| |x0 − y|−m
2
Y
j=1
|Rmj (b̃j ; x, y)| |f (y)| dy
MULTILINEAR INTEGRAL OPERATORS
≤
∞ Z
X
2k+1 Q\2k Q
k=1
+
|K(x, y) − K(x0 , y)| |x0 − y|−m
2k+1 Q\2k Q
2 X
Y
j=1
+C
αj
kD bj kBMO
∞
X
kDαj bj kBMO
Z
2k+1 Q\2k Q
∞
X
|αj |=mj
Z
k
2
k=1
X
2
Y
|Rmj (b̃j ; x, y)| |f (y)| dy
j=1
|αj |=mj
2
Y
j=1
×
2
Y
1
1
|K(x,
y)|
−
|Rmj (b̃j ; x, y)||f (y)| dy
|x − y|m
|x0 − y|m
j=1
∞ Z
X
k=1
≤C
83
k2
|x − x0 |
|f (y)| dy
|x0 − y|n+1
Z
0
|f (y)|q dy
1/q0
2k+1 Q\2k Q
k=1
1/q
|K(x, y) − K(x0 , y)|q dy
2k+1 Q\2k Q
≤C
2
Y
j=1
≤C
X
kD bj kBMO
|αj |=mj
2 X
Y
j=1
αj
∞
X
2
k (2
−k
+Ck )
Z
1
|2k+1 Q|
k=1
|f (y)|r dy
1/r
2k+1 Q
kDαj bj kBMO Mr (f )(x̃) .
|αj |=mj
For I2 , by the formula (see [5]):
Rm (b̃; x, y) − Rm (b̃; x0 , y) =
X 1
Rm−|γ| (Dγ b̃; x, x0 )(x − y)γ
γ!
|γ|<m
and Lemma 6, we have
|Rm (b̃; x, y) − Rm (b̃; x0 , y)| ≤ C
X
X
|x − x0 |m−|γ| |x − y||γ| kDα bkBMO ,
|γ|<m |α|=m
thus
|I2 | ≤ C
2 X
Y
j=1
≤C
2
Y
j=1
≤C
∞ Z
X
|αj |=mj
X
k=1
αj
kD bj kBMO
∞
X
|αj |=mj
2 X
Y
j=1
kDαj bj kBMO
k
2k+1 Q\2k Q
−k
k2
k=1
|x − x0 |
|f (y)| dy
|x0 − y|n+1
1
|2k+1 Q|
Z
2k+1 Q
Dαj bj kBMO Mr (f )(x̃) .
|αj |=mj
Similarly,
|I3 | ≤ C
2 X
Y
j=1
|αj |=mj
kDαj bj kBMO Mr (f )(x̃) .
|f (y)|r dy
1/r
84
L. LANZHE
For I4 , similar to the proof of I1 and I2 , taking 1 < p < ∞ such that 1/p+1/q+1/r =
1, we get
(x − y)α1
(x0 − y)α1 −
|K(x, y)|
m
|x0 − y|m
Rn \2Q |x − y|
X Z
|I4 | ≤ C
|α1 |=m1
× |Rm2 (b̃2 ; x, y)| |Dα1 b̃1 (y)| |f (y)| dy
X Z
|Rm2 (b̃2 ; x, y) − Rm2 (b̃2 ; x0 , y)|
+C
Rn \2Q
|α1 |=m1
|(x0 − y)α1 K(x, y)| α1
|D b̃1 (y)| |f (y)| dy
|x0 − y|m
(x − y)α1 X Z
0
|K(x, y) − K(x0 , y)|
+C
m |x
−
y|
n
0
R \2Q
×
|α1 |=m1
× |Rm2 (b̃2 ; x0 , y)| |Dα1 b̃1 (y)| |f (y)| dy
X
≤C
α2
||D b2 ||BMO
|α2 |=m2
×
1
X
Z
|α1 |=m1
|f (y)|r dy
|2k+1 Q|
0
|Dα1 b̃1 (y)|r dy
2k+1 Q
1/r
2k+1 Q
kDα2 b2 kBMO
|α2 |=m2
×k
k2
Z
1
X −k
k=1
|2k+1 Q|
+C
∞
X
∞
X X
|α1 |=m1 k=1
Z
|K(x, y) − K(x0 , y)|q dy
1/q
2k+1 Q\2k Q
×
Z
|Dα1 b̃1 (y)|p dy
1/p Z
2k+1 Q\2k Q
≤C
2 X
Y
j=1
1/r
2k+1 Q\2k Q
kDαj bj kBMO
|αj |=mj
1
|f (y)|r dy
∞
X
k 2 (2−k + Ck )
k=1
Z
1/r
|f (y)|r dy
|2k+1 Q| 2k+1 Q
2 X
Y
≤C
kDαj bj kBMO Mr (f )(x̃) .
×
j=1
|αj |=mj
Similarly,
|I5 | ≤ C
2 X
Y
j=1
|αj |=mj
kDαj bj kBMO Mr (f )(x̃) .
1/r0
MULTILINEAR INTEGRAL OPERATORS
85
For I6 , taking 1 < r1 , r2 < ∞ such that 1/q + 1/p + 1/r1 + 1/r2 = 1, then
(x − y)α1 +α2 K(x, y) (x − y)α1 +α2 K(x , y) X Z
0
0
−
|I6 | ≤ C
m
m
|x
−
y|
|x
−
y|
n
0
R \2Q
|α1 |=m1
|α2 |=m2
× |Dα1 b̃1 (y)||Dα2 b̃2 (y)||f (y)| dy
Z
∞
1/r
X X
1
−k
≤C
(2 + Ck ) k+1
|f (y)|r dy
|2
Q| 2k+1 Q
|α1 |=m1 k=1
|α2 |=m2
×
1
Z
|Dα1 b̃1 (y)|r1 dy
1/r
1
1
Z
|Dα2 b̃2 (y)|r2 dy
1/r2
|2k+1 Q| 2k+1 Q
|2k+1 Q| 2k+1 Q
2 X
∞
X
Y
≤C
kDαj bj kBMO
k 2 (2−k + Ck )Mr (f )(x̃)
j=1
≤C
2
Y
|αj |=mj
X
j=1
k=1
kDαj bj kBMO Mr (f )(x̃) .
|αj |=mj
Thus
|T b (f )(x) − T b (f )(x0 )| ≤ C
2 X
Y
j=1
kDαj bj kBMO Mr (f )(x̃) .
|αj |=mj
(II). By Lemma 6 and the following inequality, for b ∈ Lipβ (Rn ),
Z
1
kbkLipβ |x − y|β dy ≤ CkbkLipβ (|x − x0 | + d)β ,
|b(x) − bQ | ≤
|Q| Q
we get
X
|Rm (b̃; x, y)| ≤ C
kDα bkLipβ (|x − y| + d)m+β
|α|=m
and
X
|Rm (b̃; x, y) − Rm (b̃; x0 , y)| ≤ C
kDα bkLipβ (|x − y| + d)m+β ,
|α|=m
then
|I1 | ≤
∞ Z
X
k=1
+
2k+1 Q\2k Q
2
Y
1
1
−
|K(x,
y)|
|Rmj (b̃j ; x, y)| |f (y)| dy
|x − y|m
|x0 − y|m
j=1
∞ Z
X
k=1
≤C
|K(x, y) − K(x0 , y)| |x0 − y|−m
2k+1 Q\2k Q
2 X
Y
j=1
|αj |=mj
2
Y
|Rmj (b̃j ; x, y)| |f (y)| dy
j=1
αj
kD bj kLipβ
∞ Z
X
k=1
2k+1 Q\2k Q
|x − x0 |
|f (y)| dy
|x0 − y|n+1−2β
86
L. LANZHE
2 X
Y
+C
j=1
×
kDαj bj kLipβ
∞
X
|αj |=mj
Z
|2k+1 Q|2β/n
Z
0
|f (y)|q dy
1/q0
2k+1 Q\2k Q
k=1
|K(x, y) − K(x0 , y)|q dy
1/q
2k+1 Q\2k Q
≤C
2 Y
j=1
≤C
X
∞
X
αj
kD bj kLipβ
|αj |=mj
2 X
Y
j=1
(2 +Ck )
−k
k=1
1
|2k+1 Q|1−2βr/n
Z
|f (y)|r dy
1/r
2k+1 Q
kDαj bj kLipβ M2β,r (f )(x̃) ,
|αj |=mj
|I2+I3 | ≤ C
2 X
Y
∞
X
αj
kD bj kLipβ
j=1 |αj |=mj
≤C
2 X
Y
j=1
1
(2 +Ck ) k+1 1−2βr/n
|2
Q|
k=1
−k
Z
|f (y)|r dy
2k+1 Q
kDαj bj kLipβ M2β,r (f )(x̃) ,
|αj |=mj
(x − y)α1
(x0 − y)α1 −
|K(x, y)| |Rm2 (b̃2 ; x, y)|
m
|x0 − y|m
Rn \2Q |x − y|
X Z
|I4 | ≤ C
|α1 |=m1
× |Dα1 b̃1 (y)| |f (y)| dy
X Z
+C
|α1 |=m1
|Rm2 (b̃2 ; x, y) − Rm2 (b̃2 ; x0 , y)|
Rn \2Q
|(x0 − y)α1 K(x, y)|
|x0 − y|m
× |Dα1 b̃1 (y)| |f (y)| dy
(x − y)α1 0
|K(x, y − K(x0 , y)|
|Rm2 (b̃2 ; x0 , y)|
|x0 − y|m
Rn \2Q
X Z
+C
|α1 |=m1
× |Dα1 b̃1 (y)| |f (y)| dy
≤C
2 X
Y
j=1
+C
∞
X
|αj |=mj
αj
kD bj kLipβ
|αj |=mj
Z
2k+1 Q\2k Q
2−k
k=1
2 X
Y
j=1
×
kDαj bj kLipβ
∞
X
1
|2k+1 Q|1−2β/n
k+1
|2
Q|
2β/n
|f (y)| dy
1/q
2k+1 Q
Z
2k+1 Q\2k Q
k=1
|K(x, y) − K(x0 , y)|q dy
Z
0
|f (y)|q dy
1/q0
1/r
MULTILINEAR INTEGRAL OPERATORS
≤C
2 X
Y
j=1
≤C
2
Y
j=1
|I6 | ≤ C
|αj |=mj
2 X
Y
j=1
|I5 | ≤ C
kDαj bj kLipβ
∞
X
(2−k+Ck )
k=1
1
|2k+1 Q|1−2βr/n
87
Z
|f (y)|r dy
1/r
2k+1 Q
kDαj bj kLipβ M2β,r (f )(x̃) ,
|αj |=mj
X
kDαj bj kLipβ M2β,r (f )(x̃) ,
|αj |=mj
X
|α1 |=m1 ,
|α2 |=m2
(x − y)α1 +α2 K(x, y) (x − y)α1 +α2 K(x , y) 0
0
−
m
m
|x
−
y|
|x
−
y|
n
0
R \2Q
Z
× |Dα1 b̃1 (y)| |Dα2 b̃2 (y)| |f (y)| dy
≤C
2 X
Y
αj
kD bj kLipβ
j=1 |αj |=mj
≤C
2 X
Y
j=1
∞
X
(2
+Ck )
−k
k=1
1
|2k+1 Q|1−2βr/n
Z
|f (y)|r dy
1/r
2k+1 Q
kDαj bj kLipβ M2β,r (f )(x̃) .
|αj |=mj
Thus
|T b (f )(x) − T b (f )(x0 )| ≤ C
2 X
Y
j=1
kDαj bj kLipβ M2β,r (f )(x̃) .
|αj |=mj
A same argument as in the proof of (I) and (II) will give the proof of (III) and
(VI), we omit the details.
Now we are in position to prove our theorems.
Proof of Theorem 1. Without loss of generality, we may assume l = 2. We prove
the theorem in several steps. First, we prove, if Dα bj ∈ BMO(Rn ) for all α with
|α| = mj and j = 1, . . . , l,
(1)
(T b (f ))# ≤ C
2 X
Y
j=1
kDαj bj kBMO Mr (f )
|αj |=mj
√
for any r with q 0 < r <P
∞. Fix a cube Q = Q(x0 , d) and x̃ ∈ Q. Let Q̃ = 5 nQ
1
α
α
and b̃j (x) = bj (x) −
α! (D bj )Q̃ x , then Rm (bj ; x, y) = Rm (b̃j ; x, y) and
|α|=m
Dα b̃j = Dα bj − (Dα bj )Q̃ for |α| = mj . We write, for f1 = f χQ̃ and f2 = f χRn \Q̃ ,
88
L. LANZHE
Q2
Z
b
T (f )(x) =
j=1
Rmj (b̃j ; x, y)
K(x, y)f1 (y) dy
|x − y|m
Z
X
1
Rm2 (b̃2 ; x, y)(x − y)α1 Dα1 b̃1 (y)
K(x, y)f1 (y) dy
−
α1 ! Rn
|x − y|m
|α1 |=m1
Z
X
1
Rm1 (b̃1 ; x, y)(x − y)α2 Dα2 b̃2 (y)
K(x, y)f1 (y) dy
−
α2 ! Rn
|x − y|m
|α2 |=m2
Z
X
1
(x−y)α1+α2 Dα1 b̃1 (y)Dα2 b̃2 (y)
K(x, y)f1 (y) dy
+
α1 !α2 ! Rn
|x − y|m
Rn
|α1 |=m1
|α2 |=m2
Z
Q2
j=1
+
=T
Rn
Q2
j=1
Rmj +1 (bj ; x, y)
|x − y|m
Rmj (b̃j ; x, ·)
f1
K(x, y)f2 (y) dy
|x − ·|m
X
1 Rm2 (b̃2 ; x, ·)(x − ·)α1 Dα1 b̃1 −T
f1
α1 !
|x − ·|m
|α1 |=m1
−T
X
+T
X
|α2 |=m2
|α1 |=m1
|α2 |=m2
1 Rm1 (b̃1 ; x, ·)(x − ·)α2 Dα2 b̃2 f1
α2 !
|x − ·|m
1 (x − ·)α1 +α2 Dα1 b̃1 Dα2 b̃2 f1 + T b (f2 )(x) ,
α1 !α2 !
|x − ·|m
then
Q2 Rm (b̃j ; x, ·) j
j=1
f1 |T (f )(x) − T (f2 )(x0 )| ≤ T
|x − ·|m
X
1 Rm2 (b̃2 ; x, ·)(x − ·)α1 Dα1 b̃1 + T
f1 α1 !
|x − ·|m
b
b
|α1 |=m1
X
+ T
|α2 |=m2
X
+ T
|α1 |=m1
|α2 |=m2
1 Rm1 (b̃1 ; x, ·)(x − ·)α2 Dα2 b̃2 f1 α2 !
|x − ·|m
1 (x − ·)α1 +α2 Dα1 b̃1 Dα2 b̃2 f1 α1 !α2 !
|x − ·|m
+ |T b (f2 )(x) − T b (f2 )(x0 )|
= L1 (x) + L2 (x) + L3 (x) + L4 (x) + L5 (x)
MULTILINEAR INTEGRAL OPERATORS
89
and
1
|Q|
Z
Z
Z
1
1
|T b (f )(x) − T b (f2 )(x0 )| dx ≤
L1 (x) dx +
L2 (x) dx
|Q| Q
|Q| Q
Q
Z
Z
Z
1
1
1
L3 (x) dx +
L4 (x) dx +
L5 (x) dx
+
|Q| Q
|Q| Q
|Q| Q
= L1 + L2 + L3 + L4 + L5 .
Now, for L1 , if x ∈ Q and y ∈ 2Q, by using Lemma 6, we get
X
Rm (b̃; x, y) ≤ C|x − y|m
kDα bkBMO ,
|α|=m
thus, by the Lr boundedness of T (see Lemma 1) and Hölder’s inequality, we obtain
2 X
1 Z
Y
|T (f1 )(x)| dx
L1 ≤ C
kDαj bj kBMO
|Q| Q
j=1
|αj |=mj
≤C
2
Y
j=1
≤C
2
Y
j=1
≤C
2
Y
j=1
kDαj bj kBMO
|αj |=mj
2 X
Y
j=1
≤C
X
|αj |=mj
X
|αj |=mj
X
1 Z
1/r
|T (f1 )(x)|r dx
|Q| Rn
1/r
1 Z
kD bj kBMO
|f1 (x)|r dx
|Q| Rn
αj
kDαj bj kBMO
1/r
1 Z
|f (x)|r dx
|Q̃| Q̃
kDαj bj kBMO Mr (f )(x̃) .
|αj |=mj
For L2 , denoting r = uv for 1 < u, v < ∞ and 1/v + 1/v 0 = 1, we have
X
X X 1 Z
L2 ≤ C
kDα2 b2 kBMO
|T (Dα1 b̃1 f1 )(x)| dx
|Q| Q
|α2 |=m2
|α1 |=m1 |α|=m
1/u
X
X 1 Z
α2
≤C
|T (Dα1 b̃1 f1 )(x)|u dx
kD b2 kBMO
|Q| Rn
|α2 |=m2
|α1 |=m1
1/u
X
X 1 Z
≤C
kDα2 b2 kBMO
|Dα1 b̃1 (x)| |f1 (x)|u dx
|Q| Rn
|α2 |=m2
|α1 |=m1
Z
1
1/uv
X
≤C
kDα2 b2 kBMO
|f (x)|uv dx
|Q̃| Q̃
|α2 |=m2
Z
1/uv0
X 1
0
×
|Dα1 b̃1 (x) − (Dα1 b̃1 )Q̃ |uv dx
|Q̃| Q̃
|α |=m
1
1
90
L. LANZHE
≤C
2 X
Y
j=1
kDαj bj kBMO Mr (f )(x̃) .
|αj |=mj
For L3 , similar to the proof of L2 , we get
L3 ≤ C
2 X
Y
j=1
kDαj bj kBMO Mr (f )(x̃ .
|αj |=mj
Similarly, for L4 , denoting r = uw for 1 < u, v1 , v2 , w < ∞ and 1/v1 +1/v2 +1/w =
1, we obtain, by Hölder’inequality,
L4 ≤ C
X
|α1 |=m1
|α2 |=m2
≤C
1
|Q|
Z
|T (Dα1 b̃1 Dα2 b̃2 f1 )(x)| dx
Q
1/u
X 1 Z
|T (Dα1 b̃1 Dα2 b̃2 f1 )(x)|u dx
|Q| Rn
|α1 |=m1
|α2 |=m2
≤C
X
|Q|−1/u
Z
|Dα1 b̃1 (x)Dα2 b̃2 (x)f1 (x)|u dx
1/u
Rn
|α1 |=m1
|α2 |=m2
1/uv1 1 Z
1/uv2
X 1 Z
α1
uv1
|D b̃1 (x)| dx
|Dα2 b̃2 (x)|uv2 dx
≤C
|Q̃| Q̃
|Q̃| Q̃
|α |=m
1
1
|α2 |=m2
1/uw
1 Z
|f (x)|uw dx
|Q̃| Q̃
2
Y X
≤C
kDαj bj kBMO Mr (f )(x̃) .
×
j=1
|αj |=mj
For L5 , by using Key Lemma, we have
2 X
Y
L3 ≤ C
kDαj bj kBMO Mr (f )(x̃) .
j=1
|αj |=mj
We now put these estimates together and take the supremum over all Q such
that x̃ ∈ Q, we obtain
2 X
Y
(T b (f ))# (x̃) ≤ C
kDαj bj kBMO Mr (f )(x̃) .
j=1
0
|αj |=mj
Thus, taking r such that q < r < p, we obtain
MULTILINEAR INTEGRAL OPERATORS
kT b (f )kLp ≤ Ck(T b (f ))# kLp ≤ C
2 X
Y
j=1
(2)
≤C
2 X
Y
j=1
91
kDαj bj kBMO kMr (f )kLp
|αj |=mj
kDαj bj kBMO kf kLp .
|αj |=mj
Secondly, we prove that, if Dα bj ∈ Lipβ (Rn ) for all α with |α| = mj and j = 1, . . . , l,
(3)
(T b (f ))# ≤ C
2 X
Y
j=1
kDαj bj kLipβ M2β,r (f )
|αj |=mj
0
for any r with q < r < n/2β. In fact, by Lemma 6, we have, for x ∈ Q and y ∈ 2Q
X
|Rm (b̃; x, y)| ≤ C|x − y|m
sup |Dα b(z) − (Dα b)Q |
|α|=m
z∈2Q
≤ C|x − y|m |Q|β/n
X
kDα bkLipβ ,
|α|=m
similar to the proof of (1) and by Key Lemma, we obtain
Z
1
|T b (f )(x) − T b (f )(x0 )| dx
|Q| Q
Z Q2
1
j=1 Rmj (b̃j ; x, ·)
f1 dx
≤
T
m
|Q| Q
|x − ·|
Z X
1
1 Rm2 (b̃2 ; x, ·)(x − ·)α1 Dα1 b̃1 +C
f1 dx
T
|Q| Q
α1 !
|x − ·|m
|α1 |=m1
Z X
1 Rm1 (b̃1 ; x, ·)(x − ·)α2 Dα2 b̃2 1
+C
f1 dx
T
|Q| Q
α2 !
|x − ·|m
|α2 |=m2
Z X
1
1 (x−·)α1 +α2 Dα1 b̃1 Dα2 b̃2 +C
f1 dx
T
|Q| Q
α1 !α2 !
|x − ·|m
|α1 |=m1
|α2 |=m2
+
1
|Q|
Z
|T b (f2 )(x) − T b (f2 )(x0 )| dx
Q
2 X
Y
≤ C
j=1
+
≤C
1
|Q|
Z
|αj |=mj
1
|Q|1/r−2β/n
|T b (f2 )(x) − T b (f2 )(x0 )| dx
Q
2 X
Y
j=1
kDαj bj kLipβ
|αj |=mj
kDαj bj kLipβ M2β,r (f )(x̃) .
Z
Q̃
1/r
|f (x)|r dx
92
L. LANZHE
Thus, (3) holds. We take q 0 < r < p < n/2β, 1/w = 1/p − 2β/n and obtain, by
Lemma 5,
kT b (f )kLw ≤ Ck(T b (f ))# kLw ≤ C
2 X
Y
j=1
(4)
≤C
2
Y
j=1
X
||Dαj bj ||Lipβ kM2β,r (f )kLw
|αj |=mj
kDαj bj kLipβ kf kLp .
|αj |=mj
Now we verify that T b satisfies the conditions of Lemma 3. In fact, for any
1 < pi < n/2β, 1/wi = 1/pi − 2β/n(i = 1, 2) and kf kLpi ≤ 1, note that T b (f )(x) =
s
s
T b−b (f )(x) + T b (f )(x) and Dα (bs ) = (Dα b)s with Dα (bj − bsj ) ∈ BMO(Rn ) and
Dα bsj ∈ Lipβ (Rn ), by (2) and Lemma 2, we obtain
s
kT b−b (f )kLpi ≤ C
2 X
Y
j=1
≤C
≤C
|αj |=mj
2 X
Y
j=1
2
Y
j=1
kDαj (bj − bsj )kBMO kf kLpi
kDαj bj − (Dαj bj )s kBMO
|αj |=mj
X
kDαj bj kBMOϕ ϕ2 (s) ,
|αj |=mj
and by (4) and Lemma 3, we obtain
s
kT b (f )kLwi ≤ C
2 X
Y
j=1
kDαj bj kLipβ kf kLpi
|αj |=mj
≤ Cs−2β ϕ2 (s)
2 X
Y
j=1
Thus, for s = t
mT b (f )
kDαj bj kBMOϕ .
|αj |=mj
−1/n
and i = 1, 2,
(ψ 2 )−1 (t) ≤ mT b (f ) t1/pi ϕ2 (t−1/n )
≤ mT b−bs (f ) (t1/pi ϕ2 (t−1/n )/2) + mT bs (f ) t1/pi ϕ2 (t−1/n )/2
h ϕ2 (s) pi s−2β ϕ2 (s) wi i
+ 1/p 2
= Ct−1 .
≤C
t1/pi ϕ2 (s)
t i ϕ (s)
Taking 1 < p2 < p < p1 < n/2β and by Lemma 4, we obtain, for kf kLp ≤
(p/p1 )1/p ,
Z
Z ∞
ψ 2 |T b (f )(x)| dx =
mT b (f ) (ψ 2 )−1 (t) dt ≤ C ,
Rn
0
then, kT b (f )kLψ2 ≤ C.
This completes the proof of Theorem 1.
MULTILINEAR INTEGRAL OPERATORS
93
By using the same arguments as in the proof of Theorem 1 will give the proof
of Theorem 2, we omit the details.
3. Applications
In this section we shall apply the Theorems 1 and 2 to some particular operators
such as the Calderón-Zygmund singular integral operator and Littlewood-Paley
operator, Marcinkiewicz operator.
Application 1. Calderón-Zygmund singular integral operator.
Let T be the Calderón-Zygmund operator (see [7, 8, 22, 23]), the multilinear
operator related to T is defined by
Z Ql
j=1 Rmj +1 (bj ; x, y)
b
K(x, y)f (y) dy .
T (f )(x) =
|x − y|m
Rn
Then it is easily to verify that Key Lemma holds for T b , thus T satisfies the
conditions in Theorem 1 and Theorem 1 holds for T b .
Application 2. Littlewood-Paley operator.
Let ε > 0 and ψ be a fixed function which satisfies the following properties:
(1) |ψ(x)| ≤ C(1 + |x|)−(n+1) ,
(2) |ψ(x + y) − ψ(x)| ≤ C|y|ε (1 + |x|)−(n+1+ε) when 2|y| < |x|.
The multilinear Littlewood-Paley operator is defined by
Z ∞
dt 1/2
gψb (f )(x) =
|Ftb (f )(x)|2
,
t
0
where
Ftb (f )(x)
Z
=
Ql
j=1
Rn
Rmj +1 (bj ; x, y)
|x − y|m
ψt (x − y)f (y) dy
and ψt (x) = t−n ψ(x/t) for t > 0. We write that Ft (f ) = ψt ∗ f . We also define that
Z ∞
dt 1/2
gψ (f )(x) =
|Ft (f )(x)|2
,
t
0
which is the Littlewood-Paley operator (see [23]).
1/2
R∞
Let H be the space H = h : khk = 0 |h(t)|2 dt/t
< ∞ , then, for each
fixed x ∈ Rn , Ftb (f )(x) may be viewed as a mapping from [0, +∞) to H, and it is
clear that
gψ (f )(x) = kFt (f )(x)k
and gψb (f )(x) = kFtb (f )(x)k .
It is easily to see that gψb satisfies the conditions of Theorem 2 (see [11]–[16]), thus
Theorem 2 holds for gψb .
Application 3. Marcinkiewicz operator.
R
Let Ω be homogeneous of degree zero on Rn and S n−1 Ω(x0 ) dσ(x0 ) = 0. Assume
that Ω ∈ Lipγ (S n−1 ) for 0 < γ ≤ 1, that is there exists a constant M > 0 such that
94
L. LANZHE
for any x, y ∈ S n−1 , |Ω(x) − Ω(y)| ≤ M |x − y|γ . The multilinear Marcinkiewicz
operator is defined by
Z ∞
dt 1/2
µbΩ (f )(x) =
|Ftb (f )(x)|2 3
,
t
0
where
Ftb (f )(x)
Z
=
|x−y|≤t
Ω(x − y)
|x − y|n−1
Ql
j=1
Rmj +1 (bj ; x, y)
|x − y|m
f (y) dy ,
we write that
Z
Ft (f )(x) =
|x−y|≤t
Ω(x − y)
f (y) dy .
|x − y|n−1
We also define that
µΩ (f )(x) =
∞
Z
|Ft (f )(x)|2
0
dt 1/2
,
t3
which is the Marcinkiewicz operator (see [24]).
1/2
R∞
Let H be the space H = h : khk = 0 |h(t)|2 dt/t3
< ∞ . Then, it is
clear that
µΩ (f )(x) = kFt (f )(x)k
and µbΩ (f )(x) = kFtb (f )(x)k .
It is easily to see that µbΩ satisfies the conditions of Theorem 2 (see [12]–[16, 24]),
thus Theorem 2 holds for µbΩ .
Application 4. Bochner-Riesz operator.
ˆ = (1 − t2 |ξ|2 )δ fˆ(ξ) and B δ (z) = t−n B δ (z/t) for
Let δ > (n − 1)/2, Ftδ (f )(ξ)
+
t
t > 0. The maximal Bochner-Riesz operator is defined by (see [17])
Bδ,∗ (f )(x) = sup |Ftδ (f )(x)|.
t>0
Set H be the space H = {h : khk = sup |h(t)| < ∞}. The multilinear operator
t>0
related to the maximal Bochner-Riesz operator is defined by
b
b
Bδ,∗
(f )(x) = sup |Fδ,t
(f )(x)| ,
t>0
where
b
Fδ,t
(f )(x)
Ql
Z
j=1
=
Rn
Rmj +1 (bj ; x, y)
|x − y|m
Btδ (x − y)f (y) dy ,
We know
b
b
Bδ,∗
(f )(x) = kBδ,t
(f )(x)k .
b
It is easily to see that Bδ,∗
satisfies the conditions of Theorem 2 (see [12]–[13, 25]),
b
thus Theorem 2 holds for Bδ,∗
.
Acknowledgement. The author would like to express his gratitude to the referee
for his comments and suggestions.
MULTILINEAR INTEGRAL OPERATORS
95
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Department of Mathematics,
Hunan University,
Changsha 410082, P. R. of China
E-mail: lanzheliu@163.com