Weighted estimates for multilinear commutators of Marcinkiewicz integrals with bounded kernel
Let μΩ,b→ be a multilinear commutator generalized by the n-dimensional Marcinkiewicz integral with bounded kernel μ Ώ and let bj ∈OscexpLrj , 1 ≤ j ≤ m. We prove the following weighted inequalities for ω ∈ A ∞ and 0 < p < ∞: The weighted weak L(log L)1/r -type estimate is also established f...
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irk-123456789-1660632020-02-19T01:26:46Z Weighted estimates for multilinear commutators of Marcinkiewicz integrals with bounded kernel Wu, Jianglong Liu, Qingguo Статті Let μΩ,b→ be a multilinear commutator generalized by the n-dimensional Marcinkiewicz integral with bounded kernel μ Ώ and let bj ∈OscexpLrj , 1 ≤ j ≤ m. We prove the following weighted inequalities for ω ∈ A ∞ and 0 < p < ∞: The weighted weak L(log L)1/r -type estimate is also established for p =1 and ω ∈ A. Нехай μΩ,b→ — мультилінійний комутатор, що узагальнює μΏ, n-вимірний iнтеграл Марцинкевича з обмеженим ядром, та нехай bj∈OscexpLrj(1≤j≤m). Доведено такі зважені нерівності для ω∈A∞ та 0<p<∞: Зважену слабку оцінку L(log L)1/r -типу також встановлено для p=1 та ω∈A. 2014 Article Weighted estimates for multilinear commutators of Marcinkiewicz integrals with bounded kernel / Jianglong Wu, Qingguo Liu // Український математичний журнал. — 2014. — Т. 66, № 4. — С. 538–550. — Бібліогр.: 18 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166063 517.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Wu, Jianglong Liu, Qingguo Weighted estimates for multilinear commutators of Marcinkiewicz integrals with bounded kernel Український математичний журнал |
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Let μΩ,b→ be a multilinear commutator generalized by the n-dimensional Marcinkiewicz integral with bounded kernel μ Ώ and let bj ∈OscexpLrj , 1 ≤ j ≤ m. We prove the following weighted inequalities for ω ∈ A ∞ and 0 < p < ∞:
The weighted weak L(log L)1/r -type estimate is also established for p =1 and ω ∈ A. |
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Article |
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Wu, Jianglong Liu, Qingguo |
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Wu, Jianglong Liu, Qingguo |
| author_sort |
Wu, Jianglong |
| title |
Weighted estimates for multilinear commutators of Marcinkiewicz integrals with bounded kernel |
| title_short |
Weighted estimates for multilinear commutators of Marcinkiewicz integrals with bounded kernel |
| title_full |
Weighted estimates for multilinear commutators of Marcinkiewicz integrals with bounded kernel |
| title_fullStr |
Weighted estimates for multilinear commutators of Marcinkiewicz integrals with bounded kernel |
| title_full_unstemmed |
Weighted estimates for multilinear commutators of Marcinkiewicz integrals with bounded kernel |
| title_sort |
weighted estimates for multilinear commutators of marcinkiewicz integrals with bounded kernel |
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Інститут математики НАН України |
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2014 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/166063 |
| citation_txt |
Weighted estimates for multilinear commutators of Marcinkiewicz integrals with bounded kernel / Jianglong Wu, Qingguo Liu // Український математичний журнал. — 2014. — Т. 66, № 4. — С. 538–550. — Бібліогр.: 18 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT wujianglong weightedestimatesformultilinearcommutatorsofmarcinkiewiczintegralswithboundedkernel AT liuqingguo weightedestimatesformultilinearcommutatorsofmarcinkiewiczintegralswithboundedkernel |
| first_indexed |
2025-07-14T20:41:45Z |
| last_indexed |
2025-07-14T20:41:45Z |
| _version_ |
1837656396231868416 |
| fulltext |
UDC 517.5
Jianglong Wu (Mudanjiang Normal Univ., China),
Qingguo Liu (Univ. Nova Gorica, Slovenia)
WEIGHTED ESTIMATES FOR MULTILINEAR COMMUTATORS
OF MARCINKIEWICZ INTEGRALS WITH BOUNDED KERNEL*
ЗВАЖЕНI ОЦIНКИ ДЛЯ МУЛЬТИЛIНIЙНИХ КОМУТАТОРIВ
IНТЕГРАЛIВ МАРЦИНКЕВИЧА З ОБМЕЖЕНИМИ ЯДРАМИ
Let µΩ,~b be a multilinear commutator generalized by µΩ, the n-dimensional Marcinkiewicz integral with bounded kernel,
and let bj ∈ OscexpL
rj (1 ≤ j ≤ m). We prove the following weighted inequalities for ω ∈ A∞ and 0 < p <∞:
‖µΩ(f)‖Lp(ω) ≤ C‖M(f)‖Lp(ω), ‖µΩ,~b(f)‖Lp(ω) ≤ C‖ML(logL)1/r (f)‖Lp(ω).
The weighted weak L(logL)1/r -type estimate is also established for p = 1 and ω ∈ A1.
Нехай µΩ,~b — мультилiнiйний комутатор, що узагальнює µΩ, n-вимiрний iнтеграл Марцинкевича з обмеженим
ядром, та нехай bj ∈ OscexpL
rj , 1 ≤ j ≤ m. Доведено такi зваженi нерiвностi для ω ∈ A∞ та 0 < p <∞:
‖µΩ(f)‖Lp(ω) ≤ C‖M(f)‖Lp(ω), ‖µΩ,~b(f)‖Lp(ω) ≤ C‖ML(logL)1/r (f)‖Lp(ω).
Зважену слабку оцiнку L(logL)1/r-типу також встановлено для p = 1 та ω ∈ A1.
1. Introduction and main results. Suppose that Sn−1 is the unit sphere in Rn, n ≥ 2, equipped
with the normalized Lebesgue measure dσ. Let Ω ∈ L1(Sn−1) be a homogeneous function of degree
zero which satisfies the cancellation condition∫
Sn−1
Ω(x′) dx′ = 0, (1.1)
where x′ = x/|x| (∀x 6= 0).
The n-dimensional Marcinkiewicz integral corresponding to the Littlewood – Paley g-function
introduced by Stein [1] is defined by µΩ(f)(x) =
(∫ ∞
0
|FΩ,t(f)(x)|2 dt
t3
)1/2
, where FΩ,t(f)(x) =
=
∫
|x−y|≤t
Ω(x− y)
|x− y|n−1
f(y) dy.
As usual, we denote by Ap, 1 ≤ p < ∞, the Muckenhoupt’s weights class. We denote [ω]Ap as
Ap constant (see [2], Chapter V or [3], Chapter 9 for details). Operators that map Lp to Lq are called
of strong type (p, q) and operators that map Lp to Lq,∞ are called of weak type (p, q) (see [3, p. 32]).
Let
log+ t = max(log t, 0) =
log t, when t > 1,
0, when 0 ≤ t ≤ 1,
where log t = ln t, and we denote by L(logL) the set of all f with
∫
Rn
|f(x)| log+ |f(x)| dx < ∞
(see [2, p. 128], [3], § 7.5.a). Here and in what follows, ‖b‖∗ denotes the BMO-norm of b (see [3],
Chapter 7 for details).
* This work was supported in part by the Fund of Heilongjiang Provincial Education Department (No. 12531720), the
NSF (No. A200913) of Heilongjiang Province and NNSF (No. 11161042) of China.
c© JIANGLONG WU, QINGGUO LIU, 2014
538 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
WEIGHTED ESTIMATES FOR MULTILINEAR COMMUTATORS OF MARCINKIEWICZ INTEGRALS . . . 539
In 1958, Stein [1] proved that µΩ is of strong type (p, p) for 1 < p ≤ 2 and of weak type (1, 1)
when Ω ∈ Lipα, 0 < α ≤ 1, that is, there is a constant C > 0 such that∣∣Ω(x′)− Ω(y′)
∣∣ ≤ C|x′ − y′|α ∀ x′, y′ ∈ Sn−1. (1.2)
In 1990, Torchinsky and Wang [4] studied the weighted Lp-boundedness of µΩ when Ω satisfies
(1.1) and (1.2). They also considered the weighted Lp-norm inequality for the commutator of the
Marcinkiewicz integral, which is defined by
µmΩ,b(f)(x) =
∞∫
0
∣∣∣∣∣∣∣
∫
|x−y|≤t
(b(x)− b(y))mΩ(x− y)
|x− y|n−1
f(y) dy
∣∣∣∣∣∣∣
2
dt
t3
1/2
, m ∈ N.
In 2004, Ding, Lu and Zhang [5] studied the weighted weak L(logL)-type estimates for µmΩ,b,
precisely, if ω ∈ A1, b ∈ BMO, Ω satisfies (1.1) and (1.2), then, for all λ > 0, there exists a constant
C > 0, such that
ω
({
x ∈ Rn : |µmΩ,b(f)(x)| > λ
})
≤ C
∫
Rn
|f(x)|
λ
(
1 + log+ |f(x)|
λ
)m
ω(x) dx.
In 2008, Zhang [6] studied the weighted boundedness for the multilinear commutator of Marcin-
kiewicz integral µ
Ω,~b
when Ω ∈ Lipα, 0 < α ≤ 1, 0 < p < ∞ and ω ∈ A∞ (see [3], § 9.3), and
established a weighted weak L(logL)1/r-type estimate when p = 1 and ω ∈ A1, where
µ
Ω,~b
(f)(x) =
∞∫
0
∣∣∣∣∣∣∣
∫
|x−y|≤t
Ω(x− y)
|x− y|n−1
m∏
j=1
(
bj(x)− bj(y)
) f(y) dy
∣∣∣∣∣∣∣
2
dt
t3
1/2
, m ∈ N.
And in 2012, Zhang, Wu and Liu [7] establish the weighted weak L(logL)m-type estimate for µ
Ω,~b
when Ω satisfies a kind of Dini conditions.
In 2004, Lee and Rim [8] proved the Lp boundedness for µΩ when there exist constants C > 0
and ρ > 1 such that ∣∣Ω(x′)− Ω(y′)
∣∣ ≤ C(
log
1
|x′ − y′|
)ρ (1.3)
holds uniformly in x′, y′ ∈ Sn−1, and Ω ∈ L∞(Sn−1) be a homogeneous function of degree zero
with cancellation property (1.1). In 2005, Ding [9] studied the weak (1, 1)-type estimate when ρ > 2
and Ω satisfies (1.1) and (1.3).
In the following, we will always assume that Ω ∈ L∞(Sn−1) and satisfies (1.1) and (1.3), where
ρ > 2. Let m be a positive integer. For ~b = (b1, b2, . . . , bm), bj ∈ OscexpLrj , rj ≥ 1, 1 ≤ j ≤ m,
we denote
1
r
=
1
r1
+ . . .+
1
rm
, ‖~b‖ =
m∏
j=1
‖b,j‖Osc
expL
rj
. (1.4)
For the definitions of OscexpLr , ‖ · ‖OscexpLr
and ML(logL)1/r , see Section 2.
Our results can be stated as follows.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
540 JIANGLONG WU, QINGGUO LIU
Theorem 1.1. Let 0 < p < ∞ and suppose that ω ∈ A∞. For ρ > 2, Ω ∈ L∞(Sn−1) is
homogeneous of degree zero and satisfies (1.1) and (1.3). Then there is a positive constant C, such
that ∫
Rn
|µΩ(f)(x)|pω(x) dx ≤ C[ω]pA∞
∫
Rn
[M(f)(x)]pω(x) dx
for all bounded functions f with compact support.
Theorem 1.2. Let 0 < p <∞, ω ∈ A∞ and bj ∈ OscexpLrj , rj ≥ 1, 1 ≤ j ≤ m, r and ‖~b‖ be
as in (1.4). For ρ > 2, Ω ∈ L∞(Sn−1) is homogeneous of degree zero and satisfies (1.1) and (1.3).
Then there is a positive constant C, such that∫
Rn
|µ
Ω,~b
(f)(x)|pω(x) dx ≤ C‖~b‖p
∫
Rn
[ML(logL)1/r(f)(x)]pω(x) dx (1.5)
for all bounded functions f with compact support.
Since rj ≥ 1, j = 1, 2, . . . ,m, thenML(logL)1/r is pointwise smaller thanML(logL)m . Noting that
ML(logL)m is equivalent to Mm+1, the m+ 1 iterations of the Hardy – Littlewood maximal operator
M (see (21) in [10]), by using the weighted Lp-boundedness of M again, from Theorem 1.2, we
have the following result.
Corollary 1.1. Let 1 < p < ∞, ω ∈ Ap, bj ∈ OscexpLrj , rj ≥ 1, 1 ≤ j ≤ m, r and ‖~b‖ be
as in (1.4). For ρ > 2, Ω ∈ L∞(Sn−1) is homogeneous of degree zero and satisfies (1.1) and (1.3).
Then there is a positive constant C, such that∫
Rn
|µ
Ω,~b
(f)(x)|pω(x) dx ≤ C‖~b‖p
∫
Rn
|f(x)|pω(x) dx
for all bounded functions f with compact support.
Theorem 1.3. Let ω ∈ A1, bj ∈ OscexpLrj , rj ≥ 1, 1 ≤ j ≤ m, r and ‖~b‖ be as in (1.4).
For ρ > 2, Ω ∈ L∞(Sn−1) is homogeneous of degree zero and satisfies (1.1) and (1.3). Let Φ(t) =
= t log1/r(e + t). Then there is a positive constant C, for all bounded functions f with compact
support and all λ > 0, such that
ω({x ∈ Rn : µ
Ω,~b
(f)(x) > λ}) ≤ C
∫
Rn
Φ
(
‖~b‖|f(y)|
λ
)
ω(y) dy.
The remainder of the paper is organized as follows. In Section 2, we will recall some notation
and known results we need, and establish the basic estimates for sharp functions. In Section 3 we
prove Theorems 1.1 and 1.2. In the last section, we prove Theorem 1.3.
Throughout this paper, C denotes a constant that is independent of the main parameters involved
but whose value may differ from line to line. For any index p ∈ [1,∞], we denote by p′ its conjugate
index, namely, 1/p + 1/p′ = 1. For A ∼ B, we mean that there is a constant C > 0 such that
C−1B ≤ A ≤ CB.
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WEIGHTED ESTIMATES FOR MULTILINEAR COMMUTATORS OF MARCINKIEWICZ INTEGRALS . . . 541
2. Preliminaries and estimates for sharp functions. As usual, M stands for the Hardy –
Littlewood maximal operator. For a ball B in Rn, denote by fB = |B|−1
∫
B
f(y) dy. We need
the following variants of M and the Fefferman – Stein’s sharp function. For δ > 0, define
Mδ(f)(x) =
[
M(|f |δ)(x)
]1/δ
, M ]
δ(f)(x) =
[
M ](|f |δ)(x)
]1/δ
,
where
M ](f)(x) = sup
B3x
inf
c
1
|B|
∫
B
|f(y)− c| dy ≈ sup
B3x
1
|B|
∫
B
|f(y)− fB| dy.
The following relationships between M ]
δ and Mδ which will be used is a version of the classical
ones due to Fefferman and Stein (see [2, p. 153]).
Lemma 2.1 [10 – 12]. (a) Let ω ∈ A∞ and φ : (0,∞)→ (0,∞) be doubling. Then there exists
a positive constant C, depending upon the doubling condition of φ, such that, for all λ, δ > 0
sup
λ>0
φ(λ)ω({y ∈ Rn : Mδ(f)(y) > λ}) ≤ C[ω]A∞ sup
λ>0
φ(λ)ω({y ∈ Rn : M ]
δ(f)(y) > λ}),
for every function f such that the left-hand side is finite.
(b) Let ω ∈ A∞ and 0 < p, δ < ∞. Then there exists a positive constant C, depending upon p,
such that ∫
Rn
[
Mδ(f)(x)
]p
ω(x) dx ≤ C[ω]pA∞
∫
Rn
[
M ]
δ(f)(x)
]p
ω(x) dx,
for every function f such that the left-hand side is finite.
A function Φ defined on [0,∞) is said to be a Young function, if Φ is a continuous, nonnegative,
strictly increasing and convex function with Φ(0) = 0 and limt→∞Φ(t) =∞. Define the Φ-average
of a function f on a ball B by
‖f‖Φ,B = inf
{
λ > 0:
1
|B|
∫
B
Φ
(
|f(y)|
λ
)
dy ≤ 1
}
.
The maximal operator MΦ associated with the Φ-average, ‖ · ‖Φ,B, is defined by
MΦ(f)(x) = sup
B3x
‖f‖Φ,B,
where the supremum is taken over all the balls B containing x.
When Φ(t) = t logr(e + t), we denote ‖ · ‖Φ,B and MΦ by ‖ · ‖L(logL)r,B and ML(logL)r ,
respectively. When Φ(t) = et
r−1, we denote ‖ ·‖Φ,B and MΦ by ‖ ·‖expLr,B and MexpLr . If k ∈ N
then ML(logL)m ∼Mm+1 (see (21) of [10]).
We have the generalized Hölder’s inequality as follows, for details and the more general cases
see Lemma 2.3 in [11].
Lemma 2.2 [11]. Let r1, . . . , rm ≥ 1 with 1/r = 1/r1 + . . . + 1/rm and B be a ball in Rn.
Then there holds the generalized Hölder’s inequality
1
|B|
∫
B
|f1(x) . . . fm(x)g(x)| dx ≤ C‖f1‖expLr1 ,B . . . ‖fm‖expLrm ,B‖g‖L(logL)1/r,B.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
542 JIANGLONG WU, QINGGUO LIU
For r ≥ 1, we say f ∈ OscexpLr if f∈L1
loc(R
n) and ‖f‖OscexpLr
<∞, where
‖f‖OscexpLr
= sup
B
‖f − fB‖expLr,B,
and the supremum is taken over all the balls B ⊂ Rn.
By John – Nirenberg theorem (see [2] or [13]), it is not difficult to see that OscexpL1 = BMO(Rn)
and OscexpLr is contained properly in BMO(Rn) when r > 1 (see [14]). Furthermore, ‖b‖∗ ≤
≤ C‖b‖OscexpLr
when b ∈ OscexpLr and r ≥ 1 (see [6]). For more information on Orlicz space
see [15].
We will take the point of view of the vector-valued singular integral of Benedek, Calderón and
Panzone [16]. Let H be the Hilbert space defined by
H =
h : ‖h‖H =
∞∫
0
|h(t)|2
t3
dt
1/2
<∞
.
For all x ∈ Rn and t > 0, let
F
Ω,~b,t
(f)(x) =
∫
|x−y|≤t
Ω(x− y)
|x− y|n−1
m∏
j=1
(
bj(x)− bj(y)
) f(y) dy, m ∈ N.
Then for each fixed x ∈ Rn, FΩ,t(f)(x) and F
Ω,~b,t
(f)(x) can be regarded as mapping from [0,∞)
to H, and
µΩ(f)(x) = ‖FΩ,t(f)(x)‖H, µ
Ω,~b
(f)(x) = ‖F
Ω,~b,t
(f)(x)‖H.
The following pointwise estimates for the sharp function of µ come from [17].
Lemma 2.3 [17]. Let 0 < δ < 1, f, µΩ(f) be both locally integrable function. For ρ > 2,
Ω ∈ L∞(Sn−1) is homogeneous of degree zero and satisfies (1.1) and (1.3). Then there is a positive
constant C, independent of f and x, such that
M ]
δ(µΩ(f))(x) ≤ CM(f)(x), a.e. x ∈ Rn.
Some ideas for the proof of Lemma 2.3 come from [5]. For details and the more information see
Lemma 3.2.4 in [17].
For the multilinear commutators µ
Ω,~b
, there holds a similar pointwise estimate. To state it, we
first introduce some notations. For all 1 ≤ j ≤ m, we denote by Cmj the family of all finite
subsets σ = {σ(1), . . . , σ(j)} of {1, 2, . . . ,m} with j different elements. For any σ ∈ Cmj and
~b = (b1, . . . , bm), we define σ′ = {1, 2, . . . ,m} \ σ, ~bσ = (bσ(1), . . . , bσ(j)), and bσ = bσ(1) . . . bσ(j).
For any vector (rσ(1), . . . , rσ(j)) of j positive numbers and 1/rσ = 1/rσ(1) + . . .+ 1/rσ(j), we write
‖~bσ‖ = ‖~bσ‖OscexpLrσ
= ‖bσ(1)‖Osc
expL
rσ(1)
. . . ‖bσ(j)‖Osc
expL
rσ(j)
. (2.1)
For any σ = {σ(1), . . . , σ(j)} ∈ Cmj and ~bσ = (bσ(1), . . . , bσ(j)), we write
F
Ω,~bσ ,t
(f)(x) =
∫
|x−y|≤t
Ω(x− y)
|x− y|n−1
(
j∏
i=1
(
bσ(i)(x)− bσ(i)(y)
))
f(y) dy
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
WEIGHTED ESTIMATES FOR MULTILINEAR COMMUTATORS OF MARCINKIEWICZ INTEGRALS . . . 543
and
µ
Ω,~bσ
(f)(x) =
∥∥∥FΩ,~bσ ,t
(f)(x)
∥∥∥
H
.
If σ = {1, . . . ,m}, then σ′ = ∅. We understand µ
Ω,~bσ
= µ
Ω,~b
and µ
Ω,~bσ′
= µΩ.
Lemma 2.4 [17]. Let rj ≥ 1, bj ∈ OscexpLrj , 1 ≤ j ≤ m, r and ‖~b‖ be as in (1.4). For ρ > 2,
Ω ∈ L∞(Sn−1) is homogeneous of degree zero satisfying (1.1) and (1.3), then for any δ and ε with
0 < δ < ε < 1, there is a constant C > 0, depending only on δ and ε, such that, for any bounded
function f with compact support,
M ]
δ(µΩ,~b
(f))(x) ≤ C
‖~b‖ML(logL)1/r(f)(x) +
m∑
j=1
∑
σ∈Cmj
‖~bσ‖OscexpLrσ
Mε
(
µ
Ω,~bσ′
(f)
)
(x)
.
Some ideas for the proof of Lemma 2.4 come from [5, 6, 10, 11]. For details and the more
information see Lemma 3.2.5 in [17].
Remark 2.1. Noting that (1.3) is weaker than Lipα, 0 < α ≤ 1, condition, the main results in
this paper improve the main results in [6]. And the Theorem 1.3 is equivalent to the Theorem 4.1.1
in [18] when b1 = b2 = . . . = bm.
3. Proof of Theorems 1.1 and 1.2. The proof of Theorem 1.1 is similar as Theorem 1.1 in [6].
So, we omit the details and only give the proof of Theorem 1.2 here. For brevity, we write
‖h(x)‖Lp(ω) =
∫
Rn
|h(x)|pω(x) dx
1/p
for 0 < p <∞.
Proof of Theorem 1.2. Without loss of generality, we assume∫
Rn
[ML(logL)1/r(f)(x)]pω(x) dx <∞, (3.1)
since otherwise there is nothing to be proven. We divide the proof into two cases.
Case I. Suppose that ω and bj , 1 ≤ j ≤ m, are all bounded. Firstly, we take it for granted that,
for all bounded functions f with compact supports,∫
Rn
[Mδ(µΩ,~b
(f))(x)]pω(x) dx <∞ (3.2)
holds for 0 < p <∞ and appropriate δ with 0 < δ < 1.
Under the assumption of (3.2), we will proceed the proof by induction on m. For m = 1, ~b = b1,
µ
Ω,~b
= µΩ,b1 . By Lemma 2.1(b) and Lemma 2.4, for 0 < δ < ε < 1, we have
‖µΩ,b1(f)‖Lp(ω) ≤ ‖Mδ(µΩ,b1(f))‖Lp(ω) ≤ C‖M
]
δ(µΩ,b1(f))‖Lp(ω) ≤
≤ C‖b1‖OscexpLr1
(
‖ML(logL)1/r1 (f)‖Lp(ω) + ‖Mε(µΩ(f))‖Lp(ω)
)
. (3.3)
Since ω ∈ A∞, there is a p0 > 1, such that ω ∈ Ap0 . We can choose δ > 0 small enough, so
that p/δ > p0. So ω ∈ Ap/δ. Then by the definition of Mδ and the weighted Lp/δ-boundedness of
the Hardy – Littlewood maximal operator M, we get
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
544 JIANGLONG WU, QINGGUO LIU∫
Rn
[Mδ(µΩ(f))(x)]pω(x) dx =
∫
Rn
[M(|µΩ(f)|δ)(x)]p/δω(x) dx ≤
≤
∫
Rn
|µΩ(f)(x)|pω(x) dx. (3.4)
This, together with (3.3), Theorem 1.1 and the fact M(f) ≤ CML(logL)1/s(f) for any s > 0,
gives
‖µΩ,b1(f)‖Lp(ω) ≤ C‖b1‖OscexpLr1
(
‖ML(logL)1/r1 (f)‖Lp(ω) + ‖µΩ(f)‖Lp(ω)
)
≤
≤ C‖b1‖OscexpLr1
(
‖ML(logL)1/r1 (f)‖Lp(ω) + ‖M(f)‖Lp(ω)
)
≤
≤ C‖b1‖OscexpLr1
‖ML(logL)1/r1 (f)‖Lp(ω).
Now, suppose that the theorem is true for 1, 2, . . . ,m−1 and let us prove it for m. Recall that, if
σ = {σ(1), . . . , σ(j)}, 1 ≤ j ≤ m, and the corresponding satisfies 1/rσ = 1/rσ(1) + . . . + 1/rσ(j),
then σ′ = {1, . . . ,m} \ σ and the corresponding rσ′ satisfying 1/rσ′ = 1/r − 1/rσ. Reasoning as in
(3.4), for θ > 0 small enough, we obtain∫
Rn
[Mθ(µΩ,~bσ′
(f))(x)]pω(x) dx ≤ C
∫
Rn
|µ
Ω,~bσ′
(f)(x)|pω(x) dx. (3.5)
The same argument as used above and the induction hypothesis give us that
‖µ
Ω,~b
(f)‖Lp(ω) ≤ ‖Mδ(µΩ,~b
(f))‖Lp(ω) ≤ C‖M
]
δ(µΩ,~b
(f))‖Lp(ω) ≤
≤ C‖~b‖‖ML(logL)1/r(f)‖Lp(ω) + C
m∑
j=1
∑
σ∈Cmj
‖~bσ‖OscexpLrσ
‖Mε(µΩ,~bσ′
(f))‖Lp(ω) ≤
≤ C‖~b‖‖ML(logL)1/r(f)‖Lp(ω) + C
m∑
j=1
∑
σ∈Cmj
‖~bσ‖OscexpLrσ
‖µ
Ω,~bσ′
(f)‖Lp(ω) ≤
≤ C‖~b‖‖ML(logL)1/r(f)‖Lp(ω)+
+C
m∑
j=1
∑
σ∈Cmj
‖~bσ‖OscexpLrσ
‖~bσ′‖Osc
expL
rσ′
‖M
L(logL)1/rσ′ (f)‖Lp(ω) ≤
≤ C‖~b‖‖ML(logL)1/r(f)‖Lp(ω),
where the fourth inequality follows from (3.5) and the last one follows from the fact that
M
L(logL)1/rσ′ (f) ≤ML(logL)1/r(f).
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To finish the proof of this special case of Theorem 1.2, we need to check (3.2). From (3.5), it
suffices to prove ∫
Rn
|µ
Ω,~b
(f)(x)|pω(x) dx <∞, 0 < p <∞, (3.6)
whenever the weight ω and the functions bj , 1 ≤ j ≤ m, are all bounded.
Assume that suppf ⊂ B = B(0, R) for some R > 0 and write∫
Rn
|µ
Ω,~b
(f)(x)|pω(x) dx =
∫
2B
|µ
Ω,~b
(f)(x)|pω(x) dx+
∫
(2B)c
|µ
Ω,~b
(f)(x)|pω(x) dx = I + II.
Noting that ω and bj are all bounded, by the Hölder inequality, the induction hypothesis and the
fact ML(logL)m ∼Mm+1, Lp/δ-boundedness of M, there is∫
2B
|bσ(x)|p|µΩ,bσ′ (f)(x)|pω(x) dx ≤ Cω‖bσ‖pL∞(Rn)|B|
1−δ‖bσ′f‖pLp/δ(Rn)
<∞.
This and the definition of µ
Ω,~b
(f) give us that
I ≤ C
m∑
j=1
∑
σ∈Cmj
∫
2B
|bσ(x)|p|µΩ,bσ′ (f)(x)|pω(x) dx <∞. (3.7)
To deal with II , we first estimate µ
Ω,~b
(f)(x) for x ∈ (2B)c. |x|/2 ≤ |x − y| ≤ 3|x|/2 when
x ∈ (2B)c and y ∈ B. Noting that Ω ∈ L∞(Sn−1), ω and bj are bounded functions and |x| ∼ |x−y|
when x ∈ (2B)c and y ∈ B, there is a constant C
Ω,~b,ω
, depending on the L∞-norm of Ω, bj and ω,
such that
µ
Ω,~b
(f)(x) ≤ C‖Ω‖L∞(Sn−1)‖~b‖L∞(Rn)
∞∫
0
∣∣∣∣∣∣∣
∫
|x−y|≤t
|f(y)|
|x− y|n−1
dy
∣∣∣∣∣∣∣
2
dt
t3
1/2
≤
≤ C
Ω,~b,ω
∫
Rn
|f(y)|
|x− y|n−1
∫
|x−y|≤t
dt
t3
1/2
dy ≤
≤ C
Ω,~b,ω
∫
Rn
|f(y)|
|x− y|n
dy ≤ C
Ω,~b,ω
1
|2B|
∫
Rn
|f(y)| dy ≤ C
Ω,~b,ω
M(f)(x). (3.8)
By (3.8) and the fact that M(f)(x) ≤ CML(logL)1/r(f)(x), it follows from (3.1) that
II ≤ C
Ω,~b,ω
∫
(2B)c
[ML(logL)1/r(f)(x)]pω(x) dx <∞.
This together with (3.7) shows that (3.6) is true when ω and bj are bounded functions, so does
(3.2). And then Theorem 1.2 is proved for this special case.
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546 JIANGLONG WU, QINGGUO LIU
Case II. For unbounded ω and bj , we will truncate the weight ω and the functions bj , j =
= 1, . . . ,m, as follows. Let N be a positive integer, denote by ωN = inf{ω,N} and by ~bN =
= (bN1 , . . . , b
N
m), where bNj is defined by
bNj (x) =
N, when bj(x) > N,
bj(x), when |bj(x)| ≤ N,
−N, when bj(x) < −N.
By Lemma 2.4 in [11], there is a positive constant C independent of N such that
‖bNj ‖Osc
expL
rj
≤ ‖bj‖Osc
expL
rj
. (3.9)
Applying (1.5) for ~bN and ωN , and using (3.9), we have∫
Rn
|µ
Ω,~bN
(f)(x)|pωN (x) dx ≤ C‖~b‖p
∫
Rn
[ML(logL)1/r(f)(x)]pω(x) dx. (3.10)
Next, taking into account the fact that f has compact support, we deduce that bNj converges to
bj and bNσ(1) . . . b
N
σ(j)f converges to bσ(1) . . . bσ(j)f in any space Lp for p > 1 as N →∞. Recalling
the Lp-boundedness of µΩ, we claim that, at least for a subsequence,
{
|µ
Ω,~bN
(f)(x)|pωN (x)}∞N=1
converges pointwise almost everywhere to |µ
Ω,~b
(f)(x)|pω(x) as N →∞.
This fact, together with (3.10) and Fatou’s lemma, finishes the proof of Theorem 1.2.
4. Proof of Theorem 1.3. The idea of the proof of Theorem 1.3 follows that of Theorem 1.5 in
[11]. We first prove the following lemma.
Lemma 4.1. Let ω ∈ A∞, Φ(t) = t log1/r(e + t), ~b, r, and rj be the same as in Theorem 1.3.
Then for ρ > 2, Ω ∈ L∞(Sn−1) is homogeneous of degree zero and satisfies (1.1) and (1.3), there
exists a positive constant C such that
sup
t>0
ω({y ∈ Rn : M ]
δ(µΩ,~b
(f))(y) > t})
Φ(1/t)
≤ C sup
t>0
ω({y ∈ Rn : MΦ(‖~b‖f)(y) > t})
Φ(t)
(4.1)
for all bounded functions f with compact support and all 0 < δ < 1.
Proof. To use Lemma 2.1(a), we first check that
sup
t>0
1
Φ(1/t)
ω
({
x ∈ Rn : Mε(µΩ,~b
(f))(x) > t
})
<∞ (4.2)
for all bounded functions f with compact support and all δ with 0 < δ < 1.
We only prove (4.2) for the special case where ω and bj are bounded functions. For the general
case, we consider the truncations of ω and~b as in the proof of Theorem 1.2, by a limit discussion, this
time, we take into account the weak (1,1) boundedness of µΩ that gives the convergence in measure.
Then we can obtain (4.2) for all ω and ~b with the hypotheses of Lemma 4.1, we omit the details.
Assume that suppf ⊂ B = B(0, R). Then, for any 0 < ε < 1
sup
t>0
ω
({
x ∈ Rn : Mε(µΩ,~b
(f))(x) > t
})
Φ(1/t)
≤
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WEIGHTED ESTIMATES FOR MULTILINEAR COMMUTATORS OF MARCINKIEWICZ INTEGRALS . . . 547
≤ Cε sup
t>0
ω
({
x ∈ Rn : Mε(χ2BµΩ,~b
(f))(x) > t/2
})
Φ(1/t)
+
+Cε sup
t>0
1
Φ(1/t)
ω
({
x ∈ Rn : Mε(χ(2B)cµΩ,~b
(f))(x) > t/2
})
= Cε(I + II), (4.3)
where Cε is a positive constant depending on ε.
For I, making use of the weak (1,1) boundedness of M and [Φ(1/t)]−1 ≤ Ct, and noting that ω
and bj are all bounded. Then there is a positive constant Cω, depending on ω, such that
I ≤ Cω sup
t>0
t
∣∣{x ∈ Rn : Mε(χ2BµΩ,~b
(f))(x) > t/2
}∣∣ ≤
≤ Cω
∫
2B
|µ
Ω,~b
(f)(x)| dx ≤ Cω|B|1/2
∫
2B
|µ
Ω,~b
(f)(x)|2 dx
1/2
<∞,
where the last step follows as (3.7).
Recall the fact that (M(f))ε ∈ A1 for 0 < ε < 1 and f locally integrable, then
Mε(M(f))(x) =
[
M(|M(f)|ε)(x)
]1/ε ≤ CM(f)(x).
Noting that ω is bounded, it follows from (3.8) and the weak (1,1) boundedness of M that
II ≤ Cω sup
t>0
t · ω
(
{x ∈ Rn : Mε(M(f))(x) > Ct}
)
≤
≤ Cω sup
t>0
t · ω
(
{x ∈ Rn : M(f)(x) > Ct}
)
≤
≤ Cω
∫
Rn
|f(x)| dx <∞.
Combining (4.3) and the estimates for I and II, we have (4.2).
Now, let us turn to proving (4.1) by induction. For ~b ∈ OscexpLr , write b̃ = ~b/‖~b‖, then ‖b̃‖ = 1,
and µ
Ω,~b
(f)/‖~b‖ = µ
Ω,~b/‖~b‖(f) = µΩ,b̃(f). So we can assume that ‖~b‖ = 1. For m = 1, we
understand ~b = b, ‖~b‖ = ‖b‖OscexpLr
= 1, µ
Ω,~b
(f) = µΩ,b(f). Therefore, to prove (4.1), it suffices
to prove
sup
t>0
ω({y ∈ Rn : M ]
δ(µΩ,b(f))(y) > t})
Φ(1/t)
≤ C sup
t>0
ω({y ∈ Rn : ML(logL)1/r(f)(y) > t})
Φ(1/t)
(4.4)
for all bounded functions f with compact support.
Applying Lemma 2.4 for m = 1 and any ε with 0 < δ < ε < 1, it is easy to see that the left-hand
side of (4.4) is dominated by
sup
t>0
ω({y ∈ Rn : M ]
δ(µΩ,b(f))(y) > t})
Φ(1/t)
≤
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548 JIANGLONG WU, QINGGUO LIU
≤ C sup
t>0
ω({y ∈ Rn : ML(logL)1/r(f)(y) > t/2})
Φ(1/t)
+
+C sup
t>0
ω({y ∈ Rn : Mε(µΩ(f))(y) > t/2})
Φ(1/t)
.
Recall that (4.2) is valid and since [Φ(1/t)]−1 is doubling, then by Lemma 2.1(a), Lemma 2.3
and noting that M(f) ≤ML(logL)1/r(f), we have
sup
t>0
ω({y ∈ Rn : M ]
δ(µΩ,b(f))(y) > t})
Φ(1/t)
≤ C sup
t>0
ω({y ∈ Rn : ML(logL)1/r(f)(y) > t})
Φ(1/t)
+
+C sup
t>0
ω({y ∈ Rn : M ]
ε(µΩ(f))(y) > t})
Φ(1/t)
≤
≤ C sup
t>0
ω({y ∈ Rn : ML(logL)1/r(f)(y) > t})
Φ(1/t)
+ C sup
t>0
ω({y ∈ Rn : M(f)(y) > t})
Φ(1/t)
≤
≤ C sup
t>0
1
Φ(1/t)
ω({y ∈ Rn : ML(logL)1/r(f)(y) > t}).
This is (4.4), thus, we have proved (4.1) for m = 1.
Now, let us check (4.1) for the general case m ≥ 2. Suppose that (4.1) holds for m − 1, let us
prove it for m. Noting that (4.2) is true and recalling the fact that [Φ(1/t)]−1 is doubling, then by
Lemmas 2.3 and 2.4 for ε with 0 < δ < ε, Lemma 2.1(a) and the induction hypothesis on (4.1), we
obtain
sup
t>0
ω({y ∈ Rn : M ]
δ(µΩ,~b
(f))(y) > t})
Φ(1/t)
≤ C sup
t>0
ω({y ∈ Rn : MΦ(f)(y) > t/Cm})
Φ(1/t)
+
+C
m∑
j=1
∑
σ∈Cmj
sup
t>0
1
Φ(1/t)
ω({y ∈ Rn : Mε(µΩ,~bσ′
(‖~bσ‖f))(y) > t/Cm}) ≤
≤ Cm sup
t>0
1
Φ(1/t)
ω({y ∈ Rn : MΦ(f)(y) > t})+
+Cm
m∑
j=1
∑
σ∈Cmj
sup
t>0
1
Φ(1/t)
ω({y ∈ Rn : M ]
ε(µΩ,~bσ′
(‖~bσ‖f))(y) > t}) ≤
≤ Cm sup
t>0
1
Φ(1/t)
ω({y ∈ Rn : MΦ(f)(y) > t})+
+Cm
m∑
j=1
∑
σ∈Cmj
sup
t>0
1
Φ(1/t)
ω({y ∈ Rn : MΦ(‖~bσ′‖‖~bσ‖f)(y) > t}) ≤
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WEIGHTED ESTIMATES FOR MULTILINEAR COMMUTATORS OF MARCINKIEWICZ INTEGRALS . . . 549
≤ Cm sup
t>0
1
Φ(1/t)
ω({y ∈ Rn : MΦ(f)(y) > t})+
+Cm
m∑
j=1
∑
σ∈Cmj
sup
t>0
1
Φ(1/t)
ω({y ∈ Rn : MΦ(f)(y) > t}),
where ‖~bσ‖ and ‖~bσ′‖ are as in (2.1), and in the last step, we make use of the fact that ‖~bσ′‖‖~bσ‖ =
= ‖~b‖ = 1.
This concludes (4.1) for all m, so the proof of Lemma 4.1 is completed.
Lemma 4.2. Let ω ∈ A∞, Φ(t) = t log1/r(e + t), ~b, r, and rj be the same as in Theorem 1.3.
For ρ > 2, Ω ∈ L∞(Sn−1) is homogeneous of degree zero and satisfies (1.1) and (1.3), there exists
a positive constant C such that
sup
t>0
ω({y ∈ Rn : µ
Ω,~b
(f)(y) > t})
Φ(1/t)
≤ C sup
t>0
ω({y ∈ Rn : MΦ(‖~b‖f)(y) > t})
Φ(1/t)
for all bounded functions f with compact support.
The proof is similar as the proof of Lemma 4.2 in [6], we omit the details here.
To prove Theorem 1.3, we need the following weighted weak-type inequality due to Pérez and
Trujillo – González [11].
Lemma 4.3 [11]. Let ω ∈ A1, Φ(t) = t log1/r(e + t). Then there is a positive constant C, for
any λ > 0 and any locally integrable function f, such that
ω({y ∈ Rn : MΦ(f)(y) > λ}) ≤ C
∫
Rn
Φ
(
|f(y)|
λ
)
ω(y) dy.
Proof of Theorem 1.3. By homogeneity of ~b, we can assume that λ = ‖~b‖ = 1. Then we only
need to prove that
ω
({
y ∈ Rn : µ
Ω,~b
(f)(y) > 1
})
≤ C
∫
Rn
Φ
(
|f(y)|
)
ω(y) dy.
By Φ(ab) ≤ 2Φ(a)Φ(b), a, b ≥ 0 and Lemmas 4.2 and 4.3, we have
ω({y ∈ Rn : µ
Ω,~b
(f)(y) > 1}) ≤ C sup
λ>0
1
Φ(1/λ)
ω({y ∈ Rn : µ
Ω,~b
(f)(y) > λ}) ≤
≤ C sup
λ>0
ω({y ∈ Rn : MΦ(f)(y) > λ})
Φ(1/λ)
≤ C sup
λ>0
1
Φ(1/λ)
∫
Rn
Φ
(
|f(y)|
λ
)
ω(y) dy ≤
≤ C sup
λ>0
1
Φ(1/λ)
∫
Rn
Φ
(
|f(y)|
)
Φ(1/λ)ω(y) dy ≤ C
∫
: RnΦ
(
|f(y)|
)
ω(y) dy.
Theorem 1.3 is proved.
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Received 30.11.11,
after revision — 06.01.13
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