Weighted sharp boundedness for multilinear commutators
In this paper, the sharp estimates for some multilinear commutators related to certain sublinear integral operators are obtained. The operators include the Littlewood - Paley operator and the Marcinkiewicz operator. As application, we obtain the weighted $L^p (p > 1)$ inequalities and $L \lo...
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| Дата: | 2007 |
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2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509484100616192 |
|---|---|
| author | Hong, Xu Liu, Lanzhe Zeng, Jiasheng Гонг, Сюй Лю, Ланже Цзен, Цзяшен |
| author_facet | Hong, Xu Liu, Lanzhe Zeng, Jiasheng Гонг, Сюй Лю, Ланже Цзен, Цзяшен |
| author_sort | Hong, Xu |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:53:10Z |
| description | In this paper, the sharp estimates for some multilinear commutators related to certain sublinear integral operators are obtained.
The operators include the Littlewood - Paley operator and the Marcinkiewicz operator. As application, we obtain the weighted $L^p (p > 1)$ inequalities and $L \log L$-type estimate for the multilinear commutators.
|
| first_indexed | 2026-03-24T02:41:50Z |
| format | Article |
| fulltext |
UDC 517.9
Hong Xu, Jiasheng Zeng, Lanzhe Liu (Hunan Univ., China)
WEIGHTED SHARP BOUNDEDNESS
FOR MULTILINEAR COMMUTATORS
ЗВАЖЕНА ТОЧНА ОБМЕЖЕНIСТЬ
ДЛЯ МУЛЬТИЛIНIЙНИХ КОМУТАТОРIВ
In this paper, the sharp estimates for some multilinear commutators related to certain sublinear integral
operators are obtained. The operators include the Littlewood – Paley operator and the Marcinkiewicz
operator. As application, we obtain the weighted Lp (p > 1) inequalities and L log L-type estimate for
the multilinear commutators.
Одержано точнi оцiнки для деяких мультилiнiйних комутаторiв, що пов’язанi з певними сублiнiй-
ними iнтегральними операторами. Цi оператори включають в себе оператор Лiтлвуда – Палея та
оператор Марцiнкевича. Як застосування, отримано зваженi Lp (p > 1) нерiвностi та оцiнку типу
L log L для мультилiнiйних комутаторiв.
1. Introduction. Let b ∈ BMO(Rn) and T be the Calderón – Zygmund operator.
The commutator [b, T ] generated by b and T is defined by [b, T ]f(x) = b(x)Tf(x) −
− T (bf)(x). By virtue of classical result of Coifman, Rochberg and Weiss [1], we
know that the commutator [b, T ] is bounded on Lp(Rn) (1 < p < ∞). However, it
was observed that [b, T ] is not bounded, in general, from L1(Rn) to L1,∞(Rn). In
[2], the sharp inequalities for some multilinear commutators of the Calderón – Zygmund
singular integral operators are obtained. The main purpose of this paper is to prove the
sharp estimates for some multilinear commutators related to certain sublinear integral
operators. In fact, we shall establish the sharp estimates for the multilinear commutators
only under certain conditions on the size of the operators. The operators include the
Littlewood – Paley operator and the Marcinkiewicz operator. As the applications, we
obtain the weighted norm inequalities and L logL-type estimate for these multilinear
commutators. In Section 2, we will give some concepts and theorems of this paper,
whose proofs will appear in Section 3.
2. Preliminaries and theorems. First, let us introduce some notations (see [2 – 5]).
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
f#(x) = sup
x∈Q
1
|Q|
∫
Q
|f(y)− fQ|dy,
where, and in what follows, fQ = |Q|−1
∫
Q
f(x)dx. It is well known that (see [3])
f#(x) = sup
x∈Q
inf
c∈C
1
|Q|
∫
Q
|f(y)− c|dy.
We say that f belongs to BMO(Rn) if f# belongs to L∞(Rn) and ‖f‖BMO =
= ‖f#‖L∞ . For 0 < r <∞, we denote f#
r by
f#
r (x) = [(|f |r)#(x)]1/r.
c© HONG XU, JIASHENG ZENG, LANZHE LIU, 2007
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10 1419
1420 HONG XU, JIASHENG ZENG, LANZHE LIU
Let M be the Hardy – Littlewood maximal operator, i.e., M(f)(x) = sup
x∈Q
|Q|−1 ×
×
∫
Q
|f(y)|dy; we write Mp(f) = (M(fp))1/p. For k ∈ N, we denote by Mk the
operator M iterated k times, i.e., M1(f)(x) = M(f)(x) and Mk(f)(x) =
= M(Mk−1(f))(x) for k ≥ 2.
Let Φ be a Young function and Φ̃ be the complement associated with Φ. For a
function f, we denote the Φ-average by
‖f‖Φ,Q = inf
λ > 0:
1
|Q|
∫
Q
Φ
(
|f(y)|
λ
)
dy ≤ 1
and the maximal function associated to Φ by
MΦ(f)(x) = sup
x∈Q
‖f‖Φ,Q.
The main Young function to be used in this paper is Φ(t) = exp(tr) − 1 and Ψ(t) =
= t logr(t + e), the corresponding Φ-average and maximal functions are denoted by
‖ · ‖exp Lr,Q, Mexp Lr and ‖ · ‖L(log L)r,Q, ML(log L)r . We have the following inequality
for any r > 0 and m ∈ N :
M(f) ≤ML(log L)r (f), ML(log L)m(f) ∼Mm+1(f).
For r ≥ 1, we denote
‖b‖oscexp Lr = sup
Q
‖b− bQ‖exp Lr,Q;
the spaces Oscexp Lr is defined by
Oscexp Lr =
{
b ∈ L1
log(R
n) : ‖b‖oscexp Lr <∞
}
.
It is that (see [2])
‖b− b2kQ‖exp Lr,2kQ ≤ Ck‖b‖Oscexp Lr .
It is obvious that Oscexp Lr coincides with the BMO space if r = 1. For rj > 0
and bj ∈ Oscexp Lrj for j = 1, . . . ,m, we denote 1/r = 1/r1 + . . . + 1/rm and
‖b̃‖ =
∏m
j=1
‖bj‖Oscexp L
rj
. Given a positive integer m and 1 ≤ j ≤ m, we denote by
Cm
j the family of all finite subsets σ = {σ(1), . . . , σ(j)} of {1, . . . ,m} of j different
elements. For σ ∈ Cm
j , denote σc = {1, . . . ,m} \ σ. For b̃ = (b1, . . . , bm) and
σ = {σ(1), . . . , σ(j)} ∈ Cm
j , denote b̃σ = (bσ(1), . . . , bσ(j)), bσ = bσ(1) . . . bσ(j) and
‖b̃σ‖Oscexp Lrσ = ‖bσ(1)‖Osc
exp L
rσ(1)
. . . ‖bσ(j)‖Osc
exp L
rσ(j)
.
We denote the Muckenhoupt weights by Ap for 1 ≤ p <∞ (see [3]).
We are going to consider some integral operators defined below.
Let bj , j = 1, . . . ,m, be the fixed locally integral functions on Rn.
Definition 1. Let λ > 3 + 2/n, ε > 0 and ψ be a fixed function which satisfies
the following properties:
(1)
∫
ψ(x)dx = 0,
(2) |ψ(x)| ≤ C(1 + |x|)−(n+1),
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
WEIGHTED SHARP BOUNDEDNESS FOR MULTILINEAR COMMUTATORS 1421
(3) |ψ(x+ y)− ψ(x)| ≤ C|y|ε(1 + |x|)−(n+1+ε) when 2|y| < |x|.
The Littlewood – Paley multilinear commutator is defined by
gb̃
λ(f)(x) =
∫ ∫
Rn+1
+
(
t
t+ |x− y|
)nλ
|F b̃
t (f)(x, y)|2 dydt
tn+1
1/2
,
where
F b̃
t (f)(x, y) =
∫
Rn
m∏
j=1
(bj(x)− bj(z))
ψt(y − z)f(z)dz
and ψt(x) = t−nψ(x/t) for t > 0. Set Ft(f) = ψt ∗ f. We also define that
gλ(f)(x) =
∫ ∫
Rn+1
+
(
t
t+ |x− y|
)nλ ∣∣Ft(f)(y)
∣∣2 dydt
tn+1
1/2
,
which is the Littlewood – Paley function (see [6]).
Let H be the Hilbert space H =
{
h : ‖h‖ =
(∫ ∫
Rn+1
+
|h(y, t)|2dydt/tn+1
)1/2
<
<∞
}
. Then for each fixed x ∈ Rn, FA
t (f)(x, y) may be regarded as a mapping from
(0,+∞) to H, and it is clear that
gb̃
λ(f)(x) =
∥∥∥∥∥
(
t
t+ |x− y|
)nλ/2
F b̃
t (f)(x, y)
∥∥∥∥∥
and
gλ(f)(x) =
∥∥∥∥∥
(
t
t+ |x− y|
)nλ/2
Ft(f)(y)
∥∥∥∥∥ .
Definition 2. Let λ > 1, 0 < γ ≤ 1 and Ω be homogeneous of degree zero on
Rn such that
∫
Sn−1
Ω(x′)dσ(x′) = 0. Assume that Ω ∈ Lipγ(Sn−1), i.e., there exists
a constant M > 0 such that for any x, y ∈ Sn−1,
∣∣Ω(x) − Ω(y)
∣∣ ≤ M |x − y|γ . We
denote Γ(x) =
{
(y, t) ∈ Rn+1
+ : |x− y| < t
}
and the characteristic function of Γ(x) by
χΓ(x). The Marcinkiewicz multilinear commutator is defined by
µb̃
λ(f)(x) =
∫ ∫
Rn+1
+
(
t
t+ |x− y|
)nλ
|F b̃
t (f)(x, y)|2 dydt
tn+3
1/2
,
where
F b̃
t (f)(x, y) =
∫
|y−z|≤t
Ω(y − z)
|y − z|n−1
m∏
j=1
(bj(x)− bj(z))
f(z)dz.
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
1422 HONG XU, JIASHENG ZENG, LANZHE LIU
We set
Ft(f)(y) =
∫
|y−z|≤t
Ω(y − z)
|y − z|n−1
f(z)dz.
We also define
µλ(f)(x) =
∫ ∫
Rn+1
+
(
t
t+ |x− y|
)nλ
|Ft(f)(y)|2 dydt
tn+3
1/2
,
which is the Marcinkiewicz integral (see [7]).
Let H be the space H =
{
h : ‖h‖ =
(∫ ∫
Rn+1
+
|h(y, t)|2dydt/tn+3
)1/2
< ∞
}
.
Then, it is clear that
µA
λ (f)(x) =
∥∥∥∥∥
(
t
t+ |x− y|
)nλ/2
F b̃
t (f)(x, y)
∥∥∥∥∥
and
µλ(f)(x) =
∥∥∥∥∥
(
t
t+ |x− y|
)nλ/2
Ft(f)(y)
∥∥∥∥∥ .
More generally, we define the following multilinear commutator related to certain
integral operators.
Definition 3. Let F (x, y, t) be a function defined on Rn×Rn×[0,+∞), we denote
that
Ft(f)(x) =
∫
Rn
F (x, y, t)f(y)dy
and
F b̃
t (f)(x) =
∫
Rn
m∏
j=1
(bj(x)− bj(y))
F (x, y, t)f(y)dy
for every bounded and compactly supported function f.
Let H be the Banach space H =
{
h : ‖h‖ <∞
}
such that, for each fixed x ∈ Rn,
Ft(f)(x) and F b̃
t (f)(x) may be regarded as a mapping from [0,+∞) to H. Then, the
multilinear commutator related to F b̃
t is defined by
Tb̃(f)(x) =
∥∥F b̃
t (f)(x)
∥∥;
we also denote
T (f)(x) =
∥∥Ft(f)(x)
∥∥.
Note that when b1 = . . . = bm, Tb̃ is just the m order commutator. It is well known
that commutators are of great interest in harmonic analysis and have been widely studied
by many authors (see [1, 2, 4, 5, 8 – 12]). Our main purpose is to establish the sharp
inequalities for the multilinear commutator operators.
The following theorems are our main results:
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
WEIGHTED SHARP BOUNDEDNESS FOR MULTILINEAR COMMUTATORS 1423
Theorem 1. Let rj ≥ 1 and bj ∈ Oscexp Lrj for j = 1, . . . ,m. Denote 1/r =
= 1/r1 + . . .+ 1/rm. Then the following statements are frue:
(1) For any 0 < p < q < 1, there exists a constant C > 0 such that for any
f ∈ C∞
0 (Rn) and any x ∈ Rn,
(gb̃
λ(f))#p (x) ≤ C
‖b‖ML(log L)1/r (f)(x) +
m∑
j=1
∑
σ∈Cm
j
Mq(g
b̃σc
λ (f)(x)
.
(2) If 1 < p <∞ and w ∈ Ap, then∥∥gb̃
λ(f)
∥∥
Lp(w)
≤ C‖b̃‖ ‖f‖Lp(w).
(3) Denote Φ(t) = t log1/r(t + e). If w ∈ A1, then there exists a constant C > 0
such that for all λ > 0,
w
({
x ∈ Rn : gb̃
λ(f)(x) > λ
})
≤ C
∫
Rn
Φ
(
‖b̃‖|f(x)|
λ
)
w(x)dx.
Theorem 2. Let rj ≥ 1 and bj ∈ Oscexp Lrj for j = 1, . . . ,m. Denote 1/r =
= 1/r1 + . . .+ 1/rm. Then the following statements are frue:
(1) For any 0 < p < q < 1, there exists a constant C > 0 such that for any
f ∈ C∞
0 (Rn) and any x ∈ Rn,
(µb̃
λ(f))#p (x) ≤ C
‖b‖ML(log L)1/r (f)(x) +
m∑
j=1
∑
σ∈Cm
j
Mq(µ
b̃σc
λ (f)(x)
.
(2) If 1 < p <∞ and w ∈ Ap, then∥∥µb̃
λ(f)
∥∥
Lp(w)
≤ C‖b̃‖‖f‖Lp(w).
(3) Denote Φ(t) = t log1/r(t + e). If w ∈ A1, then there exists a constant C > 0
such that for all λ > 0,
w
({
x ∈ Rn : µb̃
λ(f)(x) > λ
})
≤ C
∫
Rn
Φ
(
‖b̃‖|f(x)|
λ
)
w(x)dx.
3. Proofs of theorems. We begin with a general theorem.
Main Theorem. Let rj ≥ 1 and bj ∈ Oscexp Lrj for j = 1, . . . ,m. Denote 1/r =
= 1/r1 + . . .+ 1/rm. Suppose that T is the same as in Definition 1 and such that T is
bounded on Lp(w) for all w ∈ Ap, 1 < p <∞, and weak (L1(w), L1(w)) bounded for
all w ∈ A1. If T satisfies the size condition∥∥∥Ft((b1 − (b1)2Q) . . . (bm − (bm)2Q)f)(x)−
−Ft((b1 − (b1)2Q) . . . (bm − (bm)2Q)f)(x0)
∥∥∥ ≤ CML(log L)1/r (f)(x̃)
for any cube Q = Q(x0, d) with supp f ⊂ (2Q)c and x, x̃ ∈ Q = Q(x0, d), then for
any 0 < p < q < 1, there exists a constant C > 0 such that, for any f ∈ C∞
0 (Rn) and
any x ∈ Rn,
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
1424 HONG XU, JIASHENG ZENG, LANZHE LIU
(Tb̃(f))#p (x) ≤ C
‖b‖ML(log L)1/r (f)(x) +
m∑
j=1
∑
σ∈Cm
j
Mq(Tb̃σc
(f)(x))
.
To prove the theorem, we need the following lemmas:
Lemma 1 (Kolmogorov, [3, p. 485]). Let 0 < p < q < ∞ and let f ≥ 0 be an
arbitrary function. We define, for 1/r = 1/p− 1/q,
‖f‖WLq = sup
λ>0
λ
∣∣{x ∈ Rn : f(x) > λ}
∣∣1/q
, Np,q(f) = sup
E
‖fχE‖Lp/‖χE‖Lr ,
where the sup is taken for all measurable sets E with 0 < |E| <∞. Then
‖f‖WLq ≤ Np,q(f) ≤
(
q/(q − p)
)1/p‖f‖WLq .
Lemma 2 [2]. Let rj ≥ 1 for j = 1, . . . ,m. Denote 1/r = 1/r1 + . . . + 1/rm.
Then
1
|Q|
∫
Q
∣∣f1(x) . . . fm(x)g(x)
∣∣dx ≤ ‖f‖exp Lr1 ,Q . . . ‖f‖exp Lrm ,Q‖g‖L(log L)1/r,Q.
Proof of Main Theorem. It suffices to prove that, for f ∈ C∞
0 (Rn) and some
constant C0, the following inequality holds: 1
|Q|
∫
Q
|Tb̃(f)(x)− C0|pdx
1/p
≤
≤ C
‖b‖ML(log L)1/r (f)(x̃) +
m∑
j=1
∑
σ∈Cm
j
Mq(Tb̃σc
(f))(x̃)
.
Fix a cube Q = Q(x0, d) and x̃ ∈ Q. We first consider the case m = 1. For f1 = fχ2Q
and f2 = fχRn\2Q, we write
F b1
t (f)(x) =
= (b1(x)− (b1)2Q)Ft(f)(x)− Ft((b1 − (b1)2Q)f1)(x)− Ft((b1 − (b1)2Q)f2)(x),
then∣∣Tb1(f)(x)− T (((b1)2Q − b1)f2)(x0)
∣∣ ≤ ∥∥F b1
t (f)(x)− Ft(((b1)2Q − b1)f2)(x0)
∥∥ ≤
≤
∥∥(b1(y)− (b1)2Q)Ft(f)(x)
∥∥+
∥∥Ft((b1 − (b1)2Q)f1)(x)
∥∥+
+
∥∥Ft((b1 − (b1)2Q)f2)(x)− Ft((b1 − (b1)2Q)f2)(x0)
∥∥ =
= I(x) + II(x) + III(x).
For I(x), by Hölder’s inequality for the exponent 1/l + 1/l′ = 1 with 1 < l < q/p and
pl = q, we have 1
|Q|
∫
Q
|I(x)|pdx
1/p
=
1
|Q|
∫
Q
|b1(x)− (b1)2Q|p|T (f)(x)|pdx
1/p
≤
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
WEIGHTED SHARP BOUNDEDNESS FOR MULTILINEAR COMMUTATORS 1425
≤
C
|2Q|
∫
2Q
|b1(x)− (b1)2Q|pl′
1/pl′ 1
|Q|
∫
Q
|T (f)(x)|pldx
1/pl
≤
≤ C‖b1‖Oscexp LrMpl(T (f))(x̃) ≤ C‖b1‖Oscexp LrMq(T (f))(x̃).
For II(x), by Lemma 1 and the weak type (1, 1) of T, we have 1
|Q|
∫
Q
|B(x)|pdx
1/p
=
1
|Q|
∫
Q
∣∣T ((b1 − (b1)2Q)f1)(x)
∣∣pdx
1/p
≤
≤ C|2Q|−1 ‖T ((b1 − (b1)2Q)f1)‖Lp
|2Q|1/p−1
≤
≤ C|2Q|−1‖T ((b1 − (b1)2Q)fχ2Q)‖WL1 ≤
≤ C|2Q|−1
∫
2Q
|b1(x)− (b1)2Q‖f(x)|dx ≤
≤ C‖b1 − (b1)2Q‖exp Lr,2Q‖f‖L(log L)1/r,2Q ≤
≤ C‖b1‖Oscexp LrML(log L)1/r (f)(x̃).
For III(x), using the size condition of T, we have 1
|Q|
∫
Q
|C(x)|pdx
1/p
≤ CML(log L)1/r (f)(x̃).
We now consider the case m ≥ 2. For b = (b1, . . . , bm), we write
F b̃
t (f)(x) =
∫
Rn
m∏
j=1
(bj(x)− bj(y))
F (x, y, t)f(y)dy =
=
∫
Rn
(b1(x)− (b1)2Q)− (b1(y)− (b1)2Q) . . . (bm(x)−
−(bm)2Q)− (bm(y)− (bm)2Q)F (x, y, t)f(y)dy =
=
m∑
j=0
∑
σ∈Cm
j
(−1)m−j(b(x)− (b)2Q)σ
∫
Rn
(b(y)− (b)2Q)σF (x, y, t)f(y)dy =
= (b1(x)− (b1)2Q) . . . (bm(x)− (bm)2Q)Ft(f)(x)+
+(−1)mFt((b1 − (b1)2Q) . . . (bm − (bm)2Q)f)(x)+
+
m−1∑
j=1
∑
σ∈Cm
j
(−1)m−j(b(x)− (b)2Q)σ
∫
Rn
(b(y)− b(x))σcF (x, y, t)f(y)dy =
= (b1(x)− (b1)2Q) . . . (bm(x)− (bm)2Q)Ft(f)(x)+
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
1426 HONG XU, JIASHENG ZENG, LANZHE LIU
+(−1)mFt((b1 − (b1)2Q) . . . (bm − (bm)2Q)f)(x)+
+
m−1∑
j=1
∑
σ∈Cm
j
cm,j(b(x)− (b)2Q)σF
b̃σc
t (f)(x),
whence ∣∣∣Tb̃(f)(x)− (−1)mT ((b1 − (b1)2Q) . . . (bm − (bm)2Q))f2)(x0)
∣∣∣ ≤
≤
∥∥∥Fb̃(f)(x)− (−1)mFt((b1 − (b1)2Q) . . . (bm − (bm)2Q))f2)(x0)
∥∥∥ ≤
≤
∥∥∥(b1(x)− (b1)2Q) . . . (bm(x)− (bm)2Q)Ft(f)(x)
∥∥∥+
+
m−1∑
j=1
∑
σ∈Cm
j
∥∥(b(x)− (b)2Q)σF
b̃σc
t (f)(x)
∥∥+
+
∥∥Ft((b1 − (b1)2Q) . . . (bm − (bm)2Q)f1)(x)
∥∥+
+
∥∥Ft((b1 − (b1)2Q) . . . (bm − (bm)2Q)f2)(x)−
−Ft((b1 − (b1)2Q) . . . (bm − (bm)2Q)f2)(x0)
∥∥ =
= I1(x) + I2(x) + I3(x) + I4(x).
For I1(x) and I2(x), similar to the proof of the case m = 1, we get 1
|Q|
∫
Q
(I1(x))pdx
1/p
≤ CMq(T (f))(x̃)
and 1
|Q|
∫
Q
(I2(x))pdx
1/p
≤ C
m−1∑
j=1
∑
σ∈Cm
j
ML(log L)1/r (f)(x̃).
For I3, by the weak type (1, 1) of T and Lemma 2, we obtain 1
|Q|
∫
Q
(I3(x))pdx
1/p
≤
≤ C
|2Q|
∫
2Q
|b1(x)− (b1)2Q| . . . |bm(x)− (bm)2Q‖f(x)|dx ≤
≤ C‖b1 − (b1)2Q‖exp Lr1 ,2Q . . . ‖bm − (bm)2Q‖exp Lrm ,2Q‖f‖L(log L)1/r,2Q ≤
≤ C‖b‖ML(log L)1/r (f)(x̃).
For I4, using the size condition of T, we have
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
WEIGHTED SHARP BOUNDEDNESS FOR MULTILINEAR COMMUTATORS 1427
1
|Q|
∫
Q
(I4(x))pdx
1/p
≤ CML(log L)1/r (f)(x̃).
This completes the proof of the Main Theorem.
To prove Theorems 1 and 2, it suffices to verify that gb̃
λ and µb̃
λ satisfy the size
condition in Main Theorem, that is∥∥∥∥∥
[(
t
t+ |x− y|
)nλ/2
−
(
t
t+ |x0 − y|
)nλ/2
]
×
× Ft
(
(b1 − (b1)2Q
)
. . . (bm − (bm)2Q)f)(y)
∥∥∥∥∥ ≤
≤ CML(log L)1/r (f)(x̃).
Suppose that supp f ⊂ Qc and x ∈ Q = Q(x0, d). Note that |x0 − z| ≈ |x− z| for
z ∈ Qc.
For gb̃
λ, by the condition of ψ and the inequality a1/2 − b1/2 ≤ (a − b)1/2 for
a ≥ b > 0, we get ∥∥∥∥∥
[(
t
t+ |x− y|
)nλ/2
−
(
t
t+ |x0 − y|
)nλ/2
]
×
×Ft((b1 − (b1)2Q) . . . (bm − (bm)2Q)f)(y)
∥∥∥∥∥ ≤
≤
∫ ∫
Rn+1
+
∫
(2Q)c
[
tnλ/2|x0 − x|1/2
(t+ |x0 − y|)(nλ+1)/2
∣∣b1(z)− (b1)2Q
∣∣ . . .
. . .
∣∣bm(z)− (bm)2Q
∣∣ |f(z)|
∣∣ψt(y − z)
∣∣dz]2 dydt
tn+1
1/2
≤
≤ C
∫
(2Q)c
∣∣b1(z)− (b1)2Q
∣∣ . . . ∣∣bm(z)− (bm)2Q
∣∣ |f(z)|×
×
∫ ∫
Rn+1
+
t1−n+nλ|x0 − x|dydt
(t+ |x0 − y|)nλ+1(t+ |y − z|)2n+2
1/2
dz;
noting that 2t+ |y − z| ≥ 2t+ |x0 − z| − |x0 − y| ≥ t+ |x0 − z| for |x0 − y| ≤ t and
2k+1t + |y − z| ≥ 2k+1t + |x0 − z| − |x0 − y| ≥ |x0 − z| for |x0 − y| ≤ 2k+1t and
recalling that λ > (3n+ 2)/n, we get
t−n
∫
Rn
(
t
t+ |x0 − y|
)nλ
dy
(t+ |y − z|)2n+2
=
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
1428 HONG XU, JIASHENG ZENG, LANZHE LIU
= t−n
∫
|x0−y|≤t
(
t
t+ |x0 − y|
)nλ
dy
(t+ |y − z|)2n+2
+
+t−n
∞∑
k=0
∫
2kt<|x0−y|≤2k+1t
(
t
t+ |x0 − y|
)nλ
dy
(t+ |y − z|)2n+2
≤
≤ t−n
∫
|x0−y|≤t
22n+2dy
(2t+ 2|y − z|)2n+2
+
+
∞∑
k=0
∫
|x0−y|≤2k+1t
2−knλ 2(k+2)(2n+2)dy
(2k+2t+ 2k+2|y − z|)2n+2
≤
≤ Ct−n
∫
|x0−y|≤t
dy
(2t+ |y − z|)2n+2
+
+
∞∑
k=0
∫
|x0−y|≤2k+1t
2−knλ 2k(2n+2)dy
(t+ 2k+1t+ |y − z|)2n+2
≤
≤ Ct−n
∫
|x0−y|≤t
dy
(t+ |x0 − z|)2n+2
+
+
∞∑
k=0
∫
|x0−y|≤2k+1t
2−knλ 2k(2n+2)dy
(t+ |x0 − z|)2n+2
≤
≤ Ct−n
[
tn
(t+ |x0 − z|)2n+2
+
∞∑
k=0
2k(3n+2−nλ) tn
(t+ |x0 − z|)2n+2
]
≤
≤ C
(t+ |x0 − z|)2n+2
,
since
∞∫
0
dt
(t+ |x0 − z|)2n+2
= C|x0 − z|−2n−1,
we obtain ∥∥∥∥∥
[(
t
t+ |x− y|
)nλ/2
−
(
t
t+ |x0 − y|
)nλ/2
]
×
×Ft((b1 − (b1)2Q) . . . (bm − (bm)2Q)f)(y)
∥∥∥∥∥ ≤
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
WEIGHTED SHARP BOUNDEDNESS FOR MULTILINEAR COMMUTATORS 1429
≤ C
∫
(2Q)c
∣∣b1(z)− (b1)2Q
∣∣ . . . ∣∣bm(z)− (bm)2Q
∣∣ |f(z)| |x0 − x|1/2
|x0 − z|n+1/2
dz ≤
≤ C
∞∑
k=1
∫
2k+1Q\2kQ
|x0 − x|1/2|x0 − z|−(n+1/2)
∥∥∥∥∥∥
m∏
j=1
(bj(z)− (bj)2Q)
∥∥∥∥∥∥ |f(z)|dz ≤
≤ C
∞∑
k=1
2−k/2|2k+1Q|−1
∫
2k+1Q
∥∥∥∥∥∥
m∏
j=1
(bj(z)− (bj)2Q)
∥∥∥∥∥∥ |f(z)|dz ≤
≤ C
∞∑
k=1
2−k/2
m∏
j=1
∥∥bj − (bj)2Q
∥∥
exp Lrj ,2k+1Q
‖f‖L(log L)1/r,2k+1Q ≤
≤ C
∞∑
k=1
km2−k/2
m∏
j=1
‖bj‖Oscexp L
rj
ML(log L)1/r (f)(x̃) ≤
≤ C
m∏
j=1
‖bj‖Oscexp L
rj
ML(log L)1/r (f)(x̃).
For µb̃
λ, by the condition of Ω, we get∥∥∥∥∥
[(
t
t+ |x− y|
)nλ/2
−
(
t
t+ |x0 − y|
)nλ/2
]
×
×Ft
(
(b1 − (b1)2Q
)
. . .
(
bm − (bm)2Q)f
)
(y)
∥∥∥∥∥ ≤
≤ C
∫ ∫
Rn+1
+
∫
(2Q)c
[
χΓ(z)(y, t)tnλ/2|x0 − x|1/2
(t+ |x− y|)(nλ+1)/2|y − z|n−1
×
×
m∏
j=1
|bj(z)− (bj)2Q‖f(z)|dz
]2
dydt
tn+3
1/2
≤
≤ C
∫
(2Q)c
m∏
j=1
∣∣bj(z)− (bj)2Q
∣∣|f(z)|
∫ ∫
Rn+1
+
χΓ(z)(y, t)tnλ−n−3|x0 − x|dydt
(t+ |x− y|)nλ+1|y − z|2n−2
1/2
dz;
noting that the inequalities |x− z| ≤ 2t and |y − z| ≥ |x− z| − t ≥ |x− z| − 3t hold
for |x− y| ≤ t and |y − z| ≤ t and the inequalities |x− z| ≤ t(1 + 2k+1) ≤ 2k+2t and
|y − z| ≥ |x− z| − 2k+3t hold for |x− y| ≤ 2k+1t and |y − z| ≤ t, we obtain∥∥∥∥∥
[(
t
t+ |x− y|
)nλ/2
−
(
t
t+ |x0 − y|
)nλ/2
]
×
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
1430 HONG XU, JIASHENG ZENG, LANZHE LIU
×Ft
(
(b1 − (b1)2Q
)
. . .
(
bm − (bm)2Q
)
f)(y)
∥∥∥∥∥ ≤
≤ C
∫
(2Q)c
m∏
j=1
∣∣bj(z)− (bj)2Q
∣∣ |f(z)| |x0 − x|1/2×
×
∞∫
0
∫
|x−y|≤t
(
t
t+ |x− y|
)nλ+1 χΓ(z)(y, t)t−ndydt
(|x− z| − 3t)2n+2
1/2
dz+
+ C
∫
(2Q)c
m∏
j=1
∣∣bj(z)− (bj)2Q
∣∣ |f(z)| |x0 − x|1/2×
×
∞∫
0
∞∑
k=0
∫
2kt<|x−y|≤2k+1t
(
t
t+ |x− y|
)nλ+1 χΓ(z)(y, t)t−ndydt
(|x− z| − 2k+3t)2n+2
1/2
dz ≤
≤ C
∫
(2Q)c
m∏
j=1
∣∣bj(z)− (bj)2Q
∣∣ |f(z)| |x0 − x|1/2×
×
∞∫
|x−z|/2
dt
(|x− z| − 3t)2n+2
1/2
dz +
+ C
∫
(2Q)c
m∏
j=1
∣∣bj(z)− (bj)2Q‖f(z)‖x0 − x
∣∣1/2×
×
∞∑
k=0
∞∫
2−2−k|x−z|
2−k(nλ+2)(2kt)nt−n2kdt
(|x− z| − 2k+3t)2n+2
1/2
dz ≤
≤ C
∫
(2Q)c
m∏
j=1
∣∣bj(z)− (bj)2Q
∣∣ |f(z)| |x0 − x|1/2
∣∣x− z|−n−1/2dz+
+C
∫
(2Q)c
m∏
j=1
∣∣bj(z)− (bj)2Q
∣∣ |f(z)| |x0 − x
∣∣1/2
∞∑
k=0
2k(n−nλ−2)/2|x− z|−n−1/2dz ≤
≤ C
∫
(2Q)c
m∏
j=1
∣∣bj(z)− (bj)2Q
∣∣ |f(z)| |x0 − x|1/2
|x0 − z|n+1/2
dz ≤
≤ C
m∏
j=1
‖bj‖Oscexp L
rj
ML(log L)1/r (f)(x̃).
These yields the desired results.
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
WEIGHTED SHARP BOUNDEDNESS FOR MULTILINEAR COMMUTATORS 1431
By (1) and the boundedness of gλ, µλ and ML(log L)1/r , we may obtain the conclusi-
ons (2), (3) of Theorems 1 and 2. This completes the proof of Theorems 1 and 2.
1. Coifman R., Meyer Y. Wavelets, Calderón – Zygmund and multilinear operators // Cambridge Stud.
Adv. Math. – 1997. – 48.
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Received 25.10.2005
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 10
|
| id | umjimathkievua-article-3400 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:41:50Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c0/080c8c00f3aab0a5d400e749c75f9ac0.pdf |
| spelling | umjimathkievua-article-34002020-03-18T19:53:10Z Weighted sharp boundedness for multilinear commutators Зважена точна обмеженість для мультилінійних комутаторів Hong, Xu Liu, Lanzhe Zeng, Jiasheng Гонг, Сюй Лю, Ланже Цзен, Цзяшен In this paper, the sharp estimates for some multilinear commutators related to certain sublinear integral operators are obtained. The operators include the Littlewood - Paley operator and the Marcinkiewicz operator. As application, we obtain the weighted $L^p (p > 1)$ inequalities and $L \log L$-type estimate for the multilinear commutators. Одержано точні оцінки для деяких мультилінійних комутаторів, що пов'язані з певними субліній-ними інтегральними операторами. Ці оператори включають в себе оператор Літлвуда-Палея та оператор Марцінкевича. Як застосування, отримано зважені $L^p (p > 1)$ нерівності та оцінку типу $L \log L$ для мультилінійних комутаторів. Institute of Mathematics, NAS of Ukraine 2007-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3400 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 10 (2007); 1419–1431 Український математичний журнал; Том 59 № 10 (2007); 1419–1431 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3400/3543 https://umj.imath.kiev.ua/index.php/umj/article/view/3400/3544 Copyright (c) 2007 Hong Xu; Liu Lanzhe; Zeng Jiasheng |
| spellingShingle | Hong, Xu Liu, Lanzhe Zeng, Jiasheng Гонг, Сюй Лю, Ланже Цзен, Цзяшен Weighted sharp boundedness for multilinear commutators |
| title | Weighted sharp boundedness for multilinear commutators |
| title_alt | Зважена точна обмеженість для мультилінійних комутаторів |
| title_full | Weighted sharp boundedness for multilinear commutators |
| title_fullStr | Weighted sharp boundedness for multilinear commutators |
| title_full_unstemmed | Weighted sharp boundedness for multilinear commutators |
| title_short | Weighted sharp boundedness for multilinear commutators |
| title_sort | weighted sharp boundedness for multilinear commutators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3400 |
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