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Weighted estimates for commutators of vector-valued maximal multilinear operators
Nonlinear Analysis 96 (2014) 96–108
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Weighted estimates for commutators of vector-valued maximal multilinear operators Zengyan Si a , Qingying Xue b,∗ a
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, People’s Republic of China
b
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
article
abstract
info
Let T ∗ be the maximal multilinear Calderón–Zygmund operator defined by T ∗ (⃗ f )(x) = supδ>0 |Tδ (f1 , . . . , fm )(x)|, and Tq∗ (⃗ f ) be the vector-valued version of T ∗ , Tq∗ (⃗ f )(x) =
Article history: Received 8 March 2013 Accepted 4 November 2013 Communicated by Enzo Mitidieri
∞ k=1
|T ∗ (f1k , . . . , fmk )(x)|q
1/q
, where Tδ are the smooth truncations of the multilinear
singular integral operator T . In this paper, we consider weighted norm inequalities for the iterated commutators of vector-valued maximal multilinear operators Tq∗ (⃗ f ). The weighted strong type and weighted end-point weak type estimates for the iterated commutators of Tq∗ (⃗ f ) were established respectively. © 2013 Elsevier Ltd. All rights reserved.
Keywords: Maximal multilinear operators Multilinear Calderón–Zygmund operators Commutators Vector-valued inequalities
1. Introduction Multilinear Calderón–Zygmund operators were introduced and first studied by Coifman and Meyer [1–3], and later on by Grafakos and Torres [4,5]. In recent years, the theory on multilinear Calderón–Zygmund operators and multilinear commutators have attracted much attention as a rapid developing field in harmonic analysis. Let T be a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values into the space of tempered distributions, T : S (Rn ) × · · · × S (Rn ) → S ′ (Rn ). Following [4], we say that T is a multilinear Calderón–Zygmund operator if, for some 1 ≤ qj < ∞, it extends to a bounded multilinear operator from Lq1 × · · · × Lqm to Lq , where 1q = q1 + · · · + q1 , and if there exists a function K , defined off the
diagonal x = y1 = · · · = ym in (Rn )m+1 , satisfying T⃗ f (x) = T (f1 , . . . , fm )(x) = for all x ̸∈
m
j =1
m
K (x, y1 , . . . , ym )f1 (y1 ) · · · fm (ym )dy1 · · · dym ,
(1.1)
supp fj ,
K (y0 , y1 , . . . , ym ) ≤
A m
k,l=0
∗
(Rn )m
1
mn ;
|yk − yl |
Corresponding author. Tel.: +86 10 58807735. E-mail addresses: [email protected] (Z. Si), [email protected] (Q. Xue).
0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.11.003
(1.2)
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
97
for some A > 0 and all (y0 , y1 , . . . , ym ) with y0 ̸= yj for some j. And
|K (y0 , y1 , . . . , yj , . . . , ym ) − K (y0 , y1 , . . . , y′j , . . . , ym )| ≤
A|yj − y′j |ε
mn+ε
m k,l=0
(1.3)
|yk − yl |
for some ε > 0 and all 0 ≤ j ≤ m, whenever |yj − y′j | ≤ 12 max0≤k≤m |yj − yk |. Such kernels are called multilinear Calderón–Zygmund kernels and the collection of such functions is denoted by m-CZK(A, ε) in [4]. The maximal multilinear singular integral operator is defined by T ∗ (⃗ f )(x) = sup |Tδ (f1 , . . . , fm )(x)|, δ>0
where Tδ are the smooth truncations of T given by
Tδ (⃗ f )(x) = m
2 2 i=1 |x−yi | >δ
⃗. K (x, y1 , . . . , ym )f1 (y1 ) · · · fm (ym )dy
⃗ = dy1 · · · dym . As is pointed in [5], T ∗ (⃗f )(x) is pointwise well-defined when fj ∈ Lqj (Rn ) with 1 ≤ qj ≤ ∞. Here, dy The commutators associated with T and T ∗ are defined by TΠ b⃗ (⃗ f )(x) = ∗ ⃗ TΠ ⃗ (f )(x) b
l
(Rn )m j=1
[bj (x) − bj (yj )]K (x, y1 , . . . , ym )f1 (y1 ) · · · fm (ym )dy⃗.
l m ⃗. = sup [bj (x) − bj (yj )]K (x, y1 , . . . , ym ) fj (yj )dy m 2 2 δ>0 j =1 i=1 |x−yi | >δ j=1
The theory of weighted maximal multilinear Calderón–Zygmund type operators was established in [5,6]. Recently, the weighted strong type and end-point estimates for T ∗ ⃗ with multiple weights (see Section 2, Definition 2.1 for the definition) Πb were obtained in [7] as follows: Theorem A ([7]). Let ω ⃗ ∈ Ap⃗ , 1/p = 1/p1 + · · · + 1/pm with 1 < pj < ∞, j = 1, . . . , m and b⃗ = (b1 , . . . , bm ) ∈ (BMO)m .
⃗ and ⃗f such that Then there is a constant C > 0 independent of b m
∥TΠ∗ b⃗ (⃗f )∥Lp (νω⃗ ) ≤ C
∥bj ∥BMO
j =1
m
∥fj ∥Lp (ωj ) .
j =1
Theorem B ([7]). Let ω ⃗ ∈ A(1,...,1) and b⃗ ∈ (BMO)m . Then there exists a constant C depending on b⃗ such that
νω⃗
n
x∈R :
∗ ⃗ TΠ ⃗ (f )(x) b
>t
m
≤C
m
Rn
j =1
Φ
(m)
|fj (yj )| t
1/m ωj (yj )dyj
,
m
where Φ (t ) = t (1 + log t ) and Φ +
(m)
= Φ ◦ · · · ◦ Φ.
The vector-valued multilinear Calderón–Zygmund operator Tq and vector-valued maximal multilinear operator Tq∗ associated with the operator T are defined and studied by Grafakos and Martell in [8]. Tq (⃗ f )(x) = |T (f1 , . . . , fm )(x)|q = ∥T (f1· , . . . , fm· )(x)∥lq
=
∞
1/q
|T (f1k , . . . , fmk )(x)|q
,
k=1
Tq∗ (⃗ f )(x) = |T ∗ (f1 , . . . , fm )(x)|q = ∥T ∗ (f1· , . . . , fm· )(x)∥lq
=
∞
1/q
|T (f1k , . . . , fmk )(x)| ∗
q
,
k=1
where fi = {fik }∞ k=1 for i = 1, . . . , m. Grafakos and Martell [8] obtain the following results.
98
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
Theorem C ([8]). Let T be a multilinear Calderón–Zygmund operators, and let 1/m < p < ∞, 1/p = 1/p1 + · · · + 1/pm with 1 < p1 , . . . , pm < ∞, 1/m < q < ∞ and 1/q = 1/q1 + · · · + 1/qm with 1 < q1 , . . . , qm < ∞. Then there exists a constant C > 0 such that m
∥Tq∗ (⃗f )∥Lp (Rn ) ≤ C
∥ |fj |qj ∥Lpj (Rn ) .
j =1
Cruz-Uribe, Martell and Pérez [9] obtain a weak version of Theorem C as follows: Theorem D ([9]). Let T be a multilinear Calderón–Zygmund operators, and let 1/m ≤ p < ∞, 1/p = 1/p1 + · · · + 1/pm with 1 ≤ p1 , . . . , pm < ∞, 1/m < q < ∞ and 1/q = 1/q1 + · · · + 1/qm with 1 < q1 , . . . , qm < ∞. Then there exists a constant C > 0 such that
∥Tq∗ (⃗f )∥Lp,∞ (Rn ) ≤ C
m
∥ |fj |qj ∥Lpj (Rn ) .
j=1
In this paper, we consider the vector-valued version of Theorems A and B. We will sometime use the notation ⃗ f = ⃗ = (y1 , . . . , ym ), dy⃗ = dy1 · · · dym . For the sequence (f1 , . . . , fm ), with fj = {fjk }∞ k=1 , and y ∞ {⃗fk }∞ k=1 = {f1k , . . . , fmk }k=1
of vector functions, the commutators associated with vector-valued Tq and Tq∗ can be defined by
TΠ b⃗,q (⃗ f )(x) = |TΠ b⃗ (⃗ f )(x)|q = ∥TΠ b⃗ (f1· , . . . , fm· )(x)∥lq =
∞
1/q |TΠ b⃗ (⃗fk )(x)|q
,
k=1
∗ ⃗ TΠ ⃗,q (f )(x) b
=
|TΠ∗ b⃗ (⃗f )(x)|q
=
∥TΠ∗ b⃗ (f1· , . . . , fm· )(x)∥lq
=
∞ k=1
1/q |TΠ∗ b⃗ (⃗fk )(x)|q
.
Our main results are as follows: Theorem 1.1. Let T be an m-linear Calderón–Zygmund operators and let 1/m < p < ∞,
+ · · · + p1m , with p m p ⃗ ∈ Ap⃗ , νω⃗ = i=1 ωi i . 1 < p1 , . . . , pm < ∞, 1/m < q < ∞ and q1 +· · ·+ q1 = 1q with 1 < q1 , . . . , qm < ∞. Suppose that ω m 1 bj ∈ BMO, for all j = 1, . . . , l. There then exists a constant C > 0 such that ∥TΠ∗ b⃗,q (⃗f )∥Lp (νω⃗ ) ≤
l
∥bj ∥BMO
m
j =1
1 p
=
1 p1
∥ |fj |qj ∥Lpj (wj ) .
j =1
Theorem 1.2. Let ω ⃗ ∈ A(1,...,1) and b⃗ ∈ (BMO)m . Then there exists a constant C depending on b⃗ such that
νω⃗ x ∈ R :
n
∗ ⃗ TΠ ⃗,q (f )(x) b
>t
m
≤C
m j =1
Rn
Φ
(m)
|fj |qj (yj )
t
1/m ωj (yj )dyj
,
m
where Φ (t ) = t (1 + log t ) and Φ +
(m)
= Φ ◦ · · · ◦ Φ.
2. Notations and main lemmas Throughout this paper, M always denotes the Hardy–Littlewood maximal operator. For δ > 0, Mδ is the maximal function defined by
Mδ f (x) =
1 M (|f |δ ) δ (x)
= sup Q ∋x
1
|Q |
1δ
δ
|f (y)| dy
.
Q
In addition, M ♯ is the sharp maximal function of Fefferman and Stein, 1
♯
M f (x) = sup inf Q ∋x
♯
c
|Q | 1
and Mδ f (x) = M ♯ (|f |δ ) δ (x).
|f (y) − c |dy ≈ sup Q
Q ∋x
1
|Q |
|f (y) − fQ |dy Q
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
99
We will also use the new vector valued maximal function as follows:
m 1 |fj |qj , Q ∋x j=1 |Q | Q
M (|⃗f |q )(x) = sup
where the supremum is taken over all the cubes containing x. Definition 2.1. Let 1 ≤ p1 , . . . , pm < ∞. Given ω ⃗ = (ω1 , . . . , ωm ), set νω⃗ = condition if
sup Q
m
1
|Q |
when pi = 1,
p pi
1p
ωi
Q i=1
1 |Q |
Q
m
i=1 1−p′i
ωi
1′ p i
1
|Q |
1−p′i
ωi
p p
i ⃗ satisfies the Ap⃗ i=1 ωi . We say that ω
m
1′ p i
< ∞,
Q
is understood as (infQ ωi )−1 .
To prove our results, we need the following lemmas. Lemma 2.1 ([10]). Let ω be an A∞ weight. Then there exist constant C and ρ > 0 depending upon the A∞ condition of ω such that, for all λ, ε > 0,
ω({y ∈ Rn : Mf (y) > λ, M ♯ f (y) ≤ λϵ}) ≤ C ε ρ ω
y ∈ Rn : Mf (y) >
1 2
λ
.
As a consequence, we have the following estimates for δ > 0. (1) Let ϕ : (0, ∞) → (0, ∞) be a doubling, that is, ϕ(2a) ≤ C ϕ(a) for a > 0. Then, there exists a constant C depending upon the A∞ condition of ω and doubling condition of ϕ such that ♯
sup ϕ(λ)ω({y ∈ Rn : Mδ f (y) > λ}) ≤ C sup ϕ(λ)ω({y ∈ Rn : Mδ f (y) > λ}) λ>0
(2.1)
λ>0
for every function such that the left-hand side is finite. (2) Let 0 < p < ∞. There exists a positive constant C depending upon the A∞ condition and p such that
Rn
(Mδ f (x))p ω(x)dx ≤ C
Rn
♯
(Mδ f (x))p ω(x)dx
(2.2)
for every function such that the left-hand side is finite. Lemma 2.2. Let T be a multilinear Calderón–Zygmund operator. Let 0 < δ < 1/m, 1/m < q < ∞ and 1/q = 1/q1 + · · · + 1/qm with 1 < q1 , . . . , qm < ∞. Then there exists a constant C > 0 such that ♯
f ))(x) ≤ C M (|⃗ f |q )(x) Mδ (Tq∗ (⃗ n for any smooth vector function {⃗ fk }∞ k=1 and any x ∈ R .
Proof. For simplicity, we only prove for the case m = 2, since there is no essential difference for the general case. Similarly as in [7], we choose two functions u, v ∈ C ∞ ([0, ∞)) such that |u′ (t )| ≤ Ct −1 , |v ′ (t )| ≤ Ct −1 and satisfy
χ[2,∞) (t ) ≤ u(t ) ≤ χ[1,∞) (t ),
χ[1,2] (t ) ≤ v(t ) ≤ χ[1/2,3] (t ).
We introduce two maximal operators by
m ⃗ , U (⃗f )(x) = sup K (x, y1 , . . . , ym )u( |x − y1 | + · · · + |x − ym |/η) fi (yi )dy η>0 (Rn )m i=1 m ⃗ . V ∗ (⃗f )(x) = sup K (x, y1 , . . . , ym )v( |x − y1 | + · · · + |x − ym |/η) fi (yi )dy η>0 (Rn )m i=1 m Let Ku,η (x, y1 , . . . , ym ) = K (x, y1 , . . . , ym )u( Kv,η (x, y1 , . . . , ym ) i=1 |x − yi |/η), m ( i=1 |x − yi |/η) and m ⃗ ⃗; Uη (f )(x) = Ku,η (x, y1 , . . . , ym ) fi (yi )dy ∗
(Rn )m
Vη (⃗f )(x) =
(Rn )m
i =1
Kv,η (x, y1 , . . . , ym )
m i=1
⃗. fi (yi )dy
=
K (x, y1 , . . . , ym )v
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Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
Obviously, T ∗ (⃗ f )(x) ≤ U∗ (⃗ f )(x) + V ∗ (⃗ f )(x). To prove Lemma 2.2, it suffices to show ♯
♯
Mδ (U∗q (⃗ f ))(x) ≤ C M (|⃗ f |q )(x) and where Uq (⃗ f )(x) = ∗
q ⃗ k=1 |U (fk )(x)|
∞
∗
Mδ (Vq∗ (⃗ f ))(x) ≤ C M (|⃗ f |q )(x),
1/q
and Vq (⃗ f )(x) = ∗
∗
∞
q ⃗ k=1 |V (fk )(x)|
∗
∗
1/q .
∗
We just give the proof for Uq , since the proof for Vq is almost the same as Uq . Fix x ∈ Rn and let Q be a cube of side length r centered at x. For any smooth vector function sequence {⃗ fk }∞ k=1 , set
⃗fk∞ = ⃗fk − ⃗fk0 , where ⃗fk0 = ⃗fk χ2Q = (f1k χ2Q , . . . , fmk χ2Q ). Since 0 < δ < 1/2 < 1, we have
1δ 1δ ∗ 1 δ δ ∗ δ |U (⃗f )(y) | −|c | dy ≤C |U (⃗f )(y) − c | dy q |Q | Q |Q | Q q δ 1δ 1 sup |Uη (f⃗0 )(y) + Uη (⃗f χ(2Q )c )(y) − cη | dy ≤C |Q | η 1
Q
≤C
1
q
1δ
∗ U (f⃗0 )(y)δ dy
+
q
|Q |
Q
δ 1δ sup |Uη (⃗f χ(2Q )c )(y) − cη | dy |Q | Q η q 1
= U1 + U2 q 1/q where c = supη>0 cη q = . k≥1 supη>0 cηk
(2.3)
For U1 , it is easy to see U∗ satisfy the same bounded property with T ∗ . Then we applying Kolmogorov’s inequality and Theorem D to get
1δ 2 ∗ 0 δ 1 ∗ ⃗0 ≤C U (⃗f )(y) dy ≤ C ∥ U ( f )∥ |fj |qj (z )dz q L1/2,∞ Q , |dx |Q | Q q |Q | Q Q| j =1 1
≤ C M(|⃗f |q )(x). 3 i To estimate U2 , we choose cη = i=1 cη for i = 1, 2, 3, where ∞ ∞ cη1 = Uη (f1k , f2k )(x),
0 cη2 = Uη (f1k , f2k∞ )(x),
(2.4)
∞ 0 cη3 = Uη (f1k , f2k )(x).
We may split U2 as U2 ≤ U21 + U22 + U23 , where
U21 =
1
|Q |
U22 =
1
1δ δ ∞ ∞ ∞ ∞ sup Uη (f , f )(y) − Uη (f , f )(x) dy ;
Q
1k
2k
1k
2k
q
1δ δ 0 ∞ 0 ∞ supUη (f , f )(y) − Uη (f , f )(x) dy ;
|Q |
η
Q
η
1k
2k
1k
2k
q
and
U23 =
1
|Q |
1δ δ ∞ 0 ∞ 0 sup Uη (f , f )(y) − Uη (f , f )(x) dy .
Q
η
1k
2k
1k
2k
q
For the first term U21 , we have
|Uη (f1k∞ , f2k∞ )(y) − Uη (f1k∞ , f2k∞ )(x)| ≤ C ≤C
(Rn \2Q )2 ∞ s =1
≤C
2 |Q |ε/n |f ∞ (yj )|dy⃗ |(x − y1 , x − y2 )|2n+ε j=1 jk
∞ s =1
(2s+1 Q )2 \(2s Q )2
2−εs
2 j =1
(
2s+1
1 2(s+1)n |Q |
2 |Q |ε/n |fjk (yj )|dy⃗ 1 / n 2n +ε |Q | ) j =1
2s+1 Q
|fjk (yj )|dyj .
(2.5)
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
Now let gsjk =
1 2(s+1)n |Q |
|fjk (yj )|dyj and Gsk =
2s+1 Q
|Uη (f1k∞ , f2k∞ )(y) − Uη (f1k∞ , f2k∞ )(x)| ≤ C
∞
2
j =1
101
gsjk . Then we obtain
2−εs Gsk .
s =1
For any two positive real numbers ε and q, let ρ = min{1, q}. It is easy to show that
q ∞ ∞ −ε s 2 µs ≤ C(ε,q) 2−ερ s |µs |q s=1 s=1 for any sequence µs . Applying this inequality to µs = Gsk , we get
|Uη (f1k , f2k )(y) − Uη (f1k , f2k )(x)|q ≤ C ∞
∞
∞
∞
∞
1/q −ερ s
2
s=1
q Gsk
.
k
Next, Minkowski’s inequality gives
1/qj
qj gsjk
≤
k
1 2(s+1)n |Q |
2s+1 Q
|fj |qj (yj )dyj
and then Hölder’s inequality gives
1/q
q Gsk
≤
2 j=1
k
1/qj qj gsjk
≤ M(|⃗f |q )(x).
k
∞ ∞ From which we deduce |Uη (f1k , f2k )(z ) − Uη (f1k∞ , f2k∞ )(x)|q ≤ C M(|⃗f |q )(x). Since 0 < δ < 1/2, we have
1
|Q |
δ
1δ
sup |Uη (f1k , f2k )(y) − Uη (f1k , f2k )(x)|q dy ∞
Q
∞
∞
∞
≤C
η
1
|Q |
∞ ∞ sup |Uη (f1k , f2k )(y) − Uη (f1k∞ , f2k∞ )(x)|q dy Q
η
≤ C M(|⃗f |q )(x).
(2.6)
For U22 , we observe the following fact
|x − z |ε |f2k (y2 )|dy2 2n+ε 2Q (2Q )c (|z − y1 | + |z − y2 |) ∞ |Q |ε/n |f1k (y1 )|dy1 |f2k (y2 )|dy2 ≤C (2s |Q |1/n )2n+ε 2s+1 Q 2s+1 Q s =1
|Uη (f1k0 , f2k∞ )(z ) − Uη (f1k0 , f2k∞ )(x)| ≤ C
≤C
|f1k (y1 )|dy1
∞ s =1
2−εs
2 j =1
1 2(s+1)n |Q |
2s+1 Q
|fjk (yj )|dyj .
(2.7)
The dealing of the other part is almost the same as U21 , we omit the detail. U23 can be estimated in the same way. Hence we proved Lemma 2.2. Before stating another lemma, we first introduce some notations. For 1 ≤ l ≤ m, we define MLl (log L) (|⃗ f |q ) as follows:
MLl (log L) (|⃗f |q )(x) = sup
l
Q ∋x j=1
ML(log L) (|⃗f |q )(x) = sup
m
Q ∋x j=1
∥ |fj |qj ∥L(log L),Q
m 1 |fj |qj , |Q | Q j =l +1
∥ |fj |qj ∥L(log L),Q .
It is easy to see the following inequalities hold.
M(|⃗f |q )(x) ≤ MLl (log L) (|⃗f |q )(x) ≤ ML(log L) (|⃗f |q )(x). For positive integers m and j with 1 ≤ j ≤ m, we denote by Cjm the family of all finite subsets σ = {σ (1), . . . , σ (j)} of {1, . . . , m} of j different elements, where we always take σ (k) < σ (j) if k < j. For any σ ∈ Cjm , we associated the
102
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108 m−j
complementary sequence σ ′ ∈ Cj given by σ ′ = {1, . . . , m} \ σ with the convention C0m = ∅. Given an m-tuple of m functions b and σ ∈ Cj , we also use the notation bσ for the j-tuple obtained from b given by (bσ (1) , . . . , bσ (j) ). m ∗ ⃗ ∗ ⃗ j Similarly to the above definitions for UΠ b (f )(x) and VΠ b (f )(x), if σ ∈ Cj , and bσ = (bσ (1) , . . . , bσ (j) ) in BMO , we define the following commutators:
∗ ⃗ TΠ bσ ,q (f )(x)
=
∞
1/q |TΠ∗ bσ (⃗f )(x)|q
,
k=1
j
∗ ⃗ ⃗) i=1 (bσ (i) (x) − bσ (i) (yσ (i) )) i=1 fi (yi )dy⃗. If σ is replaced by σ ′ , we denote them where TΠ bσ (f )(x) = supη>0 (Rn )m K (x, y ∗ ∗ by TΠ b ′ ,q and TΠ b ′ , similar notation will be used later for the operators U∗ and V ∗ .
σ
m
σ
Lemma 2.3. Let 0 < δ < ε < 1/m. There then exists a constant C > 0 depending only on δ and ε such that ♯
∗ ⃗ Mδ (TΠ ⃗,q f )(x) ≤ C b
l
l ∥bi ∥BMO Mε (TΠ∗ b ∥bj ∥BMO MLl (log L) (|⃗f |q )(x) + Mε (Tq∗ ⃗f )(x) + C
σ ′ ,q
j=1 σ ∈C l i∈σ j
j =1
⃗f )(x)
(2.8)
n ′ for any smooth vector function {⃗ fk }∞ k=1 and for any x ∈ R , where σ = {1, . . . , l} \ σ .
Proof. We apply a similar method as in Lemma 2.2. We first define two maximal operators as follows.
l m ⃗; = sup Ku,η (x, y1 , . . . , ym ) [bj (x) − bj (yj )] fi (yi )dy η>0 (Rn )m j=1 i=1 l m ∗ ⃗ ⃗. Kv,η (x, y1 , . . . , ym ) [bj (x) − bj (yj )] fi (yi )dy VΠ ⃗ (f )(x) = sup b η>0 (Rn )m j =1 i =1 U∗Π b⃗ (⃗f )(x)
It is easy to see T ∗ ⃗ (⃗ f )(x) ≤ U∗ ⃗ (⃗ f )(x) + V ∗ ⃗ (⃗ f )(x). It suffices to prove the conclusion in lemma (2.3) hold for U∗ ⃗ and Πb Πb Πb Π b ,q ∗ V ⃗ respectively. For similarity, we just give the proof for U∗ ⃗ , since the proof for V ∗ ⃗ is almost the same as U∗ ⃗ . Π b,q
Π b,q
Let F (⃗ y) , f1 (y1 ) · · · fm (ym ), for any λ = (λ1 , . . . , λm ) we have
U∗Π b⃗ (⃗f )(x)
Π b ,q
= sup (b1 (x) − b1 (y)) · · · (bl (x) − bl (y))Ku,η (x, y⃗)F (⃗y)dy⃗ η>0 (Rn )m = sup ((b1 (x) − λ1 ) − (b1 (y) − λ1 )) · · · ((bl (x) − λl ) − (bl (y) − λl ))Ku,η (x, y⃗)F (⃗y)dy⃗ η>0 (Rn )m l = sup (−1)l−j (bj (x) − λj ) (bj (yj ) − λj )Ku,η (x, y⃗)F (⃗y)dy⃗ n )m η>0 i=0 ( R ′ l j∈σ σ ∈Ci
j∈σ
= sup(b1 (x) − λ1 ) · · · (bl (x) − λl )Uη (⃗f )(x) + Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f )(x) η>0 l −1 l−j + (−1) (bj (x) − λj ) (bj (yj ) − λj )Ku,η (x, y⃗)F (⃗y)dy⃗. n m (R ) ′ l j∈σ i=1 σ ∈Ci
Noting the fact
Π b,q
j∈σ ′
(2.9)
j∈σ
(bj (yj ) − λj ) =
j∈σ ′
[(bj (yj ) − bj (x)) + (bj (x) − λj )]. Then,
U∗Π b⃗,q ⃗f (x) ≤ |(b1 (x) − λ1 ) · · · (bl (x) − λl )|U∗q (⃗f )(x) + |U∗ ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f )(x)|q
+ Cq,l,j
l −1
|bj (x) − λj |U∗Π b
i=1 σ ∈C l j∈σ i
σ ′ ,q
⃗f (x),
(2.10)
where Cq,l,j is an absolute constant depending only on l, j and q. Now fix x ∈ Rn , for any cube Q of side length r and centered 1 b (z )dz, for j = 1, . . . , l. Since 0 < δ < 1/m < 1, it follows that at x. Let λj = |2Q | 2Q j
1
|Q |
1/δ
∗ |U Q
Π b⃗,q
δ
δ
(f )(z ) | −|c | dz ⃗∞
δ 1/δ ∗ U (⃗f )(z ) − sup |cη | dz ≤C ⃗ Π b |Q | Q η>0 q
1
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
≤C
1
|Q |
Q
(b1 (z ) − λ1 ) · · · (bl (z ) − λl )U∗ (⃗f )(z )δ dz q
l −1
+ Cq,l,j
i=1 σ ∈C l i
+ Cq
1
|Q |
Q
δ |bj (z ) −
j∈σ
103
1/δ
λj |U∗Π b ′ ,q ⃗f (z ) σ
1/δ dz
δ 1/δ ∗ U ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f )(z ) − sup |cη | dz |Q | Q η>0 q 1
, I + II + III
(2.11)
where c = supη>0 cη q . We can choose 1 < p1 , . . . , pl < ∞ with I≤C
l
1 p1
+ ··· +
1 pl
+
1
ε
= 1δ . Since 0 < δ < ε < 1/m. Hölder’s inequality gives
∥bj ∥BMO Mε (U∗q ⃗f )(x).
j =1
Similarly, we have II ≤ C
l −1
∥bj ∥BMO Mε (U∗Π b
i=1 σ ∈C l j∈σ i
σ ′ ,q
⃗f )(x).
For III, let ⃗ fj = ⃗ fj 0 + ⃗ fj∞ , where ⃗ fj 0 = ⃗ fj χ2Q , and we choose cη =
α
α1 ,...,αm
α
|(Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )f1 1 · · · fmαm ))(x)|.
We use the notation ⃗ f α to denote f1 1 · · · fmαm . Obviously, we have
∗ U ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f )(z ) − sup |cη | η>0
q
≤ U∗q ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f 0 )(z ) + Cqm
sup |(Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗ f α ))(z )
α1 ,...,αm η>0
− (Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f α ))(x)|q ,
(2.12)
where in the last sum each αj = 0 or ∞ and in each term there is at least one αj = ∞. For the first term, we applying Kolmogorov’s estimate and Theorem D to get
1/δ
1
|Q |
|Uq ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f 0 )(z )|δ dz ∗
Q
≤ C ∥U∗q ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f 0 )∥L1/m,∞ Q , dz |Q |
l m 1 1 |bj (yj ) − λj | |fj (z )|qj dz |fj (z )|qj dz |Q | Q |Q | Q j=1 j=l+1 l m 1 ≤C ∥bj ∥BMO ∥ |fj |qj ∥L(log L),Q |fj (z )|qj dz |Q | Q j=1 j =l +1 ≤C
≤C
l
∥bj ∥BMO MLl (log L) (|⃗f |q )(x).
(2.13)
j=1
If all the αj = ∞, we have
|Q | ≤
1/δ
1
⃗α
⃗α
δ
sup |(Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )f ))(z ) − (Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )f ))(x)|q dz
Q η>0
1
|Q |
sup |(Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗ f α ))(z ) − (Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗ f α ))(x)|q dz
Q η>0
≤ (Rn \Q ∗ )m
|x − z |ε |(b1 (y1 ) − λ1 ) · · · (bl (yl ) − λl )| |f1 (yj )|qj · · · |fm (ym )|qm dy⃗ (|z − y1 | + · · · + |z − ym |)mn+ε
104
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
≤
l ∞ 1 j=1 k=1
≤
2kε 2kn |Q ∗ |
2k+1 Q ∗
2
l
2
k∥bj ∥BMO ∥ |fj |qj ∥L(log L),2k+1 Q kε m
∥bj ∥BMO ∥ |fj |qj ∥L(log L),2k+1 Q
2kn |Q ∗ |
m j=l+1
2k+1 Q ∗
|fj |qj dyj .
1
j=l+1
2kn |Q ∗ |
2k+1 Q ∗
|fj |qj dyj
1 2kn |Q ∗ |
2k+1 Q ∗
|fj |qj dyj
1
j=l+1
l
m
1
j=l+1
∥bj − λj ∥BMO ∥ |fj |qj ∥L(log L),2k+1 Q kε
j =1
≤C
m
|bj (yj ) − λj | |fj |qj dyj
l ∞ 1 j=1 k=1
≤
1
l ∞ 1 j=1 k=1
≤
2kn |Q ∗ |
2k+1 Q ∗
|fj |qj dyj
∥bj ∥BMO MLl (log L) (|⃗f |q )(x).
(2.14)
j =1
Now we estimate the typical representative of III. By Minkowski’s inequality, we have
δ 1/δ ∗ U ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )f ∞ , . . . , f ∞ , f 0 , . . . , f 0 )(z ) − sup |cη | dz 1 i i +1 m |Q | Q η>0 q C ∞ ∞ 0 0 sup |Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )f1 , . . . , fi , fi+1 , . . . , fm )(z ) ≤ |Q | Q η>0 1
0 0 − Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )f1∞ , . . . , fi∞ , fi+ 1 , . . . , fm )(x)|q dz m l |x − z |ε |(b1 (y1 ) − λ1 ) · · · (bl (yl ) − λl )| |f1 (y1 )|q1 · · · |fl (yl )|ql dy1 · · · dyl |fj (yj )|qj dyj ≤ ∗ n ∗ l (|z − y1 | + · · · + |z − ym |)mn+ε j =l +1 Q j = 1 ( R \Q ) m l |bj (yj ) − λj | |fj (yj )|qj 1 |fj (yj )|qj dyj (|Q |(m−l)/l )l dyj ≤ n ∗ |Q | Q ∗ |z − yj |n+n(m−l)/l j=l+1 j=1 (R )\Q l m 1 ≤ ∥bj ∥BMO ∥ |fj |qj ∥L(log L),2k+1 Q |fj (yj )|qj dyj |Q | Q ∗ j =1 j=l+1
≤C
l
∥bj ∥BMO MLl (log L) (|⃗f |q )(x).
(2.15)
j =1
In other cases, we can also deduce the same estimates with little modifications on the above argument. Then we proved Lemma 2.3. 3. Proof of the main results We start with the strong and weak estimates for T ∗ ⃗ . Π b ,q
Theorem 3.1. Let 0 < p < ∞, 1/m < q < ∞, and q1 + · · · + q1 = 1q with 1 < q1 , . . . , qm < ∞. Suppose that bj ∈ BMO, m 1 for j = 1, . . . , l and w ∈ A∞ , then there exists a constant C > 0, such that
|TΠ∗ b⃗,q ⃗f |p w(x)dx
Rn
≤C
l
p bj BMO
j =1
p
MLl (log L) (|⃗f |q )(x) w(x)dx,
∥ ∥
Rn
(3.1)
and sup t >0
1
Φ (m) (1/t )
ω y ∈ Rn : |TΠ∗ b⃗,q ⃗f (y)| > t m ≤ C sup t >0
for any smooth function ⃗ f with compact support. Proof. We prove (3.1) first. Assuming that
Rn
p
MLl (log L) (|⃗f |q )(x)
w(x)dx < ∞,
1
Φ (m) (1/t )
ω y ∈ Rn : MLl (log L) (|⃗f |q )(y) > t m ,
(3.2)
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
105
since otherwise there is nothing to be proved. For l = 1, by using Lemmas 2.1 and 2.3, we obtain ♯
∥TΠ∗ b⃗,q ⃗f ∥Lp (ω) ≤ ∥Mδ TΠ∗ b⃗,q ⃗f ∥Lp (ω) ≤ C ∥Mδ TΠ∗ b⃗,q ⃗f ∥Lp (ω) ≤ C ∥b1 ∥BMO [∥Mε (Tq∗ ⃗f )∥Lp (ω) + ∥MLl (log L) (|⃗f |q )∥Lp (ω) ] ≤ C ∥b1 ∥BMO [∥Tq∗ ⃗f ∥Lp (ω) + ∥MLl (log L) (|⃗f |q )∥Lp (ω) ] ≤ C ∥b1 ∥BMO [∥M(|⃗f |q )∥Lp (ω) + ∥MLl (log L) (|⃗f |q )∥Lp (ω) ] ≤ C ∥b1 ∥BMO ∥MLl (log L) (|⃗f |q )∥Lp (ω) .
(3.3)
The penultimate inequality holds by Lemmas 2.1 and 2.2. For the case when l ≥ 2, similarly to the case for l = 1, we have
∥TΠ∗ b⃗,q ⃗f ∥Lp (ω) ≤ C
l
∥bj ∥BMO ∥MLl (log L) (|⃗f |q )∥Lp (ω) .
j =1
f ∥Lp (ω) < ∞ and ∥Tq∗ ⃗ f ∥Lp (ω) < ∞. We will only To apply the inequality (2.2) in (3.3), one needs to verify now that ∥T ∗ ⃗ ⃗ Π b ,q
show the first one since the proof of another one is very similar but easier.
Suppose that the symbols bj and the weight ω are bounded functions. Since ⃗ f has compact support, we may assume supp fj ⊂ B(0, R). Then we have
∥TΠ∗ b⃗,q ⃗f ∥Lp (ω) =
|x|≤2R
|TΠ∗ b⃗,q ⃗f (x)|p ω(x)dx +
|x|>2R
|TΠ∗ b⃗,q ⃗f (x)|p ω(x)dx
= I1 + I2 . For I1 , we choose s > p, s1 , . . . , sm > 1 such that 1/s = 1/s1 + · · · 1/sm . Theorem C and Hölder’s inequality implies
I1 ≤ C |x|≤2R
≤C
m
|TΠ∗ b⃗,q ⃗f |p dx ≤ C
Rn
|Tq∗ ⃗f |s dx
p/s
Rn(s−p)/s
p Rn(s−p)/s < ∞.
∥ |fj |qj ∥Lsj
j =1
For I2 , since |x| > 2R, we have
|TΠ∗ b⃗,q ⃗f (x)|
≤ C ≤C
Thus, I2 ≤ C
Rn
(B(0,R))m
|b1 (x) − b1 (y1 )| · · · |bl (x) − bl (yl )| |f1 (y1 )| · · · |fm (ym )|dy |x − y|mn q
m 1 |fj (yj )|qj dy ≤ C M(|⃗f |q )(x) ≤ C MLl (log L) (|⃗f |q )(x). |x|n B(0,|x|) j=1 p
MLl (log L) (|⃗f |q )(x)
(3.4)
w(x)dx < ∞.
For the general case, we can check the limit (see the proof of [11], Theorem 1.1, we omit the details here). Thus, (3.1) is proved. Now, we prove (3.2). We may assume ∥bj ∥BMO = 1, for j = 1, . . . , m. By the Lebesgue differentiation theorem, it suffices to prove ∗ n l ⃗ ⃗ sup ϕ(t )ω({y ∈ Rn : |Mδ (TΠ ⃗,q f )(y)| > t }) ≤ C sup ϕ(t )ω({y ∈ R : ML(log L) (|f |q )(y) > t }), b t >0
where ϕ(t ) =
t >0
1
Φ (m) (1/t 1/m )
(3.5)
. Lemma 2.1 yields ♯
∗ n ∗ ⃗ ⃗ sup ϕ(t )ω({y ∈ Rn : |Mδ (TΠ ⃗,q f )(y)| > t }) ≤ C sup ϕ(t )ω({y ∈ R : |Mδ (Tb,q f )(y)| > t }), b t >0
t >0
(3.6)
whenever the left-hand side is finite. Therefore, (3.5) follows from ♯
∗ n l ⃗ ⃗ sup ϕ(t )ω({y ∈ Rn : |Mδ (TΠ ⃗,q f )(y)| > t }) ≤ C sup ϕ(t )ω({y ∈ R : ML(log L) (|f |q )(y) > t }). b t >0
t >0
(3.7)
106
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
We first check that the left-hand side of (3.6) is finite for all bounded functions with compact support. We assume that the bj and ω are bounded. Suppose that supp f ⊂ B(0, R). Hence, since 0 < δ < 1/m, it follows that sup ϕ(t )ω({y ∈ R : n
t >0
|Mδ (Tb∗,q ⃗f )(y)|
> t }) ≤ C sup ϕ(t )ω
Mδ (χ2B(0,R) Tb∗,q ⃗ f )(y)
t
y∈R : > 2 t + C sup ϕ(t )ω y ∈ Rn : Mδ (χRn \2B(0,R) Tb∗,q ⃗f )(y) > t >0
n
2
t >0
. = I + II.
For I, since 0 < δ < 1/m and ϕ(t ) < t 1/m , Hölder’s inequality gives
t n ∗ ⃗ I ≤ Ct y ∈ R : M1/m (χ2B(0,R) Tb,q f )(y) > 2 1/m t 1/m n ∗ ⃗ 1/m ≤ Ct y ∈ R : M (χ2B(0,R) Tb,q f ) (y) > 2 (Tb,q ⃗f (y))1/m dy ≤C 1/m
B(0,2R)
(2m−1)n/2m
≤ CR
Rn
1/2
(Tq f (y)) dy ∗⃗
2
< ∞.
For II, by (3.4) and the boundedness property of Mδ : L1,∞ (Rn ) → L1,∞ (Rn ), for any δ < 1.
m n II ≤ Ct M |fj |qj (y) > Ct y ∈ R : Mδ j =1 m ≤ Ct 1/m y ∈ Rn : M |fj |qj (y) > Ct j =1 m 1/m n 1/m εj−1 ≤C t y ∈ R : M (|fj |qj )(y) > C εj j =1 1/m
≤C
m
t 1/m
j =1
≤C
m
εj ∥ |fj |qj ∥L1 εk−1 1/m < ∞,
∥ |fj |qj ∥L1
j=1
ε
t 1/m ∥ |fj |q ∥ 1
j L where ε0 = t , ε1 , . . . , εm > 0 and εm = 1 such that jε−1 = m 1/m . j j=1 ∥ |fj |qj ∥L1 To prove (3.7) we proceed by induction on l. First, we study the case when l = 1. Let 0 < δ < ε < 1/m, Lemmas 2.2, 2.3 and Fefferman–Stein inequality yields
♯
sup ϕ(t )ω({y ∈ Rn : |Mδ (Tb∗,q ⃗ f )(y)| > t }) ≤ C sup ϕ(t )ω({y ∈ Rn : MLl (log L) (|⃗ f |q )(y) + Mε (Tq∗ ⃗ f )(y) > t }) t >0
t >0
≤ C sup ϕ(t )ω({y ∈ R : n
t >0
MLl (log L)
(|⃗f |q )(y) > t }) + C sup ϕ(t )ω({y ∈ Rn : Mε (Tq∗ ⃗f )(y) > t }) t >0
≤ C sup ϕ(t )ω({y ∈ Rn : MLl (log L) (|⃗f |q )(y) > t }) + C sup ϕ(t )ω({y ∈ Rn : M(|⃗f |q )(y) > t }) t >0
t >0
≤ C sup ϕ(t )ω({y ∈ R : n
t >0
MLl (log L)
(|⃗f |q )(y) > t }).
We now study the case when l ≥ 2. Suppose now that (3.7) holds for l − 1, let us prove it for l. By Lemmas 2.2 and 2.3, ♯
f )(y)| > t }) sup ϕ(t )ω({y ∈ Rn : |Mδ (Tb∗,q ⃗ t >0
l − 1 n l ∗ ⃗ ⃗ ≤ C sup ϕ(t )ω y ∈ R : ML(log L) (|f |q )(y) + Mε (Tb ′ f )(y) > t σ ,q t >0 j=1 σ ∈C l j
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108 l −1
≤ C sup ϕ(t )ω({y ∈ Rn : MLl (log L) (|⃗f |q )(y) > t }) +
107
sup ϕ(t )ω({y ∈ Rn : Mε (Tb∗ ′ ⃗ f )(y) > t }) σ ,q
j=1 σ ∈C l t >0 j
t >0
l −1
≤ C sup ϕ(t )ω({y ∈ Rn : MLl (log L) (|⃗f |q )(y) > t }) +
sup ϕ(t )ω({y ∈ Rn : Mε♯ (Tb∗ ′ ⃗ f )(y) > t }) σ ,q
j=1 σ ∈C l t >0 j
t >0
≤ C sup ϕ(t )ω({y ∈ Rn : MLl (log L) (|⃗f |q )(y) > t }).
(3.8)
t >0
The last inequality holds by Lemma 2.1 with the induction hypothesis on (3.7) to Tbσ ′ ,q ⃗ f . Hence, the claim (3.7) is proved. Thus, we finish the proof of Theorem 3.1. To prove the main theorems, the following lemmas are used. Lemma 3.2. Let 1 < p, q, pj , qj < ∞, j = 1, . . . , m with condition. Then there exists a constant C , such that
1 p
=
1 p1
+ ··· +
1 pm
and
1 q
=
1 q1
+ ··· +
1 , qm
and ω ⃗ satisfy the Ap⃗
∥M(|⃗f |q )∥Lp (νω⃗ ) ≤ C ∥ |fj |qj ∥Lpj (ωj ) . Proof. We exploit some ideas used in Theorem 3.3 [12] to our situation. Since ω ⃗ satisfy the Ap⃗ condition, we have
M(|⃗f |q )(x) ≤ C
m
pj
Mνc ((|fj |qj ωj /νω⃗ )t )(x)1/tpj , ω ⃗
j =1
where t = max1≤j≤m pm+(1−1/ξ )(p −1) , ξ = min1≤j≤m rj and Mνc denotes the weighted centered maximal function. Then ω ⃗ j Lemma 3.2 follows from Hölder’s inequality and the boundedness of the centered maximal operator. Since the main steps and the ideas are almost the same, we omit the proof. pm
Lemma 3.3. (i) Let 1 < p, q, pj , qj < ∞, j = 1, . . . , m with Ap⃗ condition. Then
Rn
1/p p l ≤C ML(log L) (|⃗f |q )(x) νω⃗ (x)dx
Rn
1 p
=
1 p1
+ ··· +
1 pm
and
1 q
=
1 q1
+ ··· +
1 , qm
and ω ⃗ satisfy the
1/pj |fj |q (x)pj ωj (x)dx . j
(ii) Let ω ⃗ ∈ A(1,...,1) and b⃗ ∈ (BMO)m . Then there exists a constant C depending on b⃗ such that
νω⃗ x ∈ R :
n
MLl (log L)
(|⃗f |q )(x) > t
m
≤C
m j =1
Rn
Φ
(m)
|fj |qj (yj ) t
1/m ωj (yj )dyj
,
m
where Φ (t ) = t (1 + log t ) and Φ +
(m)
= Φ ◦ · · · ◦ Φ.
Proof. We can modify the proof of Theorem 4.1 [13] to get (ii) without any difficulty. Since the main ideas are almost the same, we omit the proof here. We just prove (i). It is easy to see
MLl (log L) (|⃗f |q )(x) ≤ ML(log L) (|⃗f |q )(x) ≤ C Mr (|⃗f |q )(x), where Mr (|⃗ f |q )(x) = supQ ∋x
∥Mr (|⃗f |q )∥Lp (νω⃗ ) ≤ C
m
m
j =1
1 |Q |
Q
|fj |rqj
1r
(3.9)
, with r > 1. To prove Lemma 3.3, it suffice to prove
∥ |fj |qj ∥Lpj (ωj ) .
j =1
This is equivalent to
∥M(|⃗f |q )∥Lp/r (νω⃗ ) ≤ C
m j =1
∥ |fj |qj ∥Lpj /r (ω ) . j
This follows from Lemma 3.2, since ω ⃗ ∈ Ap⃗/r for some r > 1 (see Lemma 6.1 in [12] for details).
108
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
Proof of Theorems 1.1–1.2. Theorem 1.1 is a consequence of (3.1) and Lemma 3.3. We now prove Theorem 1.2. By homogeneity we may assume t = 1. Since Φ is submultiplicative, Theorem 3.1 and Lemma 3.3 yields
m ≤ C sup νω⃗ x ∈ Rn : TΠ∗ b⃗,q ⃗f (x) > 1 t >0
≤ C sup t >0
≤ C sup t >0
≤ C sup t >0
≤C
1
νω⃗ Φ (m) (1/t )m 1
Φ (m) (1/t )m
m
Φ (m) (1/t )m Φ (m) (1/t )m
Rn
Φ
Rn
j =1 m
1
j =1
m ∗ m ⃗ x ∈ R n : TΠ ⃗,q f (x) > t b
m νω⃗ x ∈ Rn : MLl (log L) ⃗f (x) > t m
1
m
(m)
Rn
j =1
Φ
(m)
Φ
(m)
|fj |qj (yj ) t
|fj |qj (yj )
t
(|fj |qj (yj ))Φ
ωj (yj )dyj
(m)
1 t
ωj (yj )dyj
1/m ωj (yj )dyj
.
(3.10)
This complete the proof of Theorem 1.2. Acknowledgments The authors want to express their sincere thanks to the referee for his or her valuable remarks and suggestions. The first author was supported by NSFC (Nos. 11226102, and 11226103) and Doctor Foundation of Henan Polytechnic University (No. B2012-055). The second author was supported partly by NSFC (Key program Grant No. 10931001), Program for New Century Excellent Talents in University, Program for Changjiang Scholars and Innovative Research Team in University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
R.R. Coifman, Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975) 315–331. R.R. Coifman, Y. Meyer, Commutateurs d’intégrales singulières et opérateurs multilinéaires, Ann. Inst. Fourier (Grenoble) 28 (1978) 177–202. R.R. Coifman, Y. Meyer, Au-delà des opérateurs pseudo-différentiels, Asterisque 57 (1978) 1–185. L. Grafakos, R. Torres, Multilinear Calderón–Zygmund theory, Adv. Math. 165 (2002) 124–164. L. Grafakos, R.H. Torres, Maximal operator and weighted norm inequalities for multilinear singular integrals, Indiana Univ. Math. J. 51 (5) (2002) 1261–1276. X. Chen, Weighted estimates for maximal operator of multilinear singular intergral, Bull. Pol. Acad. Sci. Math. 58 (2) (2010) 129–135. Q. Xue, Weighted estimates for the iterated commutators of multilinear maximal and fractonal type operators, Studia Math. 217 (2) (2013) 97–122. L. Grafakos, J.M. Martell, Extrapolation of weighted norm inequalities for multivariable operators and applications, J. Geom. Anal. 14 (1) (2004) 19–46. D. Cruz-Uribe, J.M. Martell, C. Pérez, Extrapolation from A∞ weights and applications, J. Funct. Anal. 213 (2004) 412–439. C. Fefferman, E. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971) 107–115. C. Pérez, R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators, J. Lond. Math. Soc. 65 (3) (2002) 672–692. A.K. Lerner, S. Ombrosi, C. Pérez, R.H. Torres, R. Trujillo-González, New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory, Adv. Math. 220 (4) (2009) 1222–1264. C. Pérez, G. Pradolini, R.H. Torres, R. Trujillo-González, End-point estimates for iterated commutators of multilinear singular integrals, Bull. London Math. Soc. (2013) http://dx.doi.org/10.1112/blms/bdt065, to appear.
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Weighted estimates for commutators of vector-valued maximal multilinear operators Zengyan Si a , Qingying Xue b,∗ a
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, People’s Republic of China
b
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
article
abstract
info
Let T ∗ be the maximal multilinear Calderón–Zygmund operator defined by T ∗ (⃗ f )(x) = supδ>0 |Tδ (f1 , . . . , fm )(x)|, and Tq∗ (⃗ f ) be the vector-valued version of T ∗ , Tq∗ (⃗ f )(x) =
Article history: Received 8 March 2013 Accepted 4 November 2013 Communicated by Enzo Mitidieri
∞ k=1
|T ∗ (f1k , . . . , fmk )(x)|q
1/q
, where Tδ are the smooth truncations of the multilinear
singular integral operator T . In this paper, we consider weighted norm inequalities for the iterated commutators of vector-valued maximal multilinear operators Tq∗ (⃗ f ). The weighted strong type and weighted end-point weak type estimates for the iterated commutators of Tq∗ (⃗ f ) were established respectively. © 2013 Elsevier Ltd. All rights reserved.
Keywords: Maximal multilinear operators Multilinear Calderón–Zygmund operators Commutators Vector-valued inequalities
1. Introduction Multilinear Calderón–Zygmund operators were introduced and first studied by Coifman and Meyer [1–3], and later on by Grafakos and Torres [4,5]. In recent years, the theory on multilinear Calderón–Zygmund operators and multilinear commutators have attracted much attention as a rapid developing field in harmonic analysis. Let T be a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values into the space of tempered distributions, T : S (Rn ) × · · · × S (Rn ) → S ′ (Rn ). Following [4], we say that T is a multilinear Calderón–Zygmund operator if, for some 1 ≤ qj < ∞, it extends to a bounded multilinear operator from Lq1 × · · · × Lqm to Lq , where 1q = q1 + · · · + q1 , and if there exists a function K , defined off the
diagonal x = y1 = · · · = ym in (Rn )m+1 , satisfying T⃗ f (x) = T (f1 , . . . , fm )(x) = for all x ̸∈
m
j =1
m
K (x, y1 , . . . , ym )f1 (y1 ) · · · fm (ym )dy1 · · · dym ,
(1.1)
supp fj ,
K (y0 , y1 , . . . , ym ) ≤
A m
k,l=0
∗
(Rn )m
1
mn ;
|yk − yl |
Corresponding author. Tel.: +86 10 58807735. E-mail addresses: [email protected] (Z. Si), [email protected] (Q. Xue).
0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.11.003
(1.2)
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
97
for some A > 0 and all (y0 , y1 , . . . , ym ) with y0 ̸= yj for some j. And
|K (y0 , y1 , . . . , yj , . . . , ym ) − K (y0 , y1 , . . . , y′j , . . . , ym )| ≤
A|yj − y′j |ε
mn+ε
m k,l=0
(1.3)
|yk − yl |
for some ε > 0 and all 0 ≤ j ≤ m, whenever |yj − y′j | ≤ 12 max0≤k≤m |yj − yk |. Such kernels are called multilinear Calderón–Zygmund kernels and the collection of such functions is denoted by m-CZK(A, ε) in [4]. The maximal multilinear singular integral operator is defined by T ∗ (⃗ f )(x) = sup |Tδ (f1 , . . . , fm )(x)|, δ>0
where Tδ are the smooth truncations of T given by
Tδ (⃗ f )(x) = m
2 2 i=1 |x−yi | >δ
⃗. K (x, y1 , . . . , ym )f1 (y1 ) · · · fm (ym )dy
⃗ = dy1 · · · dym . As is pointed in [5], T ∗ (⃗f )(x) is pointwise well-defined when fj ∈ Lqj (Rn ) with 1 ≤ qj ≤ ∞. Here, dy The commutators associated with T and T ∗ are defined by TΠ b⃗ (⃗ f )(x) = ∗ ⃗ TΠ ⃗ (f )(x) b
l
(Rn )m j=1
[bj (x) − bj (yj )]K (x, y1 , . . . , ym )f1 (y1 ) · · · fm (ym )dy⃗.
l m ⃗. = sup [bj (x) − bj (yj )]K (x, y1 , . . . , ym ) fj (yj )dy m 2 2 δ>0 j =1 i=1 |x−yi | >δ j=1
The theory of weighted maximal multilinear Calderón–Zygmund type operators was established in [5,6]. Recently, the weighted strong type and end-point estimates for T ∗ ⃗ with multiple weights (see Section 2, Definition 2.1 for the definition) Πb were obtained in [7] as follows: Theorem A ([7]). Let ω ⃗ ∈ Ap⃗ , 1/p = 1/p1 + · · · + 1/pm with 1 < pj < ∞, j = 1, . . . , m and b⃗ = (b1 , . . . , bm ) ∈ (BMO)m .
⃗ and ⃗f such that Then there is a constant C > 0 independent of b m
∥TΠ∗ b⃗ (⃗f )∥Lp (νω⃗ ) ≤ C
∥bj ∥BMO
j =1
m
∥fj ∥Lp (ωj ) .
j =1
Theorem B ([7]). Let ω ⃗ ∈ A(1,...,1) and b⃗ ∈ (BMO)m . Then there exists a constant C depending on b⃗ such that
νω⃗
n
x∈R :
∗ ⃗ TΠ ⃗ (f )(x) b
>t
m
≤C
m
Rn
j =1
Φ
(m)
|fj (yj )| t
1/m ωj (yj )dyj
,
m
where Φ (t ) = t (1 + log t ) and Φ +
(m)
= Φ ◦ · · · ◦ Φ.
The vector-valued multilinear Calderón–Zygmund operator Tq and vector-valued maximal multilinear operator Tq∗ associated with the operator T are defined and studied by Grafakos and Martell in [8]. Tq (⃗ f )(x) = |T (f1 , . . . , fm )(x)|q = ∥T (f1· , . . . , fm· )(x)∥lq
=
∞
1/q
|T (f1k , . . . , fmk )(x)|q
,
k=1
Tq∗ (⃗ f )(x) = |T ∗ (f1 , . . . , fm )(x)|q = ∥T ∗ (f1· , . . . , fm· )(x)∥lq
=
∞
1/q
|T (f1k , . . . , fmk )(x)| ∗
q
,
k=1
where fi = {fik }∞ k=1 for i = 1, . . . , m. Grafakos and Martell [8] obtain the following results.
98
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
Theorem C ([8]). Let T be a multilinear Calderón–Zygmund operators, and let 1/m < p < ∞, 1/p = 1/p1 + · · · + 1/pm with 1 < p1 , . . . , pm < ∞, 1/m < q < ∞ and 1/q = 1/q1 + · · · + 1/qm with 1 < q1 , . . . , qm < ∞. Then there exists a constant C > 0 such that m
∥Tq∗ (⃗f )∥Lp (Rn ) ≤ C
∥ |fj |qj ∥Lpj (Rn ) .
j =1
Cruz-Uribe, Martell and Pérez [9] obtain a weak version of Theorem C as follows: Theorem D ([9]). Let T be a multilinear Calderón–Zygmund operators, and let 1/m ≤ p < ∞, 1/p = 1/p1 + · · · + 1/pm with 1 ≤ p1 , . . . , pm < ∞, 1/m < q < ∞ and 1/q = 1/q1 + · · · + 1/qm with 1 < q1 , . . . , qm < ∞. Then there exists a constant C > 0 such that
∥Tq∗ (⃗f )∥Lp,∞ (Rn ) ≤ C
m
∥ |fj |qj ∥Lpj (Rn ) .
j=1
In this paper, we consider the vector-valued version of Theorems A and B. We will sometime use the notation ⃗ f = ⃗ = (y1 , . . . , ym ), dy⃗ = dy1 · · · dym . For the sequence (f1 , . . . , fm ), with fj = {fjk }∞ k=1 , and y ∞ {⃗fk }∞ k=1 = {f1k , . . . , fmk }k=1
of vector functions, the commutators associated with vector-valued Tq and Tq∗ can be defined by
TΠ b⃗,q (⃗ f )(x) = |TΠ b⃗ (⃗ f )(x)|q = ∥TΠ b⃗ (f1· , . . . , fm· )(x)∥lq =
∞
1/q |TΠ b⃗ (⃗fk )(x)|q
,
k=1
∗ ⃗ TΠ ⃗,q (f )(x) b
=
|TΠ∗ b⃗ (⃗f )(x)|q
=
∥TΠ∗ b⃗ (f1· , . . . , fm· )(x)∥lq
=
∞ k=1
1/q |TΠ∗ b⃗ (⃗fk )(x)|q
.
Our main results are as follows: Theorem 1.1. Let T be an m-linear Calderón–Zygmund operators and let 1/m < p < ∞,
+ · · · + p1m , with p m p ⃗ ∈ Ap⃗ , νω⃗ = i=1 ωi i . 1 < p1 , . . . , pm < ∞, 1/m < q < ∞ and q1 +· · ·+ q1 = 1q with 1 < q1 , . . . , qm < ∞. Suppose that ω m 1 bj ∈ BMO, for all j = 1, . . . , l. There then exists a constant C > 0 such that ∥TΠ∗ b⃗,q (⃗f )∥Lp (νω⃗ ) ≤
l
∥bj ∥BMO
m
j =1
1 p
=
1 p1
∥ |fj |qj ∥Lpj (wj ) .
j =1
Theorem 1.2. Let ω ⃗ ∈ A(1,...,1) and b⃗ ∈ (BMO)m . Then there exists a constant C depending on b⃗ such that
νω⃗ x ∈ R :
n
∗ ⃗ TΠ ⃗,q (f )(x) b
>t
m
≤C
m j =1
Rn
Φ
(m)
|fj |qj (yj )
t
1/m ωj (yj )dyj
,
m
where Φ (t ) = t (1 + log t ) and Φ +
(m)
= Φ ◦ · · · ◦ Φ.
2. Notations and main lemmas Throughout this paper, M always denotes the Hardy–Littlewood maximal operator. For δ > 0, Mδ is the maximal function defined by
Mδ f (x) =
1 M (|f |δ ) δ (x)
= sup Q ∋x
1
|Q |
1δ
δ
|f (y)| dy
.
Q
In addition, M ♯ is the sharp maximal function of Fefferman and Stein, 1
♯
M f (x) = sup inf Q ∋x
♯
c
|Q | 1
and Mδ f (x) = M ♯ (|f |δ ) δ (x).
|f (y) − c |dy ≈ sup Q
Q ∋x
1
|Q |
|f (y) − fQ |dy Q
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
99
We will also use the new vector valued maximal function as follows:
m 1 |fj |qj , Q ∋x j=1 |Q | Q
M (|⃗f |q )(x) = sup
where the supremum is taken over all the cubes containing x. Definition 2.1. Let 1 ≤ p1 , . . . , pm < ∞. Given ω ⃗ = (ω1 , . . . , ωm ), set νω⃗ = condition if
sup Q
m
1
|Q |
when pi = 1,
p pi
1p
ωi
Q i=1
1 |Q |
Q
m
i=1 1−p′i
ωi
1′ p i
1
|Q |
1−p′i
ωi
p p
i ⃗ satisfies the Ap⃗ i=1 ωi . We say that ω
m
1′ p i
< ∞,
Q
is understood as (infQ ωi )−1 .
To prove our results, we need the following lemmas. Lemma 2.1 ([10]). Let ω be an A∞ weight. Then there exist constant C and ρ > 0 depending upon the A∞ condition of ω such that, for all λ, ε > 0,
ω({y ∈ Rn : Mf (y) > λ, M ♯ f (y) ≤ λϵ}) ≤ C ε ρ ω
y ∈ Rn : Mf (y) >
1 2
λ
.
As a consequence, we have the following estimates for δ > 0. (1) Let ϕ : (0, ∞) → (0, ∞) be a doubling, that is, ϕ(2a) ≤ C ϕ(a) for a > 0. Then, there exists a constant C depending upon the A∞ condition of ω and doubling condition of ϕ such that ♯
sup ϕ(λ)ω({y ∈ Rn : Mδ f (y) > λ}) ≤ C sup ϕ(λ)ω({y ∈ Rn : Mδ f (y) > λ}) λ>0
(2.1)
λ>0
for every function such that the left-hand side is finite. (2) Let 0 < p < ∞. There exists a positive constant C depending upon the A∞ condition and p such that
Rn
(Mδ f (x))p ω(x)dx ≤ C
Rn
♯
(Mδ f (x))p ω(x)dx
(2.2)
for every function such that the left-hand side is finite. Lemma 2.2. Let T be a multilinear Calderón–Zygmund operator. Let 0 < δ < 1/m, 1/m < q < ∞ and 1/q = 1/q1 + · · · + 1/qm with 1 < q1 , . . . , qm < ∞. Then there exists a constant C > 0 such that ♯
f ))(x) ≤ C M (|⃗ f |q )(x) Mδ (Tq∗ (⃗ n for any smooth vector function {⃗ fk }∞ k=1 and any x ∈ R .
Proof. For simplicity, we only prove for the case m = 2, since there is no essential difference for the general case. Similarly as in [7], we choose two functions u, v ∈ C ∞ ([0, ∞)) such that |u′ (t )| ≤ Ct −1 , |v ′ (t )| ≤ Ct −1 and satisfy
χ[2,∞) (t ) ≤ u(t ) ≤ χ[1,∞) (t ),
χ[1,2] (t ) ≤ v(t ) ≤ χ[1/2,3] (t ).
We introduce two maximal operators by
m ⃗ , U (⃗f )(x) = sup K (x, y1 , . . . , ym )u( |x − y1 | + · · · + |x − ym |/η) fi (yi )dy η>0 (Rn )m i=1 m ⃗ . V ∗ (⃗f )(x) = sup K (x, y1 , . . . , ym )v( |x − y1 | + · · · + |x − ym |/η) fi (yi )dy η>0 (Rn )m i=1 m Let Ku,η (x, y1 , . . . , ym ) = K (x, y1 , . . . , ym )u( Kv,η (x, y1 , . . . , ym ) i=1 |x − yi |/η), m ( i=1 |x − yi |/η) and m ⃗ ⃗; Uη (f )(x) = Ku,η (x, y1 , . . . , ym ) fi (yi )dy ∗
(Rn )m
Vη (⃗f )(x) =
(Rn )m
i =1
Kv,η (x, y1 , . . . , ym )
m i=1
⃗. fi (yi )dy
=
K (x, y1 , . . . , ym )v
100
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
Obviously, T ∗ (⃗ f )(x) ≤ U∗ (⃗ f )(x) + V ∗ (⃗ f )(x). To prove Lemma 2.2, it suffices to show ♯
♯
Mδ (U∗q (⃗ f ))(x) ≤ C M (|⃗ f |q )(x) and where Uq (⃗ f )(x) = ∗
q ⃗ k=1 |U (fk )(x)|
∞
∗
Mδ (Vq∗ (⃗ f ))(x) ≤ C M (|⃗ f |q )(x),
1/q
and Vq (⃗ f )(x) = ∗
∗
∞
q ⃗ k=1 |V (fk )(x)|
∗
∗
1/q .
∗
We just give the proof for Uq , since the proof for Vq is almost the same as Uq . Fix x ∈ Rn and let Q be a cube of side length r centered at x. For any smooth vector function sequence {⃗ fk }∞ k=1 , set
⃗fk∞ = ⃗fk − ⃗fk0 , where ⃗fk0 = ⃗fk χ2Q = (f1k χ2Q , . . . , fmk χ2Q ). Since 0 < δ < 1/2 < 1, we have
1δ 1δ ∗ 1 δ δ ∗ δ |U (⃗f )(y) | −|c | dy ≤C |U (⃗f )(y) − c | dy q |Q | Q |Q | Q q δ 1δ 1 sup |Uη (f⃗0 )(y) + Uη (⃗f χ(2Q )c )(y) − cη | dy ≤C |Q | η 1
Q
≤C
1
q
1δ
∗ U (f⃗0 )(y)δ dy
+
q
|Q |
Q
δ 1δ sup |Uη (⃗f χ(2Q )c )(y) − cη | dy |Q | Q η q 1
= U1 + U2 q 1/q where c = supη>0 cη q = . k≥1 supη>0 cηk
(2.3)
For U1 , it is easy to see U∗ satisfy the same bounded property with T ∗ . Then we applying Kolmogorov’s inequality and Theorem D to get
1δ 2 ∗ 0 δ 1 ∗ ⃗0 ≤C U (⃗f )(y) dy ≤ C ∥ U ( f )∥ |fj |qj (z )dz q L1/2,∞ Q , |dx |Q | Q q |Q | Q Q| j =1 1
≤ C M(|⃗f |q )(x). 3 i To estimate U2 , we choose cη = i=1 cη for i = 1, 2, 3, where ∞ ∞ cη1 = Uη (f1k , f2k )(x),
0 cη2 = Uη (f1k , f2k∞ )(x),
(2.4)
∞ 0 cη3 = Uη (f1k , f2k )(x).
We may split U2 as U2 ≤ U21 + U22 + U23 , where
U21 =
1
|Q |
U22 =
1
1δ δ ∞ ∞ ∞ ∞ sup Uη (f , f )(y) − Uη (f , f )(x) dy ;
Q
1k
2k
1k
2k
q
1δ δ 0 ∞ 0 ∞ supUη (f , f )(y) − Uη (f , f )(x) dy ;
|Q |
η
Q
η
1k
2k
1k
2k
q
and
U23 =
1
|Q |
1δ δ ∞ 0 ∞ 0 sup Uη (f , f )(y) − Uη (f , f )(x) dy .
Q
η
1k
2k
1k
2k
q
For the first term U21 , we have
|Uη (f1k∞ , f2k∞ )(y) − Uη (f1k∞ , f2k∞ )(x)| ≤ C ≤C
(Rn \2Q )2 ∞ s =1
≤C
2 |Q |ε/n |f ∞ (yj )|dy⃗ |(x − y1 , x − y2 )|2n+ε j=1 jk
∞ s =1
(2s+1 Q )2 \(2s Q )2
2−εs
2 j =1
(
2s+1
1 2(s+1)n |Q |
2 |Q |ε/n |fjk (yj )|dy⃗ 1 / n 2n +ε |Q | ) j =1
2s+1 Q
|fjk (yj )|dyj .
(2.5)
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
Now let gsjk =
1 2(s+1)n |Q |
|fjk (yj )|dyj and Gsk =
2s+1 Q
|Uη (f1k∞ , f2k∞ )(y) − Uη (f1k∞ , f2k∞ )(x)| ≤ C
∞
2
j =1
101
gsjk . Then we obtain
2−εs Gsk .
s =1
For any two positive real numbers ε and q, let ρ = min{1, q}. It is easy to show that
q ∞ ∞ −ε s 2 µs ≤ C(ε,q) 2−ερ s |µs |q s=1 s=1 for any sequence µs . Applying this inequality to µs = Gsk , we get
|Uη (f1k , f2k )(y) − Uη (f1k , f2k )(x)|q ≤ C ∞
∞
∞
∞
∞
1/q −ερ s
2
s=1
q Gsk
.
k
Next, Minkowski’s inequality gives
1/qj
qj gsjk
≤
k
1 2(s+1)n |Q |
2s+1 Q
|fj |qj (yj )dyj
and then Hölder’s inequality gives
1/q
q Gsk
≤
2 j=1
k
1/qj qj gsjk
≤ M(|⃗f |q )(x).
k
∞ ∞ From which we deduce |Uη (f1k , f2k )(z ) − Uη (f1k∞ , f2k∞ )(x)|q ≤ C M(|⃗f |q )(x). Since 0 < δ < 1/2, we have
1
|Q |
δ
1δ
sup |Uη (f1k , f2k )(y) − Uη (f1k , f2k )(x)|q dy ∞
Q
∞
∞
∞
≤C
η
1
|Q |
∞ ∞ sup |Uη (f1k , f2k )(y) − Uη (f1k∞ , f2k∞ )(x)|q dy Q
η
≤ C M(|⃗f |q )(x).
(2.6)
For U22 , we observe the following fact
|x − z |ε |f2k (y2 )|dy2 2n+ε 2Q (2Q )c (|z − y1 | + |z − y2 |) ∞ |Q |ε/n |f1k (y1 )|dy1 |f2k (y2 )|dy2 ≤C (2s |Q |1/n )2n+ε 2s+1 Q 2s+1 Q s =1
|Uη (f1k0 , f2k∞ )(z ) − Uη (f1k0 , f2k∞ )(x)| ≤ C
≤C
|f1k (y1 )|dy1
∞ s =1
2−εs
2 j =1
1 2(s+1)n |Q |
2s+1 Q
|fjk (yj )|dyj .
(2.7)
The dealing of the other part is almost the same as U21 , we omit the detail. U23 can be estimated in the same way. Hence we proved Lemma 2.2. Before stating another lemma, we first introduce some notations. For 1 ≤ l ≤ m, we define MLl (log L) (|⃗ f |q ) as follows:
MLl (log L) (|⃗f |q )(x) = sup
l
Q ∋x j=1
ML(log L) (|⃗f |q )(x) = sup
m
Q ∋x j=1
∥ |fj |qj ∥L(log L),Q
m 1 |fj |qj , |Q | Q j =l +1
∥ |fj |qj ∥L(log L),Q .
It is easy to see the following inequalities hold.
M(|⃗f |q )(x) ≤ MLl (log L) (|⃗f |q )(x) ≤ ML(log L) (|⃗f |q )(x). For positive integers m and j with 1 ≤ j ≤ m, we denote by Cjm the family of all finite subsets σ = {σ (1), . . . , σ (j)} of {1, . . . , m} of j different elements, where we always take σ (k) < σ (j) if k < j. For any σ ∈ Cjm , we associated the
102
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108 m−j
complementary sequence σ ′ ∈ Cj given by σ ′ = {1, . . . , m} \ σ with the convention C0m = ∅. Given an m-tuple of m functions b and σ ∈ Cj , we also use the notation bσ for the j-tuple obtained from b given by (bσ (1) , . . . , bσ (j) ). m ∗ ⃗ ∗ ⃗ j Similarly to the above definitions for UΠ b (f )(x) and VΠ b (f )(x), if σ ∈ Cj , and bσ = (bσ (1) , . . . , bσ (j) ) in BMO , we define the following commutators:
∗ ⃗ TΠ bσ ,q (f )(x)
=
∞
1/q |TΠ∗ bσ (⃗f )(x)|q
,
k=1
j
∗ ⃗ ⃗) i=1 (bσ (i) (x) − bσ (i) (yσ (i) )) i=1 fi (yi )dy⃗. If σ is replaced by σ ′ , we denote them where TΠ bσ (f )(x) = supη>0 (Rn )m K (x, y ∗ ∗ by TΠ b ′ ,q and TΠ b ′ , similar notation will be used later for the operators U∗ and V ∗ .
σ
m
σ
Lemma 2.3. Let 0 < δ < ε < 1/m. There then exists a constant C > 0 depending only on δ and ε such that ♯
∗ ⃗ Mδ (TΠ ⃗,q f )(x) ≤ C b
l
l ∥bi ∥BMO Mε (TΠ∗ b ∥bj ∥BMO MLl (log L) (|⃗f |q )(x) + Mε (Tq∗ ⃗f )(x) + C
σ ′ ,q
j=1 σ ∈C l i∈σ j
j =1
⃗f )(x)
(2.8)
n ′ for any smooth vector function {⃗ fk }∞ k=1 and for any x ∈ R , where σ = {1, . . . , l} \ σ .
Proof. We apply a similar method as in Lemma 2.2. We first define two maximal operators as follows.
l m ⃗; = sup Ku,η (x, y1 , . . . , ym ) [bj (x) − bj (yj )] fi (yi )dy η>0 (Rn )m j=1 i=1 l m ∗ ⃗ ⃗. Kv,η (x, y1 , . . . , ym ) [bj (x) − bj (yj )] fi (yi )dy VΠ ⃗ (f )(x) = sup b η>0 (Rn )m j =1 i =1 U∗Π b⃗ (⃗f )(x)
It is easy to see T ∗ ⃗ (⃗ f )(x) ≤ U∗ ⃗ (⃗ f )(x) + V ∗ ⃗ (⃗ f )(x). It suffices to prove the conclusion in lemma (2.3) hold for U∗ ⃗ and Πb Πb Πb Π b ,q ∗ V ⃗ respectively. For similarity, we just give the proof for U∗ ⃗ , since the proof for V ∗ ⃗ is almost the same as U∗ ⃗ . Π b,q
Π b,q
Let F (⃗ y) , f1 (y1 ) · · · fm (ym ), for any λ = (λ1 , . . . , λm ) we have
U∗Π b⃗ (⃗f )(x)
Π b ,q
= sup (b1 (x) − b1 (y)) · · · (bl (x) − bl (y))Ku,η (x, y⃗)F (⃗y)dy⃗ η>0 (Rn )m = sup ((b1 (x) − λ1 ) − (b1 (y) − λ1 )) · · · ((bl (x) − λl ) − (bl (y) − λl ))Ku,η (x, y⃗)F (⃗y)dy⃗ η>0 (Rn )m l = sup (−1)l−j (bj (x) − λj ) (bj (yj ) − λj )Ku,η (x, y⃗)F (⃗y)dy⃗ n )m η>0 i=0 ( R ′ l j∈σ σ ∈Ci
j∈σ
= sup(b1 (x) − λ1 ) · · · (bl (x) − λl )Uη (⃗f )(x) + Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f )(x) η>0 l −1 l−j + (−1) (bj (x) − λj ) (bj (yj ) − λj )Ku,η (x, y⃗)F (⃗y)dy⃗. n m (R ) ′ l j∈σ i=1 σ ∈Ci
Noting the fact
Π b,q
j∈σ ′
(2.9)
j∈σ
(bj (yj ) − λj ) =
j∈σ ′
[(bj (yj ) − bj (x)) + (bj (x) − λj )]. Then,
U∗Π b⃗,q ⃗f (x) ≤ |(b1 (x) − λ1 ) · · · (bl (x) − λl )|U∗q (⃗f )(x) + |U∗ ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f )(x)|q
+ Cq,l,j
l −1
|bj (x) − λj |U∗Π b
i=1 σ ∈C l j∈σ i
σ ′ ,q
⃗f (x),
(2.10)
where Cq,l,j is an absolute constant depending only on l, j and q. Now fix x ∈ Rn , for any cube Q of side length r and centered 1 b (z )dz, for j = 1, . . . , l. Since 0 < δ < 1/m < 1, it follows that at x. Let λj = |2Q | 2Q j
1
|Q |
1/δ
∗ |U Q
Π b⃗,q
δ
δ
(f )(z ) | −|c | dz ⃗∞
δ 1/δ ∗ U (⃗f )(z ) − sup |cη | dz ≤C ⃗ Π b |Q | Q η>0 q
1
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
≤C
1
|Q |
Q
(b1 (z ) − λ1 ) · · · (bl (z ) − λl )U∗ (⃗f )(z )δ dz q
l −1
+ Cq,l,j
i=1 σ ∈C l i
+ Cq
1
|Q |
Q
δ |bj (z ) −
j∈σ
103
1/δ
λj |U∗Π b ′ ,q ⃗f (z ) σ
1/δ dz
δ 1/δ ∗ U ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f )(z ) − sup |cη | dz |Q | Q η>0 q 1
, I + II + III
(2.11)
where c = supη>0 cη q . We can choose 1 < p1 , . . . , pl < ∞ with I≤C
l
1 p1
+ ··· +
1 pl
+
1
ε
= 1δ . Since 0 < δ < ε < 1/m. Hölder’s inequality gives
∥bj ∥BMO Mε (U∗q ⃗f )(x).
j =1
Similarly, we have II ≤ C
l −1
∥bj ∥BMO Mε (U∗Π b
i=1 σ ∈C l j∈σ i
σ ′ ,q
⃗f )(x).
For III, let ⃗ fj = ⃗ fj 0 + ⃗ fj∞ , where ⃗ fj 0 = ⃗ fj χ2Q , and we choose cη =
α
α1 ,...,αm
α
|(Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )f1 1 · · · fmαm ))(x)|.
We use the notation ⃗ f α to denote f1 1 · · · fmαm . Obviously, we have
∗ U ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f )(z ) − sup |cη | η>0
q
≤ U∗q ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f 0 )(z ) + Cqm
sup |(Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗ f α ))(z )
α1 ,...,αm η>0
− (Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f α ))(x)|q ,
(2.12)
where in the last sum each αj = 0 or ∞ and in each term there is at least one αj = ∞. For the first term, we applying Kolmogorov’s estimate and Theorem D to get
1/δ
1
|Q |
|Uq ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f 0 )(z )|δ dz ∗
Q
≤ C ∥U∗q ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗f 0 )∥L1/m,∞ Q , dz |Q |
l m 1 1 |bj (yj ) − λj | |fj (z )|qj dz |fj (z )|qj dz |Q | Q |Q | Q j=1 j=l+1 l m 1 ≤C ∥bj ∥BMO ∥ |fj |qj ∥L(log L),Q |fj (z )|qj dz |Q | Q j=1 j =l +1 ≤C
≤C
l
∥bj ∥BMO MLl (log L) (|⃗f |q )(x).
(2.13)
j=1
If all the αj = ∞, we have
|Q | ≤
1/δ
1
⃗α
⃗α
δ
sup |(Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )f ))(z ) − (Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )f ))(x)|q dz
Q η>0
1
|Q |
sup |(Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗ f α ))(z ) − (Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )⃗ f α ))(x)|q dz
Q η>0
≤ (Rn \Q ∗ )m
|x − z |ε |(b1 (y1 ) − λ1 ) · · · (bl (yl ) − λl )| |f1 (yj )|qj · · · |fm (ym )|qm dy⃗ (|z − y1 | + · · · + |z − ym |)mn+ε
104
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
≤
l ∞ 1 j=1 k=1
≤
2kε 2kn |Q ∗ |
2k+1 Q ∗
2
l
2
k∥bj ∥BMO ∥ |fj |qj ∥L(log L),2k+1 Q kε m
∥bj ∥BMO ∥ |fj |qj ∥L(log L),2k+1 Q
2kn |Q ∗ |
m j=l+1
2k+1 Q ∗
|fj |qj dyj .
1
j=l+1
2kn |Q ∗ |
2k+1 Q ∗
|fj |qj dyj
1 2kn |Q ∗ |
2k+1 Q ∗
|fj |qj dyj
1
j=l+1
l
m
1
j=l+1
∥bj − λj ∥BMO ∥ |fj |qj ∥L(log L),2k+1 Q kε
j =1
≤C
m
|bj (yj ) − λj | |fj |qj dyj
l ∞ 1 j=1 k=1
≤
1
l ∞ 1 j=1 k=1
≤
2kn |Q ∗ |
2k+1 Q ∗
|fj |qj dyj
∥bj ∥BMO MLl (log L) (|⃗f |q )(x).
(2.14)
j =1
Now we estimate the typical representative of III. By Minkowski’s inequality, we have
δ 1/δ ∗ U ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )f ∞ , . . . , f ∞ , f 0 , . . . , f 0 )(z ) − sup |cη | dz 1 i i +1 m |Q | Q η>0 q C ∞ ∞ 0 0 sup |Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )f1 , . . . , fi , fi+1 , . . . , fm )(z ) ≤ |Q | Q η>0 1
0 0 − Uη ((b1 (·1 ) − λ1 ) · · · (bl (·l ) − λl )f1∞ , . . . , fi∞ , fi+ 1 , . . . , fm )(x)|q dz m l |x − z |ε |(b1 (y1 ) − λ1 ) · · · (bl (yl ) − λl )| |f1 (y1 )|q1 · · · |fl (yl )|ql dy1 · · · dyl |fj (yj )|qj dyj ≤ ∗ n ∗ l (|z − y1 | + · · · + |z − ym |)mn+ε j =l +1 Q j = 1 ( R \Q ) m l |bj (yj ) − λj | |fj (yj )|qj 1 |fj (yj )|qj dyj (|Q |(m−l)/l )l dyj ≤ n ∗ |Q | Q ∗ |z − yj |n+n(m−l)/l j=l+1 j=1 (R )\Q l m 1 ≤ ∥bj ∥BMO ∥ |fj |qj ∥L(log L),2k+1 Q |fj (yj )|qj dyj |Q | Q ∗ j =1 j=l+1
≤C
l
∥bj ∥BMO MLl (log L) (|⃗f |q )(x).
(2.15)
j =1
In other cases, we can also deduce the same estimates with little modifications on the above argument. Then we proved Lemma 2.3. 3. Proof of the main results We start with the strong and weak estimates for T ∗ ⃗ . Π b ,q
Theorem 3.1. Let 0 < p < ∞, 1/m < q < ∞, and q1 + · · · + q1 = 1q with 1 < q1 , . . . , qm < ∞. Suppose that bj ∈ BMO, m 1 for j = 1, . . . , l and w ∈ A∞ , then there exists a constant C > 0, such that
|TΠ∗ b⃗,q ⃗f |p w(x)dx
Rn
≤C
l
p bj BMO
j =1
p
MLl (log L) (|⃗f |q )(x) w(x)dx,
∥ ∥
Rn
(3.1)
and sup t >0
1
Φ (m) (1/t )
ω y ∈ Rn : |TΠ∗ b⃗,q ⃗f (y)| > t m ≤ C sup t >0
for any smooth function ⃗ f with compact support. Proof. We prove (3.1) first. Assuming that
Rn
p
MLl (log L) (|⃗f |q )(x)
w(x)dx < ∞,
1
Φ (m) (1/t )
ω y ∈ Rn : MLl (log L) (|⃗f |q )(y) > t m ,
(3.2)
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
105
since otherwise there is nothing to be proved. For l = 1, by using Lemmas 2.1 and 2.3, we obtain ♯
∥TΠ∗ b⃗,q ⃗f ∥Lp (ω) ≤ ∥Mδ TΠ∗ b⃗,q ⃗f ∥Lp (ω) ≤ C ∥Mδ TΠ∗ b⃗,q ⃗f ∥Lp (ω) ≤ C ∥b1 ∥BMO [∥Mε (Tq∗ ⃗f )∥Lp (ω) + ∥MLl (log L) (|⃗f |q )∥Lp (ω) ] ≤ C ∥b1 ∥BMO [∥Tq∗ ⃗f ∥Lp (ω) + ∥MLl (log L) (|⃗f |q )∥Lp (ω) ] ≤ C ∥b1 ∥BMO [∥M(|⃗f |q )∥Lp (ω) + ∥MLl (log L) (|⃗f |q )∥Lp (ω) ] ≤ C ∥b1 ∥BMO ∥MLl (log L) (|⃗f |q )∥Lp (ω) .
(3.3)
The penultimate inequality holds by Lemmas 2.1 and 2.2. For the case when l ≥ 2, similarly to the case for l = 1, we have
∥TΠ∗ b⃗,q ⃗f ∥Lp (ω) ≤ C
l
∥bj ∥BMO ∥MLl (log L) (|⃗f |q )∥Lp (ω) .
j =1
f ∥Lp (ω) < ∞ and ∥Tq∗ ⃗ f ∥Lp (ω) < ∞. We will only To apply the inequality (2.2) in (3.3), one needs to verify now that ∥T ∗ ⃗ ⃗ Π b ,q
show the first one since the proof of another one is very similar but easier.
Suppose that the symbols bj and the weight ω are bounded functions. Since ⃗ f has compact support, we may assume supp fj ⊂ B(0, R). Then we have
∥TΠ∗ b⃗,q ⃗f ∥Lp (ω) =
|x|≤2R
|TΠ∗ b⃗,q ⃗f (x)|p ω(x)dx +
|x|>2R
|TΠ∗ b⃗,q ⃗f (x)|p ω(x)dx
= I1 + I2 . For I1 , we choose s > p, s1 , . . . , sm > 1 such that 1/s = 1/s1 + · · · 1/sm . Theorem C and Hölder’s inequality implies
I1 ≤ C |x|≤2R
≤C
m
|TΠ∗ b⃗,q ⃗f |p dx ≤ C
Rn
|Tq∗ ⃗f |s dx
p/s
Rn(s−p)/s
p Rn(s−p)/s < ∞.
∥ |fj |qj ∥Lsj
j =1
For I2 , since |x| > 2R, we have
|TΠ∗ b⃗,q ⃗f (x)|
≤ C ≤C
Thus, I2 ≤ C
Rn
(B(0,R))m
|b1 (x) − b1 (y1 )| · · · |bl (x) − bl (yl )| |f1 (y1 )| · · · |fm (ym )|dy |x − y|mn q
m 1 |fj (yj )|qj dy ≤ C M(|⃗f |q )(x) ≤ C MLl (log L) (|⃗f |q )(x). |x|n B(0,|x|) j=1 p
MLl (log L) (|⃗f |q )(x)
(3.4)
w(x)dx < ∞.
For the general case, we can check the limit (see the proof of [11], Theorem 1.1, we omit the details here). Thus, (3.1) is proved. Now, we prove (3.2). We may assume ∥bj ∥BMO = 1, for j = 1, . . . , m. By the Lebesgue differentiation theorem, it suffices to prove ∗ n l ⃗ ⃗ sup ϕ(t )ω({y ∈ Rn : |Mδ (TΠ ⃗,q f )(y)| > t }) ≤ C sup ϕ(t )ω({y ∈ R : ML(log L) (|f |q )(y) > t }), b t >0
where ϕ(t ) =
t >0
1
Φ (m) (1/t 1/m )
(3.5)
. Lemma 2.1 yields ♯
∗ n ∗ ⃗ ⃗ sup ϕ(t )ω({y ∈ Rn : |Mδ (TΠ ⃗,q f )(y)| > t }) ≤ C sup ϕ(t )ω({y ∈ R : |Mδ (Tb,q f )(y)| > t }), b t >0
t >0
(3.6)
whenever the left-hand side is finite. Therefore, (3.5) follows from ♯
∗ n l ⃗ ⃗ sup ϕ(t )ω({y ∈ Rn : |Mδ (TΠ ⃗,q f )(y)| > t }) ≤ C sup ϕ(t )ω({y ∈ R : ML(log L) (|f |q )(y) > t }). b t >0
t >0
(3.7)
106
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
We first check that the left-hand side of (3.6) is finite for all bounded functions with compact support. We assume that the bj and ω are bounded. Suppose that supp f ⊂ B(0, R). Hence, since 0 < δ < 1/m, it follows that sup ϕ(t )ω({y ∈ R : n
t >0
|Mδ (Tb∗,q ⃗f )(y)|
> t }) ≤ C sup ϕ(t )ω
Mδ (χ2B(0,R) Tb∗,q ⃗ f )(y)
t
y∈R : > 2 t + C sup ϕ(t )ω y ∈ Rn : Mδ (χRn \2B(0,R) Tb∗,q ⃗f )(y) > t >0
n
2
t >0
. = I + II.
For I, since 0 < δ < 1/m and ϕ(t ) < t 1/m , Hölder’s inequality gives
t n ∗ ⃗ I ≤ Ct y ∈ R : M1/m (χ2B(0,R) Tb,q f )(y) > 2 1/m t 1/m n ∗ ⃗ 1/m ≤ Ct y ∈ R : M (χ2B(0,R) Tb,q f ) (y) > 2 (Tb,q ⃗f (y))1/m dy ≤C 1/m
B(0,2R)
(2m−1)n/2m
≤ CR
Rn
1/2
(Tq f (y)) dy ∗⃗
2
< ∞.
For II, by (3.4) and the boundedness property of Mδ : L1,∞ (Rn ) → L1,∞ (Rn ), for any δ < 1.
m n II ≤ Ct M |fj |qj (y) > Ct y ∈ R : Mδ j =1 m ≤ Ct 1/m y ∈ Rn : M |fj |qj (y) > Ct j =1 m 1/m n 1/m εj−1 ≤C t y ∈ R : M (|fj |qj )(y) > C εj j =1 1/m
≤C
m
t 1/m
j =1
≤C
m
εj ∥ |fj |qj ∥L1 εk−1 1/m < ∞,
∥ |fj |qj ∥L1
j=1
ε
t 1/m ∥ |fj |q ∥ 1
j L where ε0 = t , ε1 , . . . , εm > 0 and εm = 1 such that jε−1 = m 1/m . j j=1 ∥ |fj |qj ∥L1 To prove (3.7) we proceed by induction on l. First, we study the case when l = 1. Let 0 < δ < ε < 1/m, Lemmas 2.2, 2.3 and Fefferman–Stein inequality yields
♯
sup ϕ(t )ω({y ∈ Rn : |Mδ (Tb∗,q ⃗ f )(y)| > t }) ≤ C sup ϕ(t )ω({y ∈ Rn : MLl (log L) (|⃗ f |q )(y) + Mε (Tq∗ ⃗ f )(y) > t }) t >0
t >0
≤ C sup ϕ(t )ω({y ∈ R : n
t >0
MLl (log L)
(|⃗f |q )(y) > t }) + C sup ϕ(t )ω({y ∈ Rn : Mε (Tq∗ ⃗f )(y) > t }) t >0
≤ C sup ϕ(t )ω({y ∈ Rn : MLl (log L) (|⃗f |q )(y) > t }) + C sup ϕ(t )ω({y ∈ Rn : M(|⃗f |q )(y) > t }) t >0
t >0
≤ C sup ϕ(t )ω({y ∈ R : n
t >0
MLl (log L)
(|⃗f |q )(y) > t }).
We now study the case when l ≥ 2. Suppose now that (3.7) holds for l − 1, let us prove it for l. By Lemmas 2.2 and 2.3, ♯
f )(y)| > t }) sup ϕ(t )ω({y ∈ Rn : |Mδ (Tb∗,q ⃗ t >0
l − 1 n l ∗ ⃗ ⃗ ≤ C sup ϕ(t )ω y ∈ R : ML(log L) (|f |q )(y) + Mε (Tb ′ f )(y) > t σ ,q t >0 j=1 σ ∈C l j
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108 l −1
≤ C sup ϕ(t )ω({y ∈ Rn : MLl (log L) (|⃗f |q )(y) > t }) +
107
sup ϕ(t )ω({y ∈ Rn : Mε (Tb∗ ′ ⃗ f )(y) > t }) σ ,q
j=1 σ ∈C l t >0 j
t >0
l −1
≤ C sup ϕ(t )ω({y ∈ Rn : MLl (log L) (|⃗f |q )(y) > t }) +
sup ϕ(t )ω({y ∈ Rn : Mε♯ (Tb∗ ′ ⃗ f )(y) > t }) σ ,q
j=1 σ ∈C l t >0 j
t >0
≤ C sup ϕ(t )ω({y ∈ Rn : MLl (log L) (|⃗f |q )(y) > t }).
(3.8)
t >0
The last inequality holds by Lemma 2.1 with the induction hypothesis on (3.7) to Tbσ ′ ,q ⃗ f . Hence, the claim (3.7) is proved. Thus, we finish the proof of Theorem 3.1. To prove the main theorems, the following lemmas are used. Lemma 3.2. Let 1 < p, q, pj , qj < ∞, j = 1, . . . , m with condition. Then there exists a constant C , such that
1 p
=
1 p1
+ ··· +
1 pm
and
1 q
=
1 q1
+ ··· +
1 , qm
and ω ⃗ satisfy the Ap⃗
∥M(|⃗f |q )∥Lp (νω⃗ ) ≤ C ∥ |fj |qj ∥Lpj (ωj ) . Proof. We exploit some ideas used in Theorem 3.3 [12] to our situation. Since ω ⃗ satisfy the Ap⃗ condition, we have
M(|⃗f |q )(x) ≤ C
m
pj
Mνc ((|fj |qj ωj /νω⃗ )t )(x)1/tpj , ω ⃗
j =1
where t = max1≤j≤m pm+(1−1/ξ )(p −1) , ξ = min1≤j≤m rj and Mνc denotes the weighted centered maximal function. Then ω ⃗ j Lemma 3.2 follows from Hölder’s inequality and the boundedness of the centered maximal operator. Since the main steps and the ideas are almost the same, we omit the proof. pm
Lemma 3.3. (i) Let 1 < p, q, pj , qj < ∞, j = 1, . . . , m with Ap⃗ condition. Then
Rn
1/p p l ≤C ML(log L) (|⃗f |q )(x) νω⃗ (x)dx
Rn
1 p
=
1 p1
+ ··· +
1 pm
and
1 q
=
1 q1
+ ··· +
1 , qm
and ω ⃗ satisfy the
1/pj |fj |q (x)pj ωj (x)dx . j
(ii) Let ω ⃗ ∈ A(1,...,1) and b⃗ ∈ (BMO)m . Then there exists a constant C depending on b⃗ such that
νω⃗ x ∈ R :
n
MLl (log L)
(|⃗f |q )(x) > t
m
≤C
m j =1
Rn
Φ
(m)
|fj |qj (yj ) t
1/m ωj (yj )dyj
,
m
where Φ (t ) = t (1 + log t ) and Φ +
(m)
= Φ ◦ · · · ◦ Φ.
Proof. We can modify the proof of Theorem 4.1 [13] to get (ii) without any difficulty. Since the main ideas are almost the same, we omit the proof here. We just prove (i). It is easy to see
MLl (log L) (|⃗f |q )(x) ≤ ML(log L) (|⃗f |q )(x) ≤ C Mr (|⃗f |q )(x), where Mr (|⃗ f |q )(x) = supQ ∋x
∥Mr (|⃗f |q )∥Lp (νω⃗ ) ≤ C
m
m
j =1
1 |Q |
Q
|fj |rqj
1r
(3.9)
, with r > 1. To prove Lemma 3.3, it suffice to prove
∥ |fj |qj ∥Lpj (ωj ) .
j =1
This is equivalent to
∥M(|⃗f |q )∥Lp/r (νω⃗ ) ≤ C
m j =1
∥ |fj |qj ∥Lpj /r (ω ) . j
This follows from Lemma 3.2, since ω ⃗ ∈ Ap⃗/r for some r > 1 (see Lemma 6.1 in [12] for details).
108
Z. Si, Q. Xue / Nonlinear Analysis 96 (2014) 96–108
Proof of Theorems 1.1–1.2. Theorem 1.1 is a consequence of (3.1) and Lemma 3.3. We now prove Theorem 1.2. By homogeneity we may assume t = 1. Since Φ is submultiplicative, Theorem 3.1 and Lemma 3.3 yields
m ≤ C sup νω⃗ x ∈ Rn : TΠ∗ b⃗,q ⃗f (x) > 1 t >0
≤ C sup t >0
≤ C sup t >0
≤ C sup t >0
≤C
1
νω⃗ Φ (m) (1/t )m 1
Φ (m) (1/t )m
m
Φ (m) (1/t )m Φ (m) (1/t )m
Rn
Φ
Rn
j =1 m
1
j =1
m ∗ m ⃗ x ∈ R n : TΠ ⃗,q f (x) > t b
m νω⃗ x ∈ Rn : MLl (log L) ⃗f (x) > t m
1
m
(m)
Rn
j =1
Φ
(m)
Φ
(m)
|fj |qj (yj ) t
|fj |qj (yj )
t
(|fj |qj (yj ))Φ
ωj (yj )dyj
(m)
1 t
ωj (yj )dyj
1/m ωj (yj )dyj
.
(3.10)
This complete the proof of Theorem 1.2. Acknowledgments The authors want to express their sincere thanks to the referee for his or her valuable remarks and suggestions. The first author was supported by NSFC (Nos. 11226102, and 11226103) and Doctor Foundation of Henan Polytechnic University (No. B2012-055). The second author was supported partly by NSFC (Key program Grant No. 10931001), Program for New Century Excellent Talents in University, Program for Changjiang Scholars and Innovative Research Team in University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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