Commutator of Marcinkiewicz Integral Operators on Herz-Morrey-Hardy Spaces with Variable Exponents ()
1. Introduction
Firstly in 1938, Marcinkiewicz [1] introduced the Marcinkiewicz integral. Next, the Marcinkiewicz integral operator has been studied extensively by many mathematicians in various fields. For example, Stain in [2] introduced the Marcinkiewicz integral operator related to the littlewood-Paley
g function on
ℝn and proved that
μΩ is of type
(p,p) for
1<p≤2 and of week type
(1,1). In [3], Ding, Fan and Pan improved the above result and obtained the
Lp(1<p<∞) and weighted
Lp(1<p<∞) boundedness of the Marcinkiewicz cussed the boundedness for the commutator generated by the Marcinkiintegral
μ under some weak conditions. Torchinsky and Wang in [4] discussed integral
μΩ and
BMO(ℝn) function on Lebesgue spaces
Lp(ℝn).
On the other hand, a class of functional spaces called Herz-Morrey-Hardy spaces with variable exponent has attracted great interest in recent years. We find that in successive studies in this field, in [5] [6] Xu, Yang introduced Herz-Morrey-Hardy spaces with variable exponents and their some applications. He obtained that certain singular integral operators are bounded from Herz-Morrey-Hardy spaces with variable exponents into Herz-Morrey spaces with variable exponents as an application of the atomic characterization. Also, he established their molecular decomposition, and by using their atomic and molecular decompositions, he gave the boundedness of a convolution type singular integral on Herz-Morrey-Hardy spaces with variable exponents. Omer in [7] proved the boundedness of commutators generated by the Calderón-Zygmund and used properties of variable exponent, BMO(Rn) function and Lipschitz function to prove this boundedness. Also, Yang in [8] established some boundedness for
TDγ−DγT and
(T*−T#)Dγ on the homogeneous Morrey-Herz-type Hardy spaces with variable exponents and studied Boundedness of Calderón-Zygmund operator on these spaces.
Suppose
Sn−1(n≥2) denotes the unit sphere in
ℝn equipped with the normalized measure
dσ. Let
Ω be homogenous function of degree zero and satisfies
∫Sn−1 Ω(x′)dσ(x′)=0, (1.1)
where
x′=x/|x| for any
x≠0.
Then the Marcinkiewicz integral operator
μΩ is defined by
μΩ(f)(x)=(∫∞0|FΩ,t(f)(x)|2dtt3)1/2, (1.2)
where
FΩ,t(f)(x)=∫|x−y|≤1Ω(x−y)|x−y|n−1f(y)dy. (1.3)
Let
b∈Lipγ(ℝn) and
b∈BMO be a locally integrable function on
ℝn, the commutator generated by the Marcinkiewicz integral
μΩ and b is defined by
[b,μΩ]=(∫∞0|∫|x−y|≤tΩ(x−y)|x−y|n−1[b(x)−b(y)]f(y)dy|2dtt3)1/2. (1.4)
Motivated by [6] and [7], the aim of this paper is to study the boundedness for the commutator of Marcinkiewicz integral operator
[b,μΩ] on the Herz-Morrey-Hardy space with variable exponent where
Ω∈Ls(Sn−1) for
s≥1, with BMO function and Lipschitz function, we will define The definitions of the Morrey-Herz spaces with variable exponents, the Morrey-Herz-Hardy spaces with variable exponents (which will be defined in the next section), and the preliminary lemmas are presented in Section 2. In Section 3, we will prove the boundedness of the commutator of Marcikiewicz integrals on Herz-Morrey-Hrdy spaces with variable exponent with
b∈Lipγ(ℝn). Lastly, in Section 4 we will prove the boundedness of the commutator of Marcikiewicz integrals on Herz-Morrey-Hrdy spaces with variable exponent with function
b∈BMO(ℝn).
A given open set
Ω⊂ℝn and a measurable function
p(⋅):Ω→[1,∞),
Lp(⋅)(Ω) denotes the set of measurable function f on
Ω such that for some
λ>0,
Lp(⋅)(Ω)={f is measurable:∫Ω(|f(x)|η)p(x)dx<∞ for some constant η>0}, (1.5)
the space
Lp(⋅)Loc(Ω) is defined by
Lp(⋅)Loc(Ω)={f ismeasurable:f∈Lp(⋅)(K) forallcompact K⊂Ω}. (1.6)
The Lebesgue spaces
Lp(⋅)(Ω) is Banach spaces with the norm defined by
‖f‖Lp(⋅)(Ω)=inf{η>0:∫Ω(|f(x)|η)p(x)dx≤1}, (1.7)
where
p−=essinf{p(x):x∈Ω}>1,
p+=esssup{p(x):x∈Ω}<∞.
Denotes
p′(x)=p(x)/(p(x)−1). Let M be the Hardy-Littlewood maximal operator. We denote
B(Ω) to be the set of all functions
p(⋅)∈P(Ω) satisfying the M is bounded on
Lp(⋅)(Ω).
Definition 1.1. [6]
Let
0<q≤∞,
p(⋅)∈P(ℝn),
0≤λ<∞. Let
α(⋅) be a bounded real-valued measurable function on
ℝn. The nonhomogeneous Morrey-Herz space
MKα(⋅),qp(⋅),λ(ℝn) and homogeneous Morrey-Herz space with variable exponents
M˙Kα(⋅),qp(⋅),λ(ℝn) are respectively defined by
MKα(⋅),qp(⋅),λ:={f∈Lp(⋅)loc(ℝn\{0}):‖f‖MKα(⋅),qp(⋅),λ<∞}, (1.8)
and
M˙Kα(⋅),qp(⋅),λ:={f∈Lp(⋅)Loc(ℝn\{0}):‖f‖M˙Kα(⋅),qp(⋅),λ<∞}, (1.9)
where
‖f‖MKα(⋅),qp(⋅),λ:=supL∈ℕ02−Lλ(L∑k=0‖2kα(⋅)f˜χk‖qLp(⋅))1/q, (1.10)
‖f‖M˙Kα(⋅),qp(⋅),λ:=supL∈ℤ2−Lλ(L∑k=−∞‖2kα(⋅)fχk‖qLp(⋅))1/q. (1.11)
Definition 1.2. [9]
For all
0<γ≤1, the Lipschitz space
Lipγ(ℝn) is defined by
Lipγ={f:‖f‖Lipγ=supx,y∈ℝn;x≠y|f(x)−f(y)||x−y|γ<∞}. (1.12)
Definition 1.3. [5]
Let
α(⋅)∈L∞(ℝn), p(⋅)∈P(ℝn), 0<q≤∞, 0≤λ<∞ and
N>n+1. The nonhomogeneous Herz-Morrey-Hardy space with variable exponent
HMKα(⋅),qp(⋅)λ(ℝn) and homogeneous Herz-Morrey-Hardy space with variable exponents
HM˙Kα(⋅),qp(⋅)λ(ℝn) are respectively defined by
HMKα(⋅),qp(⋅)λ(ℝn):={f∈S′(ℝn):‖f‖HMKα(⋅),qp(⋅)λ:=‖GNf‖MKα(⋅),qp(⋅)λ<∞}, (1.13)
HM˙Kα(⋅),qp(⋅)λ(ℝn):={f∈S′(ℝn):‖f‖HM˙Kα(⋅),qp(⋅)λ:=‖GNf‖M˙Kα(⋅),qp(⋅)λ<∞}. (1.14)
Definition 1.4. [10] (Hölder’s inequality) Let
α>1 and
1/α+1/β=1. Then the discrete and integral forms of Hölder’s inequality are given as
∫ba|f(x)g(x)|dx≤(∫ba|f(x)|α)1/α(∫ba|g(x)|β)1/β, (1.15)
for continuous function f and g on
[a,b].
Definition 1.5. [10] (Minkowski’s inequality) Let
u>1. Then the discrete and integral forms of Minkowski’s inequality are given as
(∫ba|f(x)+g(x)|udx)1/u≤(∫ba|f(x)|u)1/u+(∫ba|g(x)|u)1/u, (1.16)
for continuous function f and g on
[a,b]. for more general functions can be obtained naturally. A further generalization is: If
u>1, then
(∫(∫|f(x,y)|dy)udx)1/u≤∫(∫|f(x)|udx)1/udy. (1.17)
2. Preliminaries
In this section, we give some preliminaries which we used to prove theorems.
Lemma 2.1. [11] Let
p(⋅)∈P(ℝn). Then for any
f∈Lp(⋅) and
g∈Lp′(⋅), we have
∫ℝn|f(x)g(x)|dx≤Cp‖f‖Lp(⋅)(ℝn)‖g‖Lp′(⋅)(ℝn),
where
Cp=1+1p−−1p+.
This inequality is called the generalized Hölder inequality with respect to the variable
Lp(⋅) spaces.
Lemma 2.2. [12] Given
p(⋅), p1(⋅), p2(⋅)∈P(ℝn), for any
f∈Lp1(⋅)(ℝn), g∈Lp2(⋅)(ℝn), when
1p(⋅)=1p2(⋅)+1p1(⋅), we get
‖f(x)g(x)‖Lp(⋅)(ℝn)≤C‖f‖Lp1(⋅)(ℝn)‖g‖Lp2(ℝn),
where
Cp1, p2=[1+1p1−−1p1+]1p−.
Proposition 2.3. [13] If
q(⋅)∈P(ℝn) satisfies
|q(x)−q(y)|≤−Clog(|x−y|), |x−y|≤1/2,
|q(x)−q(y)|≤Clog(e+|x|), |y|≥|x|,
then
q(⋅)∈B(ℝn).
Lemma 2.4. [14] Let k be a positive integer and B be a ball in
ℝn. Then we have that for all
b∈BMO(ℝn) and
i,j∈ℤ with
i<j, we have
1)
C−1‖b‖k∗≤supB1‖χB‖Lp(⋅)(ℝn)‖(b−bB)χB‖Lp(⋅)(ℝn)≤C‖b‖k∗,
2)
‖(b−bBi)kχBj‖Lp(⋅)(ℝn)≤C(j−i)K‖b‖k∗‖χBj‖Lq(⋅)(ℝn),
where
Bi={x∈ℝn:|x|≤2i} and
Bj={x∈ℝn:|x|≤2j}.
Lemma 2.5. [15] Let
q(⋅)∈B(ℝn), then there exist positive constants
C>0, such that for all balls
B⊂ℝn and all measurable subset
R⊂B,
‖χR‖Lq(⋅)(ℝn)‖χB‖Lq(⋅)(ℝn)≤C(|R||B|)δ1, ‖χR‖Lq′(⋅)(ℝn)‖χB‖Lq′(⋅)(ℝn)≤C(|R||B|)δ2,
where
δ1,δ2 are constants with
0<δ1,δ2<1.
Lemma 2.6. [16] If
q(⋅)∈B(ℝn), then there exists a constant
C>0 such that for any balls B in
ℝn,
1|B|‖χB‖Lq(⋅)(ℝn)‖χB‖Lq′(⋅)(ℝn)≤C.
Lemma 2.7. [6] Let
0<q<∞, p(⋅)∈B(ℝn), 0<λ<∞, and
α(⋅)∈L∞(ℝn) be log-Hölder continuous both at the origin and infinity,
2λ≤α(⋅),nδ2≤α(0),α∞<∞,
δ2 as in lemma 2.4. Then
f∈HM˙Kα(⋅),qp(⋅)λ(ℝn) (or
HMKα(⋅),qp(⋅)λ(ℝn) ) if and only if
f=∑∞k=−∞ λkfk (or
f=∑∞k=0 λkfk ), in the sense of
f∈S′(ℝ)n, where each
ak is a central
(α(⋅),p(⋅)) atom with support contained in
Bk and
supL≤0,L∈Z2−Lλ∑Lk=−∞|λk|q<∞ or
(supL≤0,L∈Z2−Lλ∑Lk=0|λk|q),
moreover
‖f‖HM˙Kα(⋅),qp(⋅)λ≈infsupL≤0,L∈Z2−Lλ(∑Lk=−∞|λk|q)1/q
or
‖f‖HMKα(⋅),qp(⋅)λ≈infsupL≤0,L∈Z2−Lλ(∑Lk=0|λk|q)1/q,
where infimum is taken over all above decomposition of f.
Lemma 2.8. [17] Let
q(⋅)∈P(ℝn),q∈(0,∞] and
λ∈[0,∞). If
α(⋅)∈L∞(ℝn)∩Plog0(ℝn)∩Plog∞(ℝn), then
‖f‖qM˙Kα(⋅),qp(⋅),λ(ℝn)=max{supL≤0,L∈ℤ2−LλqL∑k=−∞ 2kqα(0)‖fχk‖qLp(⋅), supL≤0,L∈Z2−Lλq(−1∑k=−∞2kqα(0)‖fχk‖qLp(⋅)+L∑k=0 2kqα(∞)‖fχk‖qLp(⋅))}.
Lemma 2.9. [18] Let
Ω satisfies
Lr -Dini condition with
r∈[1,∞). If there exist constants
C>0 and
R>0 such that
|y|<R/2, then for every
x∈ℝn, we have
(∫R<|R|<2R|Ω(x−y)|x−y|−Ω(x)|x||rdx)1/r≤CR(nr−n){|y|R+∫|y|/2R<δ<|y|/Rωr(δ)δdδ}.
Lemma 2.10. [15] Given E, let
q(⋅)∈P(E),f:E×E→ℝn be a measurable function (with respect to product measure) such that for almost every
y∈E,f(.,y)∈Lq(⋅)(E). Then
‖∫E f(.,y)dy‖Lq(⋅)(E)≤C∫E‖f(.,y)‖Lq(⋅)(E)dy.
Lemma 2.11. [19] If
a>0, 1≤s≤∞, 0≤d≤s and
−n+(n−1)d/s<v<∞, then
(∫|y|≤a|x||y|v|Ω(x−y)|ddy)1/d≤C|x|(v+n)/d‖Ω‖Ls(Sn−1).
Lemma 2.12. [19] Let
q(⋅)∈P satisfies Proposition 2.3. Then
‖χQ‖Lq(⋅)(ℝn)≈(|Q|1q(x)if |Q|≤2n and x∈Q|Q|1q(∞)if |Q|≥1
for every cube (or ball)
Q∈ℝn, where
p(∞)=limx→∞p(x).
3. Lipschitz Boundedness for the Commutator of Marcikiewicz Integrals Operator
In this section, we prove the boundedness of the commutator of Marcikiewicz integrals on Herz-Morrey-Hrdy spaces with variable exponent so when
b∈Lipγ(ℝn) under some conditions.
Theorem 3.1.
Suppose that
b∈Lipγ(ℝn) with
0<γ≤1. If
q1(⋅)∈P(ℝn) satisfies proposition 2.3 with
q+1<n/γ,1/q1(x)−1/q2(x)=γ/n,
Ω∈Ls(Sn−1)(s>q+2) with
1≤s′<q−1 and satisfies
∫10Ωs(δ)δ1+γdδ<∞,
let
0<p1≤q2<∞ and
nδ2≤α<nδ2+γ or (
0<max(nδ2,α2)≤α1<nδ2+γ ). Then the commutator
[b,μΩ] is bounded from
HM˙Kα(⋅),qp(⋅)λ(ℝn) (or
HMKα(⋅),qp(⋅)λ(ℝn) ) to
M˙Kα(⋅),qp(⋅)λ(ℝn) (or
MKα(⋅),qp(⋅)λ(ℝn) ).
To the proof the above theorem, we will recall the following lemma.
Lemma 3.1. [15]
Suppose that
b∈Lipγ(ℝn) with
0<γ≤1. If
q1(⋅)∈P(ℝn) satisfies Proposition 2.3 with
q+1<n/γ,1/q1(x)−1/q2(x)=γ/n with
Ω∈Ls(Sn−1)(s>q+2). Then the commutator
[b,μΩ] is bounded from
Lq1(⋅)(ℝn) to
Lq1(⋅)(ℝn).
Next, we will give the Lipschitz estimate about the commutator
[b,μΩ] on Herz-Morrey-Hardy spaces with variable exponent.
Proof Theorem 3.1:
To prove this theorem, we only prove the homogeneous case. Let
f∈HMKα(⋅),qp(⋅)λ(ℝn). By lemma 2.6 we have
f=∑∞j=−∞ λjfj converged in
S′(ℝ)n, where each
bj is a central
(α(⋅),p(⋅)) atom with support contained in
Bj and
‖f‖HMKα(⋅),qp(⋅)λ≈infsupL≤0,L∈Z2−Lλ(L∑j=−∞|λj|q)1/q.
Here we denote
Δ=supL≤0,L∈Z2−Lλ∑Lk=−∞|λk|q. By lemma 2.8 we have
‖[b,μΩ](f)‖qMKα(⋅),qp(⋅),λ(ℝn)=max{supL≤0,L∈ℤ2−LλqL∑k=−∞ 2kqα(0)‖[b,μΩ](f)χk‖qLp(⋅), supL≤0,L∈Z2−Lλq(−1∑k=−∞ 2kqα(0)‖[b,μΩ](f)χk‖qLp(⋅) +L∑k=0 2kqα(∞)‖[b,μΩ](f)χk‖qLp(⋅))}.
I=supL≤0,L∈ℤ2−LλqL∑k=−∞ 2kqα(0)‖[b,μΩ](f)χk‖qLp(⋅),II=−1∑k=−∞ 2kqα(0)‖[b,μΩ](f)χk‖qLp(⋅),III=supL≤0,L∈ℤ2−LλqL∑k=0 2kqα(∞)‖[b,μΩ](f)χk‖qLp(⋅).
In beginning, we examine a function which we will use in proving
|[b,μΩ](bj)(x)|≤{∫|x|0|∫|x−y|≤t|Ω(x−y)||x−y|n−1[b(x)−b(y)]bj(y)dy|2dtt3}1/2 +{∫∞|x||∫|x−y|≤t|Ω(x−py)||x−y|n−1[b(x)−b(y)]bj(y)dy|2dtt3}1/2:≃ϒ1+ϒ2.
When
x∈Ak and
|x−y|≤t with
t≤|x|, it follows from
j≤k−2 that
|x−y|∼|x|. We have
|1|x−y|2−1|x|2|≤|y||x−y|3. (3.1)
Then by (3.1), the Minkowski’s inequality, the generalized Hölder’s inequality and the vanishing of the moment of
bj we have
ϒ1≤C∫ℝn||Ω(x−y)||x−y|2−|Ω(x)||x|2||b(x)−b(y)||bj(y)|(∫|x||x−y|dtt3)1/2dy≤C∫ℝn||Ω(x−y)||x−y|2−|Ω(x)||x|2||b(x)−b(y)||bj(y)||1|x−y|2−1|x|2|1/2dy
≤C∫ℝn||Ω(x−y)||x−y|2−|Ω(x)||x|2||b(x)−b(y)||bj(y)||y|1/2|x−y|3/2dy≤C2(j−k)/2∫Bj||Ω(x−y)||x−y|2−|Ω(x)||x|2||b(x)−b(y)||bj(y)|dy.
Similarly, we consider
ϒ2. Noting that
|x−y|∼|x|. By the Minkowski’s inequality, the generalized Hölder’s inequality and the vanishing moments of
bj we have
ϒ2≤C∫ℝn||Ω(x−y)||x−y|n−1−|Ω(x)||x|n−2||b(x)−b(y)||bj(y)|(∫∞|x|dtt3)1/2dy≤C∫Bj||Ω(x−y)||x−y|2−|Ω(x)||x|2||b(x)−b(y)||bj(y)|dy.
So we have
|[b,μΩ](bj)(x)|≤C∫Bj||Ω(x−y)||x−y|n−|Ω(x)||x|n||b(x)−b(y)||bj(y)|dy.
From lemma 2.10 and the Minkowski’s inequality we have
‖[b,μΩ](bj)χk‖Lp(⋅)(ℝn)≤C∫Bj‖||Ω(x−y)||x−y|n−|Ω(x)||x|n||b(x)−b(y)|χk(⋅)‖Lp(⋅)(ℝn)|bj(y)|dy≤C∫Bj‖||Ω(⋅−y)||⋅−y|n−|Ω(⋅)|| ⋅ |n||b(⋅)−b(0)|χk(⋅)‖Lp(⋅)(ℝn)|bj(y)| +C∫Bj‖||Ω(⋅−y)||⋅−y|n−|Ω(⋅)|| ⋅ |n|χk(⋅)‖Lp(⋅)(ℝn)|b(0)−b(y)||bj(y)|:=ϒ∗1+ϒ∗2.
For
ϒ∗1, noting
s>p′, we denote
˜p′(⋅)>1 and
1p(x)=1˜p′(x)+1s. By lemma 2.2 we have
‖||Ω(⋅−y)||⋅−y|n−|Ω(⋅)|| ⋅ |n||b(⋅)−b(0)|χk(⋅)‖Lp(⋅)(ℝn)≤‖||Ω(⋅−y)||⋅−y|n−|Ω(⋅)|| ⋅ |n|χk(⋅)‖Ls(⋅)(ℝn)‖|b(0)−b(y)||bj(y)|‖Lp′(⋅)(ℝn)≤C‖b‖Lipγ2kγ‖||Ω(⋅−y)||⋅−y|n−|Ω(⋅)|| ⋅ |n|χk(⋅)‖Ls(⋅)(ℝn)‖χBK‖Lp′(⋅)(ℝn).
When
|Bk|≤2n and
xk∈Bk, by Lemma 2.12 we have
‖χBK‖Lp′(⋅)(ℝn)≃|Bk|1p′(xk)≈‖χBK‖Lp(⋅)(ℝn)|Bk|−1s−−γn.
When
|Bk|≥1 we have
‖χBK‖Lp′(⋅)(ℝn)≃|Bk|1p′(∞)≈‖χBK‖Lp(⋅)(ℝn)|Bk|−1s−−γn.
So we obtain
‖χBK‖Lp′(⋅)(ℝn)≈‖χBK‖Lp(⋅)(ℝn)|Bk|−1s−−γn.
By lemma 2.9 we have
‖||Ω(⋅−y)||⋅−y|n−|Ω(⋅)|| ⋅ |n|χk(⋅)‖Ls(⋅)(ℝn)≤2(k−1)(ns−n){|y|2k+∫|y|2k−1|y|2kωs(δ)δdδ}≤2(k−1)(ns−n){2j−k+1+2(j−k+1)γ∫10ωs(δ)δdδ}≤2(k−1)(ns−n)2(j−k)γ.
Now, by using the generalized Hölder’s inequality we get:
ϒ*1≤∫Bj‖||Ω(⋅−y)||⋅−y|n−|Ω(⋅)|| ⋅ |n||b(⋅)−b(y)|χk(⋅)‖Lp(⋅)(ℝn)|bj(y)|dy≤C‖b‖Lipγ2−kn+(j−k)γ−kγ‖χBK‖Lp(⋅)(ℝn)|Bk|−1s−−γn∫Bj|bj(y)|dy≤C‖b‖Lipγ2−kn+(j−k)γ−kγ‖χBK‖Lp(⋅)(ℝn)‖bj‖Lp(⋅)(ℝn)‖χBj‖Lp′(⋅)(ℝn). (3.2)
For
ϒ*2 similar to the method of
ϒ*1 we have
‖||Ω(⋅−y)||⋅−y|n−|Ω(⋅)|| ⋅ |n|χk(⋅)‖Lp(⋅)(ℝn)≤‖||Ω(⋅−y)||⋅−y|n−|Ω(⋅)|| ⋅ |n|χk(⋅)‖Ls(⋅)(ℝn)‖χk(⋅)‖Lp′(⋅)(ℝn)≤‖||Ω(⋅−y)||⋅−y|n−|Ω(⋅)|| ⋅ |n|χk(⋅)‖Ls(⋅)(ℝn)‖χBk(⋅)‖Lp′(⋅)(ℝn)≤2(k−1)(ns−n)2(j−k)γ‖χBK‖Lp′(⋅)(ℝn)≤2−kn+(j−k)γ−kγ‖χBk‖Lp(⋅)(ℝn).
Now, by using the generalized Hölder’s inequality we get:
ϒ*2≤∫Bj‖||Ω(⋅−y)||⋅−y|n−|Ω(⋅)|| ⋅ |n|χk(⋅)‖Lp(⋅)(ℝn)|b(0)−b(y)||bj(y)|dy≤C‖b‖Lipγ2−kn+2(j−k)γ‖χBk‖Lp(⋅)(ℝn)‖bj‖Lp(⋅)(ℝn)‖χBj‖Lp′(⋅)(ℝn)≤C‖b‖Lipγ2−kn+(j−k)γ‖χBk‖Lp(⋅)(ℝn)‖bj‖Lp(⋅)(ℝn)‖χBj‖Lp′(⋅)(ℝn). (3.3)
Now by (3.3), (3.4), and lemmas 2.5 and 2.6, we have
‖[b,μΩ](bj)χk‖Lp(⋅)(ℝn)≤C‖b‖Lipγ2−kn+(j−k)γ‖χBk‖Lp(⋅)(ℝn)‖bj‖Lp(⋅)(ℝn)‖χBj‖Lp′(⋅)(ℝn)≤C‖b‖Lipγ2(j−k)γ‖bj‖Lp(⋅)(ℝn)‖χBj‖Lp′(⋅)(ℝn)‖χBk‖Lp′(⋅)(ℝn)≤C2−jα+(j−k)(γ+nδ2)‖b‖Lipγ.
Firstly we estimate I. We need to show that there exists a positive constant C, such that
I≤CΔ, we consider
I=supL≥0,L∈ℤ2−LλqL∑k=−∞ 2kqα(0)|λj|‖[b,μΩ](f)χk‖qLp(⋅)(ℝn)≤supL≥0,L∈ℤ2−LλqL∑k=−∞ 2kqα(0)(∞∑j=K|λj|‖[b,μΩ]χk‖Lp(⋅)(ℝn))q +supL≥0,L∈ℤ2−LλqL∑k=−∞ 2kqα(0)(k−1∑j=−∞|λj|‖[b,μΩ]χk‖Lp(⋅)(ℝn))q:=I1+I2.
By the
(Lp(⋅)(ℝn),Lq(⋅)(ℝn)), bounbedness of the commutator
[b,μΩ] on
Lp(⋅) (see [15] ), we have the following. Therefore, when
0<q≤1
I1=supL≥0,L∈ℤ2−LλqL∑k=−∞ 2kqα(0)(∞∑j=K‖[b,μΩ]χk‖Lp(⋅)(ℝn))q≤supL≤0,L∈ℤ2−LλqL∑k=−∞ 2kqα(0)(∞∑j=k|λj|2−jαj)q≤supL≤0,L∈ℤ2−LλqL∑k=−∞ 2kqα(0)(−1∑j=k|λj|2−jα(0)jq+∞∑j=0|λj|2−jα∞q)≤supL≤0,L∈ℤ2−LλqL∑k=−∞ −1∑j=k|λj|q2α(0)(k−j)q +supL≤0,L∈ℤ2−LλqL∑k=−∞ 2kqα(0)∞∑j=0|λj|q2−jα∞q
≤supL≤0,L∈ℤ2−Lλq−1∑j=−∞|λj|qj∑k=−∞ 2α(0)(k−j)q +supL≤0,L∈ℤ∞∑j=0 2−jλq|λj|q2(λ−α∞)jq2−LλqL∑k=−∞ 2kqα( 0 )
(3.4)
When
, let
we have
(3.5)
We estimate
by lemma 2.1 when
by
, we get
(3.6)
When
, let
. Since
, by Hölder’s inequality, we have
(3.7)
Secondly we estimate
. We need to show that there exists a positive constant C, such that
, we consider
When
, we get
(3.8)
When
, let
we have
(3.9)
For
, when
, by
we get
(3.10)
When
, let
. Since
, by Hölder’s inequality, we have
(3.11)
Thirdly, we estimate
, we need to show that there exists a positive constant C, such that
When
, by the boundedness of
in
( [20] ), we have
(3.12)
When
, by
and the boundedness of
in
( [20] ) and Hölder’s inequality, we get
(3.13)
When
, by
we get
(3.14)
When
, let
. Since
, and by Hölder’s inequality, we have
(3.15)
Joint the estimates for I, II and III, we obtain
Then we complete the proof of Theorem 3.1.
4. BMO Boundedness for the Commutator of Marcikiewicz Integrals Operator
In this section, we prove the boundedness of the commutator of Marcikiewicz integrals on Herz-Morrey-Hrdy spaces with variable exponent with function
.
Theorem 4.1.
Suppose that
with
. If
satisfies proposition 2.3 and
. Let
and
(or
). Then
is bounded from
(or
) to
(or
).
proof:
In a way similar to theorem (3.2) we only prove the homogeneous case. Let
and
. Let us write
Then we have
From the Hölder’s inequality, we have
Noting
, we denote
and
. By lemmas 3.2, 3.10 we have
By lemma (2.12), when
and when
respectively we have
,
and
we obtain
.
So we have
(4.1)
Similarly by lemma 2.4 we have
(4.2)
Now, by (4.1), (4.2), lemmas 2.4, 2.5 and 2.3, we have
(4.3)
By the boundedness of
in
see [7], we have
So we have
Firstly we estimate H. We need to show that there exists a positive constant C, such that
Consider
By boundedness of
in
, see ( [20] ), when
we have
(4.4)
When
and
, and let
, we have