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Commutator of Marcinkiewicz Integral Operators on Herz-Morrey-Hardy Spaces with Variable Exponents

Abstract

In this paper, our aim is to prove the boundedness of commutators generated by the Marcinkiewicz integrals operator [b,μΩ] and obtain the result with Lipschitz function and BMO function f on the Herz-Morrey-Hardy spaces with variable exponents .

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Khalil, O. , Tao, S. and Mahamat, B. (2021) Commutator of Marcinkiewicz Integral Operators on Herz-Morrey-Hardy Spaces with Variable Exponents. Applied Mathematics, 12, 421-448. doi: 10.4236/am.2021.125030.

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 n and proved that μΩ is of type (p,p) for 1<p2 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(n) function on Lebesgue spaces Lp(n).

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 TDγDγT 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 Sn1(n2) denotes the unit sphere in n equipped with the normalized measure dσ. Let Ω be homogenous function of degree zero and satisfies

Sn1Ω(x)dσ(x)=0, (1.1)

where x=x/|x| for any x0.

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)=|xy|1Ω(xy)|xy|n1f(y)dy. (1.3)

Let bLipγ(n) and bBMO be a locally integrable function on n, the commutator generated by the Marcinkiewicz integral μΩ and b is defined by

[b,μΩ]=(0||xy|tΩ(xy)|xy|n1[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 ΩLs(Sn1) for s1, 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 bLipγ(n). 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 bBMO(n).

A given open set Ωn 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)={fismeasurable:Ω(|f(x)|η)p(x)dx<forsomeconstantη>0}, (1.5)

the space Lp()Loc(Ω) is defined by

Lp()Loc(Ω)={fismeasurable:fLp()(K)forallcompactKΩ}. (1.6)

The Lebesgue spaces Lp()(Ω) is Banach spaces with the norm defined by

fLp()(Ω)=inf{η>0:Ω(|f(x)|η)p(x)dx1}, (1.7)

where p=essinf{p(x):xΩ}>1, p+=esssup{p(x):xΩ}<.

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(n), 0λ<. Let α() be a bounded real-valued measurable function on n. The nonhomogeneous Morrey-Herz space MKα(),qp(),λ(n) and homogeneous Morrey-Herz space with variable exponents M˙Kα(),qp(),λ(n) are respectively defined by

MKα(),qp(),λ:={fLp()loc(n\{0}):fMKα(),qp(),λ<}, (1.8)

and

M˙Kα(),qp(),λ:={fLp()Loc(n\{0}):fM˙Kα(),qp(),λ<}, (1.9)

where

fMKα(),qp(),λ:=supL02Lλ(Lk=02kα()f˜χkqLp())1/q, (1.10)

fM˙Kα(),qp(),λ:=supL2Lλ(Lk=2kα()fχkqLp())1/q. (1.11)

Definition 1.2. [9]

For all 0<γ1, the Lipschitz space Lipγ(n) is defined by

Lipγ={f:fLipγ=supx,yn;xy|f(x)f(y)||xy|γ<}. (1.12)

Definition 1.3. [5]

Let α()L(n),p()P(n),0<q,0λ< and N>n+1. The nonhomogeneous Herz-Morrey-Hardy space with variable exponent HMKα(),qp()λ(n) and homogeneous Herz-Morrey-Hardy space with variable exponents HM˙Kα(),qp()λ(n) are respectively defined by

HMKα(),qp()λ(n):={fS(n):fHMKα(),qp()λ:=GNfMKα(),qp()λ<}, (1.13)

HM˙Kα(),qp()λ(n):={fS(n):fHM˙Kα(),qp()λ:=GNfM˙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(n). Then for any fLp() and gLp(), we have

n|f(x)g(x)|dxCpfLp()(n)gLp()(n),

where Cp=1+1p1p+.

This inequality is called the generalized Hölder inequality with respect to the variable Lp() spaces.

Lemma 2.2. [12] Given p(),p1(),p2()P(n), for any fLp1()(n),gLp2()(n), when 1p()=1p2()+1p1(), we get

f(x)g(x)Lp()(n)CfLp1()(n)gLp2(n),

where Cp1,p2=[1+1p11p1+]1p.

Proposition 2.3. [13] If q()P(n) satisfies

|q(x)q(y)|Clog(|xy|),|xy|1/2,

|q(x)q(y)|Clog(e+|x|),|y||x|,

then q()B(n).

Lemma 2.4. [14] Let k be a positive integer and B be a ball in n. Then we have that for all bBMO(n) and i,j with i<j, we have

1) C1bksupB1χBLp()(n)(bbB)χBLp()(n)Cbk,

2) (bbBi)kχBjLp()(n)C(ji)KbkχBjLq()(n),

where Bi={xn:|x|2i} and Bj={xn:|x|2j}.

Lemma 2.5. [15] Let q()B(n), then there exist positive constants C>0, such that for all balls Bn and all measurable subset RB,

χRLq()(n)χBLq()(n)C(|R||B|)δ1,χRLq()(n)χBLq()(n)C(|R||B|)δ2,

where δ1,δ2 are constants with 0<δ1,δ2<1.

Lemma 2.6. [16] If q()B(n), then there exists a constant C>0 such that for any balls B in n,

1|B|χBLq()(n)χBLq()(n)C.

Lemma 2.7. [6] Let 0<q<,p()B(n),0<λ<, and α()L(n) be log-Hölder continuous both at the origin and infinity, 2λα(),nδ2α(0),α<, δ2 as in lemma 2.4. Then fHM˙Kα(),qp()λ(n) (or HMKα(),qp()λ(n) ) if and only if f=k=λkfk (or f=k=0λkfk ), in the sense of fS()n, where each ak is a central (α(),p()) atom with support contained in Bk and

supL0,LZ2LλLk=|λk|q< or (supL0,LZ2LλLk=0|λk|q),

moreover

fHM˙Kα(),qp()λinfsupL0,LZ2Lλ(Lk=|λk|q)1/q

or

fHMKα(),qp()λinfsupL0,LZ2Lλ(Lk=0|λk|q)1/q,

where infimum is taken over all above decomposition of f.

Lemma 2.8. [17] Let q()P(n),q(0,] and λ[0,). If α()L(n)Plog0(n)Plog(n), then

fqM˙Kα(),qp(),λ(n)=max{supL0,L2LλqLk=2kqα(0)fχkqLp(),supL0,LZ2Lλq(1k=2kqα(0)fχkqLp()+Lk=02kqα()fχkqLp())}.

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 xn, we have

(R<|R|<2R|Ω(xy)|xy|Ω(x)|x||rdx)1/rCR(nrn){|y|R+|y|/2R<δ<|y|/Rωr(δ)δdδ}.

Lemma 2.10. [15] Given E, let q()P(E),f:E×En be a measurable function (with respect to product measure) such that for almost every yE,f(.,y)Lq()(E). Then

Ef(.,y)dyLq()(E)CEf(.,y)Lq()(E)dy.

Lemma 2.11. [19] If a>0,1s,0ds and n+(n1)d/s<v<, then

(|y|a|x||y|v|Ω(xy)|ddy)1/dC|x|(v+n)/dΩLs(Sn1).

Lemma 2.12. [19] Let q()P satisfies Proposition 2.3. Then

χQLq()(n)(|Q|1q(x)if|Q|2nandxQ|Q|1q()if|Q|1

for every cube (or ball) Qn, where p()=limxp(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 bLipγ(n) under some conditions.

Theorem 3.1.

Suppose that bLipγ(n) with 0<γ1. If q1()P(n) satisfies proposition 2.3 with q+1<n/γ,1/q1(x)1/q2(x)=γ/n, ΩLs(Sn1)(s>q+2) with 1s<q1 and satisfies

10Ωs(δ)δ1+γdδ<,

let 0<p1q2< and nδ2α<nδ2+γ or ( 0<max(nδ2,α2)α1<nδ2+γ ). Then the commutator [b,μΩ] is bounded from HM˙Kα(),qp()λ(n) (or HMKα(),qp()λ(n) ) to M˙Kα(),qp()λ(n) (or MKα(),qp()λ(n) ).

To the proof the above theorem, we will recall the following lemma.

Lemma 3.1. [15]

Suppose that bLipγ(n) with 0<γ1. If q1()P(n) satisfies Proposition 2.3 with q+1<n/γ,1/q1(x)1/q2(x)=γ/n with ΩLs(Sn1)(s>q+2). Then the commutator [b,μΩ] is bounded from Lq1()(n) to Lq1()(n).

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 fHMKα(),qp()λ(n). By lemma 2.6 we have f=j=λjfj converged in S()n, where each bj is a central (α(),p()) atom with support contained in Bj and

fHMKα(),qp()λinfsupL0,LZ2Lλ(Lj=|λj|q)1/q.

Here we denote Δ=supL0,LZ2LλLk=|λk|q. By lemma 2.8 we have

[b,μΩ](f)qMKα(),qp(),λ(n)=max{supL0,L2LλqLk=2kqα(0)[b,μΩ](f)χkqLp(),supL0,LZ2Lλq(1k=2kqα(0)[b,μΩ](f)χkqLp()+Lk=02kqα()[b,μΩ](f)χkqLp())}.

I=supL0,L2LλqLk=2kqα(0)[b,μΩ](f)χkqLp(),II=1k=2kqα(0)[b,μΩ](f)χkqLp(),III=supL0,L2LλqLk=02kqα()[b,μΩ](f)χkqLp().

In beginning, we examine a function which we will use in proving

|[b,μΩ](bj)(x)|{|x|0||xy|t|Ω(xy)||xy|n1[b(x)b(y)]bj(y)dy|2dtt3}1/2+{|x|||xy|t|Ω(xpy)||xy|n1[b(x)b(y)]bj(y)dy|2dtt3}1/2:ϒ1+ϒ2.

When xAk and |xy|t with t|x|, it follows from jk2 that |xy||x|. We have

|1|xy|21|x|2||y||xy|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

ϒ1Cn||Ω(xy)||xy|2|Ω(x)||x|2||b(x)b(y)||bj(y)|(|x||xy|dtt3)1/2dyCn||Ω(xy)||xy|2|Ω(x)||x|2||b(x)b(y)||bj(y)||1|xy|21|x|2|1/2dy

Cn||Ω(xy)||xy|2|Ω(x)||x|2||b(x)b(y)||bj(y)||y|1/2|xy|3/2dyC2(jk)/2Bj||Ω(xy)||xy|2|Ω(x)||x|2||b(x)b(y)||bj(y)|dy.

Similarly, we consider ϒ2. Noting that |xy||x|. By the Minkowski’s inequality, the generalized Hölder’s inequality and the vanishing moments of bj we have

ϒ2Cn||Ω(xy)||xy|n1|Ω(x)||x|n2||b(x)b(y)||bj(y)|(|x|dtt3)1/2dyCBj||Ω(xy)||xy|2|Ω(x)||x|2||b(x)b(y)||bj(y)|dy.

So we have

|[b,μΩ](bj)(x)|CBj||Ω(xy)||xy|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)χkLp()(n)CBj||Ω(xy)||xy|n|Ω(x)||x|n||b(x)b(y)|χk()Lp()(n)|bj(y)|dyCBj||Ω(y)||y|n|Ω()|||n||b()b(0)|χk()Lp()(n)|bj(y)|+CBj||Ω(y)||y|n|Ω()|||n|χk()Lp()(n)|b(0)b(y)||bj(y)|:=ϒ1+ϒ2.

For ϒ1, 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()(n)||Ω(y)||y|n|Ω()|||n|χk()Ls()(n)|b(0)b(y)||bj(y)|Lp()(n)CbLipγ2kγ||Ω(y)||y|n|Ω()|||n|χk()Ls()(n)χBKLp()(n).

When |Bk|2n and xkBk, by Lemma 2.12 we have

χBKLp()(n)|Bk|1p(xk)χBKLp()(n)|Bk|1sγn.

When |Bk|1 we have

χBKLp()(n)|Bk|1p()χBKLp()(n)|Bk|1sγn.

So we obtain

χBKLp()(n)χBKLp()(n)|Bk|1sγn.

By lemma 2.9 we have

||Ω(y)||y|n|Ω()|||n|χk()Ls()(n)2(k1)(nsn){|y|2k+|y|2k1|y|2kωs(δ)δdδ}2(k1)(nsn){2jk+1+2(jk+1)γ10ωs(δ)δdδ}2(k1)(nsn)2(jk)γ.

Now, by using the generalized Hölder’s inequality we get:

ϒ*1Bj||Ω(y)||y|n|Ω()|||n||b()b(y)|χk()Lp()(n)|bj(y)|dyCbLipγ2kn+(jk)γkγχBKLp()(n)|Bk|1sγnBj|bj(y)|dyCbLipγ2kn+(jk)γkγχBKLp()(n)bjLp()(n)χBjLp()(n). (3.2)

For ϒ*2 similar to the method of ϒ*1 we have

||Ω(y)||y|n|Ω()|||n|χk()Lp()(n)||Ω(y)||y|n|Ω()|||n|χk()Ls()(n)χk()Lp()(n)||Ω(y)||y|n|Ω()|||n|χk()Ls()(n)χBk()Lp()(n)2(k1)(nsn)2(jk)γχBKLp()(n)2kn+(jk)γkγχBkLp()(n).

Now, by using the generalized Hölder’s inequality we get:

ϒ*2Bj||Ω(y)||y|n|Ω()|||n|χk()Lp()(n)|b(0)b(y)||bj(y)|dyCbLipγ2kn+2(jk)γχBkLp()(n)bjLp()(n)χBjLp()(n)CbLipγ2kn+(jk)γχBkLp()(n)bjLp()(n)χBjLp()(n). (3.3)

Now by (3.3), (3.4), and lemmas 2.5 and 2.6, we have

[b,μΩ](bj)χkLp()(n)CbLipγ2kn+(jk)γχBkLp()(n)bjLp()(n)χBjLp()(n)CbLipγ2(jk)γbjLp()(n)χBjLp()(n)χBkLp()(n)C2jα+(jk)(γ+nδ2)bLipγ.

Firstly we estimate I. We need to show that there exists a positive constant C, such that ICΔ, we consider

I=supL0,L2LλqLk=2kqα(0)|λj|[b,μΩ](f)χkqLp()(n)supL0,L2LλqLk=2kqα(0)(j=K|λj|[b,μΩ]χkLp()(n))q+supL0,L2LλqLk=2kqα(0)(k1j=|λj|[b,μΩ]χkLp()(n))q:=I1+I2.

By the (Lp()(n),Lq()(n)), bounbedness of the commutator [b,μΩ] on Lp() (see [15] ), we have the following. Therefore, when 0<q1

I1=supL0,L2LλqLk=2kqα(0)(j=K[b,μΩ]χkLp()(n))qsupL0,L2LλqLk=2kqα(0)(j=k|λj|2jαj)qsupL0,L2LλqLk=2kqα(0)(1j=k|λj|2jα(0)jq+j=0|λj|2jαq)supL0,L2LλqLk=1j=k|λj|q2α(0)(kj)q+supL0,L2LλqLk=2kqα(0)j=0|λj|q2jαq

supL0,L2Lλq1j=|λj|qjk=2α(0)(kj)q+supL0,Lj=02jλq|λj|q2(λα)jq2LλqLk=2kqα( 0 )

sup L 0 , L 2 L λ q j = L | λ j | q + sup L 0 , L 2 L λ q j = L 1 | λ j | q k = j 2 α ( 0 ) ( k j ) q + Δ sup L 0 , L j = 0 2 ( λ α ) j q k = L 2 ( α ( 0 ) k L λ ) q Δ + sup L 0 , L j = L 1 2 j λ q | λ j | q 2 ( j L ) λ q k = j 2 α ( 0 ) ( k j ) q + Δ Δ + sup L 0 , L j = L 1 2 j λ q | λ j | q 2 ( j L ) λ q k = j 2 α ( 0 ) ( k j ) q Δ . (3.4)

When 0 < q < , let 1 / q + 1 / q = 1 we have

I 1 = sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = K | λ j | [ b , μ Ω ] χ k L p ( ) ( n ) ) q sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k | λ j | 2 j α ) q sup L 0 , L 2 L λ q k = L ( j = k 1 | λ j | 2 α ( 0 ) ( k j ) ) q + sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = 0 | λ j | 2 j α ) q

sup L 0 , L 2 L λ q k = L ( j = k 1 | λ j | q 2 α ( 0 ) ( k j ) q / 2 ) ( j = k 1 2 α ( 0 ) ( k j ) q / 2 ) q / q + sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = 0 | λ j | q 2 j α q / 2 ) ( j = 0 2 j α q / 2 ) q / q sup L 0 , L 2 L λ q k = L j = k 1 | λ j | q 2 ( j k ) q / 2 + sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) j = 0 | λ j | q 2 j α q / 2

sup L 0 , L 2 L λ q j = 1 | λ j | q k = j 2 α ( 0 ) ( j k ) q / 2 + sup L 0 , L j = 0 2 j λ q | λ j | q 2 ( λ α / 2 ) j q 2 L λ q k = L 2 k q α ( 0 ) sup L 0 , L 2 L λ q j = L | λ j | q + sup L 0 , L 2 L λ q j = L 1 | λ j | q j = j 2 α ( 0 ) ( k j ) q / 2 + Δ sup L 0 , L j = 0 2 ( λ α / 2 ) j / 2 q k = L 2 k q α ( 0 ) L λ q

Δ + sup L 0 , L j = L 1 2 j λ q | λ j | q 2 ( j L ) λ q j = j 2 α ( 0 ) ( k j ) q / 2 + Δ Δ + Δ sup L 0 , L j = L 1 2 ( j L ) λ q j = j 2 α ( 0 ) ( k j ) q / 2 Δ . (3.5)

We estimate I 2 by lemma 2.1 when 0 < q 1 by n δ 2 α ( 0 ) < γ + n δ 2 , we get

I 2 = sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k 1 | λ j | [ b , μ Ω ] χ k L p ( ) ( n ) ) q C b L i p γ q sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k 1 2 j α + ( j k ) ( γ + n δ 2 ) | λ j | ) p C b L i p γ q sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k 1 | λ j | p 2 ( j α + ( j k ) ( γ + n δ 2 ) ) q ) C b L i p γ q sup L 0 , L 2 L λ q j = L | λ j | p k = j + 1 1 2 q ( j k ) [ γ + n δ 2 α ( 0 ) ] C b L i p γ q Δ . (3.6)

When 0 < q < , let 1 / q + 1 / q = 1 . Since n δ 2 α ( 0 ) < γ + n δ 2 , by Hölder’s inequality, we have

I 2 = sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k 1 | λ j | [ b , μ Ω ] χ k L p ( ) ( n ) ) q C b L i p γ q sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k 1 2 j α + ( j k ) ( γ + n δ 2 ) | λ j | ) p C b L i p γ q sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k 1 | λ j | p 2 ( j α + ( j k ) ( γ + n δ 2 ) ) q / 2 ) × ( j = k 1 2 ( j α + ( j k ) ( γ + n δ 2 ) ) q / 2 ) q / q

C b L i p γ q sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k 1 | λ j | p 2 ( j α + ( j k ) ( γ + n δ 2 ) ) q / 2 ) C b L i p γ q sup L 0 , L 2 L λ q j = L | λ j | p k = j + 1 1 2 q / 2 ( j k ) [ γ + n δ 2 α ( 0 ) ] C b L i p γ q Δ . (3.7)

Secondly we estimate I I . We need to show that there exists a positive constant C, such that I I C Δ , we consider

I I = k = 1 2 k q α ( 0 ) [ b , μ Ω ] ( f ) χ k L p ( ) q k = 1 2 k q α ( 0 ) ( j = k | λ j | [ b , μ Ω ] ( b j ) χ k L p ( ) ) q + k = 1 2 k q α ( 0 ) ( j = k 1 | λ j | [ b , μ Ω ] ( b j ) χ k L p ( ) ) q : = I I 1 + I I 2 .

When 0 < q 1 , we get

I I 1 = k = 1 2 k q α ( 0 ) ( j = k | λ j | [ b , μ Ω ] ( b j ) χ k L p ( ) ) q k = 1 2 k q α ( 0 ) ( j = k 1 2 j α j | λ j | ) p

k = 1 2 k q α ( 0 ) ( j = k 1 | λ j | q 2 j q α ( 0 ) + j = 0 | λ j | q 2 j q α ) k = 1 j = k 1 | λ j | q 2 α ( 0 ) ( k j ) q + k = 1 2 k q α ( 0 ) j = 0 | λ j | q 2 j α q

k = 1 | λ j | q k = j 2 q α ( 0 ) ( k j ) + j = 0 | λ j | q 2 j q α k = 1 2 k q α ( 0 ) k = 1 | λ j | q + j = 0 2 j λ q | λ j | q 2 j q α k = 1 2 k q α ( 0 ) Δ + Δ i = j | λ i | q j = 0 2 ( λ α ) j q k = j 2 k q α ( 0 ) Δ . (3.8)

When 0 < q < , let 1 / q 1 + 1 / q 1 = 1 we have

I I 1 = k = 1 2 k q α ( 0 ) ( j = k | λ j | [ b , μ Ω ] ( b j ) χ k L p ( ) ) q k = 1 2 k q α ( 0 ) ( j = k | λ j | 2 j α ) q k = 1 ( j = k 1 | λ j | 2 α ( 0 ) ( j k ) ) q + k = 1 2 k q α ( 0 ) ( j = 0 | λ j | 2 j α ) q k = 1 ( j = k 1 | λ j | q 2 q / 2 α ( 0 ) ( j k ) ) ( j = k 1 2 α ( 0 ) ( j k ) q / 2 ) q / q + k = 1 2 k q α ( 0 ) ( j = 0 | λ j | q 2 q / 2 j α ) ( j = 0 2 q / 2 j α ) q / q

k = 1 | λ j | q k = j 2 q / 2 α ( 0 ) ( j k ) + k = 1 2 k q α ( 0 ) j = 0 | λ j | q 2 q / 2 j α k = 1 | λ j | q + j = 0 2 ( λ α / 2 ) j q 2 λ j q i = j | λ i | q k = 1 2 k q α ( 0 ) Δ + Δ j = 0 2 ( λ α / 2 ) j q k = 1 2 k q α ( 0 ) Δ . (3.9)

For I I 2 , when 0 < q 1 , by n δ 2 α ( 0 ) < γ + n δ 2 we get

I I 2 = k = 1 2 k q α ( 0 ) ( j = k 1 | λ j | [ b , μ Ω ] ( b j ) χ k L p ( ) ) q k = 1 2 k q α ( 0 ) ( C b L i p γ j = k 1 | λ j | 2 j α + ( j k ) ( γ + n δ 2 ) ) q C b L i p γ q k = 1 2 k q α ( 0 ) ( j = k 1 | λ j | q 2 [ j α + ( j k ) ( γ + n δ 2 ) ] q ) C b L i p γ q j = 1 | λ j | q k = j + 1 1 2 [ j α + ( j k ) ( γ + n δ 2 ) ] q C b L i p γ q Δ . (3.10)

When 1 < q < , let 1 / q + 1 / q = 1 . Since n δ 2 α ( 0 ) < γ + n δ 2 , by Hölder’s inequality, we have

I I 2 = k = 1 2 k q α ( 0 ) ( j = k 1 | λ j | [ b , μ Ω ] ( b j ) χ k L p ( ) ) q k = 1 2 k q α ( 0 ) ( C b L i p γ j = k 1 | λ j | 2 j α + ( j k ) ( γ + n δ 2 ) ) q C b L i p γ q k = 1 2 k q α ( 0 ) ( j = k 1 | λ j | q 2 [ j α + ( j k ) ( γ + n δ 2 ) ] q / 2 ) × ( j = k 1 | λ j | q 2 [ j α + ( j k ) ( γ + n δ 2 ) ] q / 2 ) q / q

C b L i p γ q k = 1 2 k q α ( 0 ) ( j = k 1 | λ j | q 2 [ j α + ( j k ) ( γ + n δ 2 ) ] q / 2 ) C b L i p γ q j = 1 | λ j | q k = j + 1 1 2 ( j k ) [ γ + n δ 2 α ( 0 ) ] q / 2 C b L i p γ q Δ . (3.11)

Thirdly, we estimate I I I , we need to show that there exists a positive constant C, such that I I I C Δ

I I I = sup L 0 , L 2 L λ q k = 0 L 2 k q α ( ) [ b , μ Ω ] ( f ) χ k L p ( ) q sup L 0 , L 2 L λ q k = 0 L 2 k q α ( ) ( j = k | λ j | [ b , μ Ω ] ( b j ) χ k L p ( ) ) q + sup L 0 , L 2 L λ q k = 0 L 2 k q α ( ) ( j = k 1 | λ j | [ b , μ Ω ] L p ( ) χ k L p ( ) ) q : = I I I 1 + I I I 2 .

When 0 < q 1 , by the boundedness of [ b , μ Ω ] in L p ( ) ( [20] ), we have

I I I 1 = sup L 0 , L 2 L λ q k = 0 L 2 k q α ( j = k | λ j | [ b , μ Ω ] ( b j ) χ k L p ( ) ) q sup L 0 , L 2 L λ q k = 0 L 2 k q α j = k | λ j | q [ b , μ Ω ] ( b j ) χ k L p ( ) q sup L 0 , L 2 L λ q k = 0 L 2 k q α j = k | λ j | q 2 α j j q sup L 0 , L 2 L λ q k = 0 L 2 k q α j = k | λ j | q 2 α j q = sup L 0 , L 2 L λ q j = 0 L | λ j | q k = 0 j 2 α ( k j ) q + sup L 0 , L 2 L λ q j = L | λ j | q k = 0 L 2 α ( k j ) q

sup L 0 , L 2 L λ q k = 0 L | λ j | q + sup L 0 , L j = L 2 ( j L ) q λ 2 j λ q i = j | λ i | q k = 0 L 2 α ( k j ) q Δ + Δ sup L 0 , L j = L 2 ( j L ) q λ 2 j λ q 2 α ( L j ) q Δ + Δ sup L 0 , L j = L 2 ( j L ) q ( λ α ) Δ . (3.12)

When 0 < q , by n δ 2 α ( 0 ) , α < γ + n δ 2 and the boundedness of [ b , μ Ω ] in L p ( ) ( [20] ) and Hölder’s inequality, we get

I I I 1 = sup L 0 , L 2 L λ q k = 0 L 2 k q α ( ) ( j = k | λ j | [ b , μ Ω ] ( b j ) χ k L p ( ) ) q sup L 0 , L 2 L λ q k = 0 L 2 k q α ( j = k | λ j | q [ b , μ Ω ] ( b j ) χ k L p ( ) q / 2 ) × ( j = k [ b , μ Ω ] ( b j ) χ k L p ( ) q / 2 ) q / q sup L 0 , L 2 L λ q k = 0 L 2 k q α ( j = k | λ j | q b j L p ( ) q / 2 ) ( j = k b j L p ( ) q / 2 ) q / q

sup L 0 , L 2 L λ q k = 0 L 2 k q α ( j = k | λ j | q | B j | α j q / ( 2 n ) ) ( j = k | B j | α j q / ( 2 n ) ) q / q sup L 0 , L 2 L λ q k = 0 L 2 k q α ( j = k | λ j | q | B j | α j q / ( 2 n ) ) sup L 0 , L 2 L λ q k = 0 L 2 α k q / 2 ( j = k | λ j | q | B j | α j q / ( 2 n ) ) = sup L 0 , L 2 L λ q j = 0 L | λ j | q k = 0 j 2 ( k j ) α q / 2 + sup L 0 , L 2 L λ q j = L | λ j | q k = 0 L 2 ( k j ) α q / 2

sup L 0 , L 2 L λ q j = 0 L | λ j | q + sup L 0 , L j = L 2 ( j L ) λ q 2 j λ q i = j | λ j | q k = 0 L 2 ( k j ) α q / 2 Δ + Δ sup L 0 , L j = L 2 ( j L ) λ q 2 ( L j ) α q / 2 Δ + Δ sup L 0 , L j = L 2 ( j L ) q ( α α / 2 ) Δ . (3.13)

When 0 < q 1 , by n δ 2 α ( 0 ) , α < γ + n δ 2 we get

I I I 2 = sup L 0 , L 2 L λ q k = 0 L 2 k q α ( j = k 1 | λ j | [ b , μ Ω ] ( b j ) χ k L p ( ) ) q sup L 0 , L 2 L λ q k = 0 L 2 k q α ( C b L i p γ q j = k 1 | λ j | q 2 [ j α j + ( j k ) ( γ + n δ 2 ) ] q ) = C b L i p γ q sup L 0 , L 2 L λ q k = 0 L 2 k q α ( j = 1 | λ j | q 2 [ j α ( 0 ) + ( j k ) ( γ + n δ 2 ) ] q ) + C b L i p γ q sup L 0 , L 2 L λ q k = 0 L 2 k q α ( ) ( j = 0 k 1 | λ j | q 2 [ j α + ( j k ) ( γ + n δ 2 ) ] q )

C b L i p γ q sup L 0 , L 2 L λ q k = 0 L 2 k q [ α + γ + n δ 2 ] j = 1 | λ j | q 2 [ γ + n δ 2 + α ( 0 ) ] j q + C b L i p γ q sup L 0 , L 2 L λ q j = 0 L | λ j | q k = j + 1 2 [ γ + n δ 2 α ] ( j k ) q C b L i p γ q sup L 0 , L 2 L λ q j = 1 | λ j | q + C b L i p γ q sup L 0 , L 2 L λ q j = 0 L 1 | λ j | q C b L i p γ q Δ . (3.14)

When 1 < q < , let 1 / q + 1 / q = 1 . Since n δ 2 α ( 0 ) , α < γ + n δ 2 , and by Hölder’s inequality, we have

I I I 2 = sup L 0 , L 2 L λ q k = 0 L 2 k q α ( ) ( j = k 1 | λ j | [ b , μ Ω ] ( b j ) χ k L p ( ) ) q sup L 0 , L 2 L λ q k = 0 L 2 k q α ( ) ( j = 1 C b L i p γ q | λ j | 2 [ j α j + ( j k ) ( γ + n δ 2 ) ] ) q C b L i p γ q sup L 0 , L 2 L λ q k = 0 L 2 k q α ( ) ( j = 1 | λ j | 2 [ j α ( 0 ) + ( j k ) ( γ + n δ 2 ) ] ) q + C b L i p γ q sup L 0 , L 2 L λ q k = 0 L 2 k q α ( ) ( j = 0 k 1 | λ j | 2 [ j α + ( j k ) ( γ + n δ 2 ) ] ) q

C b L i p γ q sup L 0 , L 2 L λ q k = 0 L 2 k q [ α ( γ + n δ 2 ) ] ( j = 1 | λ j | 2 [ ( γ + n δ 2 ) α ( 0 ) ] j ) q + C b L i p γ q sup L 0 , L 2 L λ q k = 0 L ( j = 0 k 1 | λ j | 2 ( j k ) [ γ + n δ 2 α ] ) q ( C b L i p γ q sup L 0 , L 2 L λ q j = 1 | λ j | q 2 [ ( γ + n δ 2 ) α ( 0 ) ] j q / 2 ) × ( j = 1 | λ j | q 2 [ ( γ + n δ 2 ) α ( 0 ) ] j q / 2 ) q / q

+ C b L i p γ q sup L 0 , L 2 L λ q k = 0 L ( j = 0 k 1 | λ j | q 2 ( j k ) [ γ + n δ 2 α ] q / 2 ) × ( j = 0 k 1 | λ j | 2 ( j k ) [ γ + n δ 2 α ] q / 2 ) q / q C b L i p γ q sup L 0 , L 2 L λ q j = 1 | λ j | q 2 [ ( γ + n δ 2 ) α ( 0 ) ] j q / 2 + C b L i p γ q sup L 0 , L 2 L λ q k = 0 L j = 0 k 1 | λ j | q 2 ( j k ) [ γ + n δ 2 α ] q / 2

C b L i p γ q sup L 0 , L 2 L λ q j = 1 | λ j | q + C b L i p γ q sup L 0 , L 2 L λ q j = 0 L 1 | λ j | q k = j + 1 L 2 ( j k ) [ γ + n δ 2 α ] q / 2 C b L i p γ q sup L 0 , L 2 L λ q j = 1 | λ j | q + C b L i p γ q sup L 0 , L 2 L λ q j = 0 L 1 | λ j | q C b L i p γ q Δ . (3.15)

Joint the estimates for I, II and III, we obtain

[ b , μ Ω ] ( f ) M K p ( ) , λ α ( ) , q ( n ) q C b L i p γ q f H M K p ( ) , λ α ( ) , q .

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 b B M O ( n ) .

Theorem 4.1.

Suppose that b B M O ( n ) with 0 < γ 1 . If p ( ) P ( n ) satisfies proposition 2.3 and Ω L s ( S n 1 ) ( s > q ) . Let 0 < p 1 p 2 < and

0 < λ < α < n δ 2 γ n s (or 0 < λ < α 1 α 1 < n δ 2 γ n s ). Then [ b , μ Ω ] is bounded from H M K ˙ p ( ) , λ α ( ) , q ( n ) (or H M K p ( ) , λ α ( ) , q ( n ) ) to M K ˙ p ( ) , λ α ( ) , q ( n ) (or M K p ( ) , λ α ( ) , q ( n ) ).

proof:

In a way similar to theorem (3.2) we only prove the homogeneous case. Let b B M O ( n ) and f H M K p ( ) , λ α ( ) , q ( n ) . Let us write

f ( x ) = j = 0 f ( x ) χ j ( x ) = j = 0 f j ( x ) .

Then we have

[ b , μ Ω ] ( f ) M K p ( ) , λ α ( ) , q ( n ) q = max { sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) [ b , μ Ω ] ( f ) χ k L p ( ) q , sup L 0 , L Z 2 L λ q ( k = 1 2 k q α ( 0 ) [ b , μ Ω ] ( f ) χ k L p ( ) q + k = 0 L 2 k q α ( ) [ b , μ Ω ] ( f ) χ k L p ( ) q ) } : = max { H , H H + H H H } .

H = sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) [ b , μ Ω ] ( f ) χ k L p ( ) q , H H = k = 1 2 k q α ( 0 ) [ b , μ Ω ] ( f ) χ k L p ( ) q , H H H = sup L 0 , L 2 L λ q k = 0 L 2 k q α ( ) [ b , μ Ω ] ( f ) χ k L p ( ) q .

From the Hölder’s inequality, we have

| [ b , μ Ω ] ( b j ) χ k | L p ( ) ( n ) C B j | Ω ( x y ) | | x y | n | b ( x ) b ( y ) | | f j ( y ) | d y C 2 k n B j | Ω ( x y ) | | b ( x ) b ( y ) | | f j ( y ) | d y

C 2 k n ( | b ( x ) b B j | B j | Ω ( x y ) | | f j ( y ) | d y + B j | Ω ( x y ) | | b B j b ( y ) | | f j ( y ) | d y ) C 2 k n ( | b ( x ) b B j | Ω ( x ) χ j ( ) L q ( ) ( n ) f j L q ( ) ( n ) d y + Ω ( x ) ( b B j b ( ) ) χ j ( ) L q ( ) ( n ) f j L q ( ) ( n ) d y ) .

Noting s > q , we denote q ˜ ( ) > 1 and 1 q ( x ) = 1 q ˜ ( x ) + 1 s . By lemmas 3.2, 3.10 we have

Ω ( x ) χ j ( ) L q ( ) ( n ) Ω ( x ) χ j ( ) L s ( ) ( n ) χ j ( ) L q ˜ 2 ( ) ( n ) Ω ( x ) χ j ( ) L s ( ) ( n ) χ B j ( ) L q ˜ 2 ( ) ( n ) 2 j γ ( A j | y | s γ | Ω ( x y ) | s d y ) 1 s χ B j ( ) L q ˜ 2 ( ) ( n ) C 2 j γ | 2 | k ( γ + n s ) Ω L s ( S n 1 ) χ B j ( ) L q ˜ 2 ( ) ( n ) .

By lemma (2.12), when | B j | 2 n , x j B j and when | B k | 1 respectively we have

χ B j L q ˜ ( ) ( n ) | B k | 1 q ˜ ( x k ) χ B j L q ( ) ( n ) | B j | 1 s ,

and

χ B j L q ˜ ( ) ( n ) | B j | 1 q ˜ ( ) χ B j L q ( ) ( n ) | B j | 1 s ,

we obtain χ B j L q ˜ ( ) ( n ) χ B j L q ( ) ( n ) | B j | 1 s .

So we have

Ω ( x ) χ j ( ) L q ( ) ( n ) C 2 ( k j ) ( γ + n s ) Ω L s ( S n 1 ) χ B j ( ) L q ( ) ( n ) . (4.1)

Similarly by lemma 2.4 we have

Ω ( x ) ( b B j b ( ) ) χ j ( ) L q ( ) ( n ) Ω ( x ) χ j ( ) L s ( n ) ( b B j b ( ) ) χ j ( ) L q ˜ ( ) ( n ) C b χ B j ( ) L q ˜ 2 ( ) ( n ) Ω ( x ) χ j ( ) L s ( n ) C b 2 ( k j ) ( γ + n s ) Ω L s ( S n 1 ) χ B j L q ( ) ( n ) . (4.2)

Now, by (4.1), (4.2), lemmas 2.4, 2.5 and 2.3, we have

[ b , μ Ω ] ( f j ) χ k L q ( ) ( n ) C 2 k n ( 2 ( k j ) ( γ + n s ) Ω L s ( S n 1 ) χ B j L q ( ) ( n ) ( b ( ) b B j ) χ k ( ) L q ( ) ( n ) f j L q ( ) ( n ) + b 2 ( k j ) ( γ + n s ) Ω L s ( S n 1 ) χ B j L q ( ) ( n ) χ k L q ( ) ( n ) f j L q ( ) ( n ) d y ) C 2 k n ( ( k j ) b 2 ( k j ) ( γ + n s ) Ω L s ( S n 1 ) χ B j L q ( ) ( n ) χ B k L q ( ) ( n ) f j L q ( ) ( ℝ n )

+ b 2 ( k j ) ( γ + n s ) Ω L s ( S n 1 ) χ B j L q ( ) ( n ) χ B k L q ( ) ( n ) f j L q ( ) ( n ) d y ) C ( k j ) b 2 k n 2 ( k j ) ( γ + n s ) Ω L s ( S n 1 ) χ B j L q ( ) ( n ) χ B k L q ( ) ( n ) f j L q ( ) ( n ) C ( k j ) b 2 ( k j ) ( γ + n s ) Ω L s ( S n 1 ) f j L q ( ) ( n ) χ B j L q ( ) ( n ) χ B k L q ( ) ( n ) C b ( k j ) 2 ( k j ) ( n δ 2 γ n s ) Ω L s ( S n 1 ) f j L q ( ) ( n ) . (4.3)

By the boundedness of μ Ω in L p ( ) see [7], we have

( μ Ω f j ) χ k L p ( ) f j L p ( ) | B j | α j / n = 2 j α j .

So we have

[ b , μ Ω ] ( f j ) χ k L q ( ) ( n ) b ( k j ) Ω L s ( S n 1 ) 2 ( k j ) ( n δ 2 γ n s ) j α j .

Firstly we estimate H. We need to show that there exists a positive constant C, such that H C Δ Consider

H = sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) [ b , μ Ω ] ( f ) χ k L p ( ) q sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k | λ j | [ b , μ Ω ] ( f j ) χ k L p ( ) q ) q + sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k 1 | λ j | [ b , μ Ω ] ( f j ) χ k L p ( ) q ) q : = H 1 + H 2 .

By boundedness of [ b , μ Ω ] in L p ( ) , see ( [20] ), when 0 < q 1 we have

H 1 = sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k | λ j | [ b , μ Ω ] ( f j ) χ k L p ( ) q ) q sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k | λ j | 2 j α j ) q sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k 1 | λ j | 2 j α ( 0 ) j q + j = 0 | λ j | 2 j α q ) sup L 0 , L 2 L λ q k = L j = k 1 | λ j | q 2 α ( 0 ) ( k j ) q + sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) j = 0 | λ j | q 2 j α q

sup L 0 , L 2 L λ q j = 1 | λ j | q k = j 2 α ( 0 ) ( k j ) q + sup L 0 , L j = 0 2 j λ q | λ j | q 2 ( λ α ) j q 2 L λ q k = L 2 k q α ( 0 )

sup L 0 , L 2 L λ q j = L | λ j | q + sup L 0 , L 2 L λ q j = L 1 | λ j | q k = j 2 α ( 0 ) ( k j ) q + Δ sup L 0 , L j = 0 2 ( λ α ) j q k = L 2 ( α ( 0 ) k L λ ) q Δ + sup L 0 , L j = L 1 2 j λ q | λ j | q 2 ( j L ) λ q k = j 2 α ( 0 ) ( k j ) q + Δ Δ + sup L 0 , L j = L 1 2 j λ q | λ j | q 2 ( j L ) λ q k = j 2 α ( 0 ) ( k j ) q Δ . (4.4)

When 1 < q < and 1 / q + 1 / q = 1 , and let γ + n δ 2 α > 0 , we have

H 1 = sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k | λ j | [ b , μ Ω ] ( f j ) χ k L p ( ) q ) q sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = k | λ j | 2 j α ) q sup L 0 , L 2 L λ q k = L ( j = k 1 | λ j | 2 α ( 0 ) ( k j ) ) q + sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = 0 | λ j | 2 j α ) q

sup L 0 , L 2 L λ q k = L ( j = k 1 | λ j | q 2 α ( 0 ) ( k j ) q / 2 ) ( j = k 1 2 α ( 0 ) ( k j ) q / 2 ) q / q + sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) ( j = 0 | λ j | q 2 j α q / 2 ) ( j = 0 2 j α q / 2 ) q / q sup L 0 , L 2 L λ q k = L j = k 1 | λ j | q 2 ( j k ) q / 2 + sup L 0 , L 2 L λ q k = L 2 k q α ( 0 ) j = 0 | λ j | q 2 j α q / 2

sup L 0 , L 2 L λ q j = 1 | λ j | q k = j 2 α ( 0 ) ( j k ) q / 2 + sup L 0 , L j = 0 2 j λ q | λ j | q 2 ( λ α / 2 ) j q 2 L λ q k = L 2 k q α ( 0 ) sup L 0 , L 2 L λ q j = L | λ j | q + sup L 0 , L 2 L λ q j = L 1 | λ j | q j = j 2 α ( 0 ) ( k j ) q / 2 + Δ sup L 0 , L j = 0 2 ( λ α / 2 ) j / 2 q k = L 2 k q α ( 0 ) L λ q

Δ + sup L 0 , L j = L 1 2 j λ q | λ j | q 2 ( j L ) λ q j = j 2 α ( 0 ) ( k j ) q / 2 + Δ Δ + Δ sup L 0 , L j = L 1 2 ( j L ) λ q j = j 2 α ( 0 ) ( k j ) q / 2

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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