# Commutators of the Fractional Hardy Operator on Weighted Variable Herz-Morrey Spaces

Authors: Amjad Hussain; Muhammad Asim; Muhammad Aslam; Fahd Jarad
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In the present paper, our aim is to establish the boundedness of commutators of the fractional Hardy operator and its adjoint operator on weighted Herz-Morrey spaces with variable exponentsMK̇p,q⋅α⋅,λw.


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## Body

## 1. Introduction

Hardy operators and related commutators play an indispensable role in the theory of partial differential equations [1, 2] and the characterization of function spaces [3–5]. Without going into much details, let us first define the fractional Hardy operators [3]
(1)Hgz=1zn−β∫t≤zgtdt,H∗gz=∫t>zgttn−βdt,z∈ℝn/0and related commutators:
(2)b,Hβg=bHg−Hbg,b,Hβ∗g=bH∗g−H∗bg.It is important to note that takingβ=0 in (1), we get multidimensional Hardy operator defined and studied in [6, 7]. Also, (1) reduces to the one dimensional Hardy operator [8] if we choose β=0 and n=1. Here, we cite some important literature with regards to the study of Hardy-type operators on different function spaces which include [9–15].The new development of variable exponent commenced with the work of Kov’aˇcik and R’akosn’ık in [16], where a class of function spaces having variable exponent was defined, and basic properties of variable exponent Lebesgue space were explored. Recently, the theory of variable exponent analysis is modeled in terms of the boundedness of the Hardy Littlewood maximal operator M [17–21]:
(3)Mgz=supB:ball,z∈B1B∫B∫Bgtdt.Besides, MuckenhouptAp theory [22] is generalized in the recent span of time with regard to variable exponent spaces ([23–28]). By taking into account the generalization of function spaces with variable exponents and the same with weights, many results like duality, boundedness of sublinear operators, the wavelet characterization, and commutators of fractional and singular integrals have been studied [29–38].Recently, authors have studied generalized Herz space in terms of both Muckenhoupt weights and variable exponent [39–41]. Moreover, an idea of combining two function spaces to develop a new one is also an interesting problem in Harmonic analysis. One such problem is considered in [42] in which Herz-Morrey space was defined. Although, the weighted versions of Herz-Morrey spaces were introduced recently in [43, 44].In this piece of work, our main focus is on establishing the boundedness of commutators of fractional Hardy operators on a class of function spaces called the weighted Herz-Morrey space with variable exponents. We seek to find the boundedness of these commutators with symbol functions in BMO (bounded mean oscillation) spaces. In establishing such a boundedness, we make use of the boundedness of the fractional integral operatorIβ(4)Iβgz=∫ℝngtz−tn−βdton weighted Lebesgue space which was done in [39].In the rest of this paper, the symbolC expresses a constant whose value may differ at all of its occurrences. The Greek letter χS denotes the characteristics function of a sphere S where S is a measurable subset of Rn and ∣S∣ represents its Lebesgue measure. Before turning to our key results, let us first define the relevant variable exponent function spaces.


## 2. Preliminaries

Let us consider a measurable functionp· on ℝn having range 1,∞. The Lebesgue space with variable exponent Lp⋅ℝn is the set of all measurable function f such that
(5)Lp⋅ℝn=f:∫ℝnfxσpxdx<∞,forsomeσ>0.The spaceLp⋅ℝn turns out to be Banach function space under the norm:
(6)fLp⋅ℝn=infσ>0:∫ℝnfxσpxdx≤1.We denote byPℝn the set of all measurable functions p⋅:ℝn⟶1,∞ such that
(7)1<p−≤px≤p+<∞,where
(8)p−≔essinfx∈ℝnpx,p+≔esssupx∈ℝnpx.Definition 1.
Supposep· is a real valued function on ℝn. We say that
(i)
Cloclogℝn is the set of all local log-Holder continuous functions p· satisfying(9)px−py≲−Clogx−y,x−y<12,x,y∈ℝn.(ii)
C0logℝn is the set of all local log-Holder continuous function p· satisfying at the origin(10)px−p0≲Cloge+1/x,x−y<12,x∈ℝn.(iii)
C∞logℝn is the set of all log-Holder continuous functions satisfying at infinity(11)px−p∞≤C∞loge+x,x∈ℝn.(iv)
Clogℝn=C∞log∩Cloclog denotes the set of all global log-Holder continuous functions p·.It was proved in [21] that if p⋅∈Pℝn∩Clogℝn, then Hardy-Littlewood maximal operator M is bounded on Lp⋅ℝn.Supposewx is a weight function on ℝn, which is nonnegative and locally integrable on ℝn. Let Lp⋅w be the space of all complex-valued functions f on ℝn such thatfw1/p⋅∈Lp⋅ℝn. The space Lp⋅w is a Banach function space equipped with the norm:
(12)fLp⋅w=fw1p⋅Lp⋅.Benjamin Muckenhoupt introduced the theory ofAp1<p<∞ weights on ℝn in [22]. Recently, in [39, 40], Izuki and Noi generalized the Muckenhoupt Ap class by taking p as a variable.Definition 2.
Letp⋅∈Pℝn. A weight w is an Ap⋅ weight if
(13)supB1Bw1/p⋅χBLp⋅w−1/p⋅χBLp′⋅<∞.
In [25], the authors proved that w∈Ap⋅ if and only if M is bounded on the space Lp⋅.Remark 3 (see [39]).
Supposep⋅,q⋅∈Pℝn∩Clogℝn and p·≤q·, then we have
(14)A1⊂Ap⋅⊂Aq⋅.Definition 4.
Supposep1⋅,p2⋅∈Pℝn and β∈0,n such that 1/p2x=1/p1x−β/n. A weight w is said to be Ap1⋅,p2⋅ weight if
(15)χBLp2⋅wp2⋅χBLp1⋅wp1⋅′≤CB1−βn.Definition 5 (see [39]).
Supposep1⋅,p2⋅∈Pℝn and β∈0,n such that 1/p2x=1/p1x−β/n. Then, w∈Ap1⋅,p2⋅ if and only if wp2⋅∈A1+p2⋅/p1′⋅.Now, we define the variable exponent weighted Morrey-Herz spaceMK̇q,p⋅α⋅,λw. Let Bk=x∈ℝn:x≤2k, Ak=Bk/Bk−1, and χk=χAk for k∈ℤ.Definition 6.
Letw be a weight on ℝn,λ∈0,∞,q∈0,∞,p⋅∈Pℝn and α⋅:ℝn⟶ℝ with α⋅∈L∞ℝn. The space MK̇q,p⋅α⋅,λw is the set of all measurable functions which is given by
(16)MK̇q,p⋅α⋅,λw=f∈Llocp⋅ℝn/0,w:fMK̇q,p⋅α⋅,λw<∞,where
(17)fMK̇q,p⋅α⋅,λw=supk0∈Z2−k0λ∑k=−∞k02kα⋅qfχkLp⋅wq1/q.
Obviously,MK̇q,p⋅α⋅,0w=K̇q,p⋅α⋅w is the weighted Herz space with variable exponent (see [30]). Here, it is important to refer to some of the pioneering studies of the Herz space with constant exponents made in [45, 46].


## 3. Some Useful Lemmas

We start this section with some useful lemmas that will be helpful in proving our main results.Lemma 7 (see [47]).
IfX is Banach function space, then
(i)
The associated spaceX′ is also Banach function space(ii)
⋅X′′ and ⋅X are equivalent(iii)
Ifg∈X and f∈X′, then(18)∫ℝnfxgx≤gXfX′is the generalized Hölder inequality.Lemma 8 (see [39]).
SupposeX is a Banach function space. Then, we have that for all balls B,
(19)1≤1BχBXχBX′.Lemma 9 (see [28, 39]).
LetX be a Banach function space. Suppose that the Hardy Littlewood maximal operator M is weakly bounded on X; that is,
(20)χMf>σX≲σ−1fXis true for σ>0 and for all f∈X. Then, we have
(21)supB:ball1BχBXχBX′<∞.Lemma 10 (see [39, 48]).
(1)
Xℝn,W is Banach function space equipped with the norm(22)fXℝn,W=fwX,where
(23)Xℝn,W=f∈M:fW∈X:.(2)
The associate spaceX′ℝn,W−1 is also a Banach function spaceLemma 11 (see [39]).
LetX be a Banach function space. Assume that M is bounded on X′, then there exists a constant δ∈0,1 for all B⊂ℝn and E⊂B,
(24)χEXχBX≲EBδ.
The paper [16] shows that Lp⋅ℝn is a Banach function space and the associated space Lp′⋅Rn with equivalent norm.Remark 12.
Letp⋅∈Pℝn, and by comparing the Lebesgue space Lp⋅wp⋅ and Lp′⋅w−p′⋅ with the definition of Xℝn,W, we have
(1)
If we takeW=w and X=Lp⋅ℝn, then we get Lp⋅ℝn,w=Lp⋅wp⋅(2)
If we considerW=w−1 and X=Lp′⋅ℝn, then we have Lp′⋅w−p′⋅=Lp′⋅ℝn,w−1
By virtue of Lemma10, we get Lp⋅ℝn,w′=Lp⋅wp⋅′=Lp′⋅w−p′⋅=Lp′⋅ℝn.w−1. Next, in view of Lemma 11 and Remark 12, we have the following Lemma.Lemma 13 (see [41]).
Letp·∈PRn∩Clogℝn be a Log Hölder continuous function both at infinity and at origin, if wp2·∈Ap2· implies w−p2′∈Ap2′·. Thus, the Hardy Littlewood operator is bounded on Lp2′∙wp2′∙, and there exist constants δ1,δ2∈0,1 such that
(25)χELP2∙wp2·χBLP2∙wp2·=χELP2′∙w−P2′∙′χBLP2′∙w−P2′∙′≲EBδ1,χELP2∙wp2·′χBLP2∙wp2·′≲EBδ1,for all balls B and all measurable sets E⊂B.Lemma 14 (see [39]).
Letp1·∈Pℝn∩ClogRn and 0<β<n/p1+ and 1/p2·=1/p1∙−β/n. If ∈Ap1·,p2·, then Iβ is bounded from Lp1·wP1· to Lp2·wP2·.


## 4. Main Results and their Proofs

Definition 15.
Letf∈Lloc1Rn and set
(26)bBMO=supB∫Bbx−bBdx,where the supremum is taken all over the balls B∈ℝn and bB=B−1∫Bbydy. The function b is a bounded mean oscillation if bBMO<∞ and BMOℝn consist of all f∈Lloc1ℝn with BMOℝn<∞. For a comprehensive review of the BMO space, we suggest the reader to follow the books [49, 50].Lemma 16.
Letq·∈Pℝn and w be an Aq· weight. Then, for all b∈BMO and all l,i∈ℤ with l>i, we have
(27)bBMO∼supB:Ball1χBLq∙wq·b−bBχBLq∙wq·,(28)b−bBiχBlLq∙wq·≤Cl−ibBMOχBlLq∙wq·.Proof.
First part of this lemma is a consequence of [[41], Theorem 18]. Next, we will prove (28), for all l,i∈ℤ with l>i(29)b−bBiχBlLq∙wq·≤Cb−bBl+b−bBlχBlLq∙wq·≤Cb−bBlχBlLq∙wq·+bBl−bBiχBlLq∙wq·.
In the view of (27), we have
(30)b−bBiχBlLq∙wq·≤CbBMOχBlLq∙wq·.
Also, it is easy to see that(31)bBl−bBi≤∑n=1l−1bn+1−bn≤∑n=1l−11Bn∫Bnbn+1−bxdx≤C∑n=1l−11Bn+1∫Bnbn+1−bxdx=Cl−ibBMOℝn.
Combining (29), (30), and (31), we get (28).Proposition 17.
Letq·∈Pℝn, 0<p<∞, and 0≤λ<∞. If α·∈L∞ℝn∩Clogℝn, then
(32)fMK̇p,q·α·,λwq·p=supk0∈Z2−k0λp∑j=−∞k02jα·pfχjLq∙wq·p≤maxsupk0∈Zk0<02−k0λp∑j=−∞k02jα·pfχjLq∙wq·p,supk0∈Zk0<02−k0λp∑j=−∞−12jα·pfχjLq∙wq·p+2−k0λp∑j=0k02jα∞pfχjLq∙wq·p.Proof.
The proof is similar to the proof of Proposition17 in [44]. So, we omit the details.Theorem 18.
Let0<p1≤p2<∞, q2·∈Pℝn∩Clogℝn, and q1· be such that 1/q1·=1/q2∙−β/n..
Also, letwq2·∈A1, b∈BMOℝn, λ>0, and α·∈L∞ℝn∩Clogℝn be log Hölder continuous at the origin, with α0≤α∞<λ+nδ2−β, where 0<δ2<1, then
(33)b,HβfMK̇p2,q2·α·,λwq2·≤CbBMOfMK̇p1,q1·α·,λwq2·.Proof.
For anyf∈MK̇p1,q1·α·,λwq1·, if we denote fl=f·χl=f·χAl, and for each l∈ℤ,
(34)fx=∑1=−∞∞fx·χlx=∑1=−∞∞flx,then it is not difficult to see that
(35)b,Hβfxχjx≤1xn−β∫Bjbx−byfydy·χjx≤2−jn−β∑l=−∞j∫Bjbx−byfydy·χjx≤2−jn−β∑l=−∞j∫Bjbx−bBlfydy·χjx+2−jn−β∑l=−∞j∫Bjbx−bBlfydy·χjx=E1+E2.
The generalized Hölder inequality (Lemma7) yields the following inequality for E1:
(36)E1=2−jn−β∑l=−∞j∫Blbx−bBlfydy·χjx≤2−jn−β∑l=−∞jbx−bBl·χjxflLq1∙wq1·χBlLq1∙wq1·′.
Applying the norm on both sides and using Lemma16, we get
(37)E1Lq2∙wq2·≤2−jn−β∑l=−∞jbx−bBl·χBjLq2∙wq2·flLq1∙wq1·χBlLq1∙wq1·′≤2−jn−β∑l=−∞jj−lbBMOχBlLq2∙wq2·′flLq1∙wq1·χBlLq1∙wq1·′.
Now, we turn to estimateE2. For this, we have
(38)E2≤2−jn−β∑l=−∞jby−bBl·χlLq1∙wq1·′flLq1∙wq1··χjx≤2−jn−β∑l=−∞jby−bBl·χBlLq1∙wq1·′flLq1∙wq1··χjx.
Similar to the estimation forE1, we take the norm on both sides of above inequality and use Lemma 16 to obtain
(39)E2Lq2∙wq2·≤2−jn−β∑l=−∞jby−bBl·χlLq1∙wq1·′flLq1∙wq1··χjLq2∙wq2·≤2−jn−β∑l=−∞jbBMOχBlLq1∙wq1·′flLq1∙wq1·χBjLq2∙wq2·.
Hence, from inequalities (35), (37), and (39), one has b,HβfχjLq2∙wq2·≤2−jn−βbBMO∑l=−∞jj−1flLq1∙wq1·χBjLq2∙wq2·χBlLq1∙wq1·′, which by virtue of Lemma 9 reduces to
(40)b,HβfχjLq2∙wq2·≤2jβbBMO∑l=−∞jj−lflLq1∙wq1·χBlLq1∙wq1·′χBjLq2∙wq2·′−1.
Now using Lemma13, we learn
(41)b,HβfχjLq2∙wq2·≤2jβbBMO∑l=−∞jj−lflLq1∙wq1·χBlLq1∙wq1·′χBlLq2∙wq2·′χBlLq2∙wq2·′χBjLq2∙wq2·′≤2jβbBMO∑l=−∞jj−l2l−jnδ2bBMOflLq1∙wq1·χBlLq1∙wq1·′χBlLq2∙wq2·′.
In the definition of the fraction integralIβ, we replace f by χBl to obtain
(42)IβχBlx≥C2lβχBlx,from which we infer that
(43)χBlx≤C2−lβIβχBlx.
Taking the norm on both sides and using Lemmas14 and 9, respectively, we get
(44)χBlLq2∙wq2·≤C2lβIβχBlLq2∙wq2·≤C2lβχBlLq1∙wq1·≤C2ln−βχBlLq1∙wq1·′−1.
In view of Lemmas8 and 9, the use of (44) into (41) results in the following inequality:
(45)b,HβfχjLq2∙wq2·≤CbBMO∑l=−∞j2ln−β2jβj−l2l−jnδ2flLq1∙wq1·χlLq2∙wq2·χlLq2∙wq2·′−1≤CbBMO∑l=−∞j2j−lβ−nδ2j−lflLq1∙wq1·2−lnχlLq2∙wq2·χlLq2∙wq2·′−1≤CbBMO∑l=−∞j2j−lβ−nδ2j−lflLq1∙wq1·.
Now, by virtue of the conditionp1≤p2 and Proposition 17, we have
(46)b,HβfχjMK̇p2,q2·α·,λwq2·p1≤maxsupk0∈Zk0<02−k0λp1∑j=−∞k02jα0p1b,HβfχjLq2∙wq2·p1,supk0∈Zk0≥02−k0λp1∑j=−∞k02jα0p1b,HβfχjLq2∙wq2·p1+2−k0λp1∑j=0k02jα∞p1b,HβfχjLq2∙wq2·p1=maxX1,X2,X3,where
(47)X1=supk0∈Zk0<02−k0λp1∑j=−∞k02jα0p1b,HβfχjLq2∙wq2·p1,X2=supk0∈Zk0≥02−k0λp1∑j=−∞−12jα0p1b,HβfχjLq2∙wq2·p1,X3=supk0∈Zk0≥02−k0λp1∑j=0k02jα0p1b,HβfχjLq2∙wq2·p1.
To estimateX1, X2, and X3, we make use of the conditions on α·, such that for l<0, we have
(48)flLq1∙wq1·=2−lα02jα0p1flLq1∙wq1·p11p1≤2−lα0∑1=−∞l2iα0p1fiLq1∙wq1·p11p1≤2lλ−α02−lλ∑1=−∞l2iα·p1fiLq1∙wq1·p11p1≤C2lλ−α0fMK̇p1,q1·α·,λwq1·,and for l≥0, we obtain
(49)flLq1∙wq1·=2−lα∞2lα∞p1flLq1∙wq1·p11p1≤2−lα∞∑1=−∞l2iα∞p1fiLq1∙wq1·p11p1≤2lλ−α∞2−lλ∑1=−∞l2iα·p1fiLq1∙wq1·p11p1≤C2lλ−α∞fMK̇p1,q1·α·,λwq1·.
In order to estimateX1, we need to use α0≤α∞<nδ2+λ−β.
(50)X1≤supk0∈Zk0<02−k0λp1∑j=−∞k02jα0p1∑l=−∞jj−l2j−lβ−nδ2bBMOflLq1∙wq1·p1≤Csupk0∈Zk0<02−k0λp1∑j=−∞k02jα0p1∑l=−∞jj−l2j−lβ−nδ22lλ−α0bBMOfMK̇p1,q1·α·,λwq1·p1≤Csupk0∈Zk0<02−k0λp1∑j=−∞k02jα0p1∑l=−∞jj−l2j−lβ−nδ22lλ−α0p1bBMOp1fMK̇p1,q1·α·,λwq1·p1≤Csupk0∈Zk0<02−k0λp1∑j=−∞k02jλp1∑l=−∞jj−l2l−j−β+nδ2−α0+λp1bBMOp1fMK̇p1,q1·α·,λwq1·p1≤CbBMOp1fMK̇p1,q1·α·,λwq1·p1.
The result ofX2 is similar to that of X1. Next, we will estimate X3 below
(51)X3≤supk0∈Zk0<02−k0λp1∑j=−∞k02jα∞p1∑l=−∞jj−l2j−lβ−nδ2bBMOflLq1∙wq1·p1≤Csupk0∈Zk0<02−k0λp1∑j=−∞k02jα∞p1∑l=−∞jj−l2j−lβ−nδ22lλ−α∞bBMOfMK̇p1,q1·α·,λwq2·p1≤Csupk0∈Zk0<02−k0λp1∑j=−∞k02jα∞p1∑l=−∞jj−l2j−lβ−nδ22lλ−α∞p1bBMOp1fMK̇p1,q1·α·,λwq2·p1≤Csupk0∈Zk0<02−k0λp1∑j=−∞k02jλp1∑l=−∞jj−l2l−j−β+nδ2−α∞+λp1bBMOp1fMK̇p1,q1·α·,λwq2·p1≤CbBMOp1fMK̇p1,q1·α·,λwq2·p1.
Finally, we combine the estimates forXii=1,2,3, to have the desired result.Theorem 19.
Letp1,p2,q1·,q2·,β,α· and w be as in Theorem 18. In addition, if λ−nδ1<α0≤α∞, where 1<δ1<0, then
(52)b,Hβ∗fMK̇p2,q2·α·,λwq2·≤CbBMOfMK̇p1,q1·α·,λwq2·.Proof.
We write(53)b,Hβ∗fxχjx≤∫ℝn\Bjyβ−nbx−byfydy·χjx≤∑l=j+1∞∫Blyβ−nbx−byfydy·χjx≤∑l=j+1∞∫Blyβ−nbx−bBlfydy·χjx+≤∑l=j+1∞∫Blyβ−nby−bBlfydy·χjx=F1+F2.
We estimateF1 and F2 separately. A use of generalized inequality results in the following:
(54)F1≤C∑l=j+1∞2−ln−β∫Blbx−bBlfydy·χjx≤C∑l=j+1∞2−ln−βfiLq1∙wq1·χBlLq1∙wq1·′bx−bBl·χj.
Applying the weighted Lebesgue space norm on both sides and using Lemma16, we obtain
(55)F1Lq2∙wq2·≤C∑l=j+1∞2−ln−βbx−bBjχjLq2∙wq2·fiLq1∙wq1·χBlLq1∙wq1·′≤C∑l=j+1∞2−ln−βbBMObx−bBjχjLq2∙wq2·fiLq1∙wq1·χBlLq1∙wq1·′.
Similarly,(56)F2≤C∑l=j+1∞2−ln−β∫Blby−bBlfydy·χjx≤C∑l=j+1∞2−ln−βby−bBl·χjLq1∙wq1·flLq1∙wq1·′·χjxx.
In view of the weighted Lebesgue norm and Lemma16, we get
(57)F2Lq2∙wq2·≤C∑l=j+1∞2−ln−βby−bBl·χjLq1∙wq1·′flLq1∙wq1··χjLq2∙wq2·≤C∑l=j+1∞2−ln−βl−jbBMOχlLq1∙wq1·′flLq1∙wq1··χjLq2∙wq2·.
Hence, from (53), (55), and (57), we obtain
(58)b,Hβ∗fχjLq2∙wq2·≤∑l=j+1∞2−ln−βl−jbBMOχjLq2∙wq2·flLq1∙wq1··χBlLq1∙wq1·′≤∑l=j+1∞2−ln−βl−jbBMOflLq1∙wq1·χBlLq1∙wq1·′χlLq2∙wq2·χjLq2∙wq2·χlLq2∙wq2·≤∑l=j+1∞2nδj−12−ln−βl−jbBMOf1Lq1∙wq1·χBlLq1∙wq1·′χlLq2∙wq2·.
Using the condition ofAq1·,q2· weights given in the Definition 4, the above inequality reduces to
(59)b,Hβ∗fχjLq2∙wq2·≤∑l=j+1∞2nδ1j−ll−jbBMOflLq1∙wq1·.
Next, the conditionp1<p2 and Proposition 17 help us to write
(60)b,Hβ∗fχjMK̇p2,q2·α·,λwq2·p1=maxY1,Y2,Y3,where
(61)Y1=supk0∈Zk0<02−k0λp1∑j=−∞k02jα0p1b,Hβ∗f·χjLq2∙wq2·p1,Y2=supk0∈Zk0≥02−k0λp1∑j=−∞−12jα0p1b,Hβ∗f·χjLq2∙wq2·p1,Y3=supk0∈Zk0≥02−k0λp1∑j=0k02jα∞p1b,Hβ∗f·χjLq2∙wq2·p1.
Lastly, in view of the condition−nδ1+λ<α0≤α∞, we estimate Yi,i=1,2,3, as we estimated Xi,i=1,2,3, in Theorem 18. Hence, we finish the proof.



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*Source: 9705250-2021-10-19.xml*