# Estimates for Commutators of Bilinear Fractionalp-Adic Hardy Operator on Herz-Type Spaces

Authors: Amjad Hussain; Naqash Sarfraz; Ilyas Khan; Aisha M. Alqahtani
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In the current article, we investigate the boundedness of commutators of the bilinear fractionalp-adic Hardy operator on p-adic Herz spaces and p-adic Morrey-Herz spaces by considering the symbol function from central bounded mean oscillations and Lipschitz spaces.


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## Body

## 1. Introduction

For everyx≠0, there is a unique γ=γx∈ℤ such that x=pγm/n, where p≥2 is a fixed prime number which is coprime to m,n∈ℤ. The mapping ·p:ℚ→ℝ+ defines a norm on ℚ with a range
(1)0∪pγ:γ∈ℤ.It follows from Ostrowski’s theorem (see [1]) that each nontrivial absolute value on ℚ is either the p-adic absolute value ·p or usual absolute value ·. The p-adic norm ·p is an ultrametric on ℚ, that is
(2)x+yp≤maxxp,yp.The field ofp-adic numbers is represented by ℚp and is the completion of rational numbers with respect to the p-adic norm ·p. Any p-adic number is written in series form (see [2]) as
(3)⋯+d4p4+d3p3+d2p2+d1p+d0+d−1p+⋯+d−j+1pj−1+d−jpj,where dk∈ℤ/pℤ. Hence, each member of ℚp is written in the form
(4)⋯dkdk−1⋯d1d0.d−1d−2⋯d−j.The higher dimensional vector spaceℚpn consists of tuples x=x1,⋯,xn, where xk∈ℚp,k=1,⋯,n, with the following norm
(5)xp=max1≤k≤nxkp.Forγ∈ℤ and a=a1,a2,⋯,an∈ℚpn, we represent by
(6)Bγa=x∈ℚpn:x−ap≤pγ,the closed ball with the center a and radius pγ and by
(7)Sγa=x∈ℚpn:∣x−ap=pγ,the corresponding sphere. For a=0, we write Bγ0=Bγ and Sγ0=Sγ. It is easy to see that the equalities
(8)a0+Bγ=Bγa0,a0+Sγ=Sγa0,Sγa0=Bγa0\Bγ−1a0,hold for all a0∈ℚpn and γ∈ℤ.Since the spaceℚpn is locally compact commutative group under addition, so it leads to a translation-invariant Haar measure dx which is normalized as follows
(9)∫B0dx=B0H=1,where EH denotes the Haar measure of a measurable subset E of ℚpn. In addition, it is not hard to see that Bγa=pnγ, Sγa=pnγ1−p−n, for any a∈ℚpn.Recently,p-adic analysis has taken considerable attention in harmonic analysis defined on the p-adic field [3–9] and mathematical physics [10, 11]. Furthermore, applications of p-adic analysis have been found in quantum gravity [12, 13], string theory [14], spring glass theory [15], and quantum mechanics [11].The Hardy operator was taken into consideration in [16] and is given as below:
(10)Hfx=1x∫0xftdt,x>0,satisfying the following inequality:
(11)HfLpℝ+≤pp−1fLpℝ+,1<p<∞.The generalization of (10) to n-dimensional Euclidian space was made in [17], which is given by:
(12)Hfx=1xn∫t≤xftdt,where f∈Lloc1ℝn and x=x1,⋯,xn. The boundedness of Hardy operator on Lpℝn was investigated in [18]. Without going into the detailed history regarding the boundedness of Hardy-type operators and their commutators on function spaces, we refer the readers to see [19–25] and the references therein.Fractional calculus is one of the major fields in the modern ages due to its numerous applications in science and engineering, see for instance [26–29]. Also, fractional integral operators are an integral part of the mathematical analysis. In this sense, Wu [30] defined the p-adic fractional Hardy operator as:
(13)Hβpfx=1xpn−β∫tp≤xpftdt,x∈ℚpn\0,where f∈Llocℚpn and 0≤β<n. Also, he gave the following definition of its commutators:
(14)Hβ,bp=bHβpf−Hβpbf.Ifβ=0, the fractional p-adic Hardy type operator is thep-adic Hardy operator [31, 32]. The commutator estimates of fractional Hardy-type operators on Herz spaces were obtained in [30, 32]. The articles [33, 34] are also important with regard to the study of p-adic Hardy operators on function spaces.Multilinear operators are studied in the analysis because of their natural appearance in numerous physical phenomenons and their purpose is not merely to generalize the theory of linear operators. We refer articles [35–37] for better comprehension of multilinear operators. The m-linear Hardy operator was defined by Fu et al. [19] and is given by:
(15)Hmf1,⋯,fmx=1xmn∫t1,⋯,tm<xf1t1⋯fmtmdt1⋯dtm,x∈ℝn\0,for f1,⋯,fm in Lloc1ℝn. In the same paper, they worked out the precise norm of the very operator on Lebesgue spaces with power weights.Now, we introduce the definition ofm-linear fractional p-adic Hardy operator as
(16)Hβp,mf1,⋯,fmx=1xpmn−β∫t1,⋯,tmp≤xpf1t1⋯fmtmdt1⋯dtm,x∈ℚpn\0, for f1,⋯,fm in Lloc1ℚpn. The 2-linear fractional p-adic Hardy operator will be referred to as a bilinear fractional p-adic Hardy operator. If β=0, we get the m-linear p-adic Hardy operator, see [38], where the authors obtained the sharp bounds of the m-linear p-adic Hardy operator and Hardy-Littlewood-Pólya operator on Lebesgue spaces with power weights. In [33], sharp bounds for the m-linear p-adic Hardy operator on the product of p-adic Lebesgue spaces have been obtained in an efficient way. Next, we define the commutator generated by the m-linear fractional p-adic Hardy operator as follows. Let bi∈Lloc1ℚpn for i=1,⋯,m, then
(17)Hβ,b→p,mf1,⋯,fmx=∑i=1mHβ,bip,mf1,⋯,fmx,where
(18)Hβ,bip,mf1,⋯,fmx=bixHβp,mf1,⋯,fmx−Hβp,mf1,⋯,fi−1,fibi,fi+1,⋯,fmx.Ifβ=0, we get the commutator operator defined in [39] with ℝn as underlying space.The aim of this article is to establish the CMO (central bounded mean oscillation) and Lipschitz estimates for commutators of a bilinear fractionalp-adic Hardy operator on p-adic function spaces such as p-adic Herz spaces and Morrey-Herz spaces. Before moving to our main results, let us specify that χk is the characteristic function of a sphere Sk, and C is a constant free from essential variables and its value may change at its multiple occurrences. It is imperative to recall the definition of homogeneous p-adic Herz spaces and homogeneous p-adic Morrey-Herz spaces, p-adic CMO spaces, and p-adic Lipschitz spaces.Definition 1 [31].
Suppose0<q,r<∞ and α∈ℝ. The homogeneous p-adic Herz space K̇qα,rℚpn is defined by
(19)Kqα,rℚpn=f∈Lqℚpn:fKqα,rℚpn<∞,where
(20)fKqα,rℚpn=∑k=−∞∞pkαrfχkLqℚpnr1/q.Obviously,K̇q0,qℚpn=Lqℚpn and K̇qα/q,qℚpn=Lqxpα.Definition 2 [40].
Suppose0<q,r<∞,λ≥0, and α∈ℝ. The homogeneous p-adic Morrey-Herz space is defined as:
(21)MK̇r,qα,λℚpn=f∈Llocqℚpn\0:fMK̇r,qα,λℚpn<∞,where
(22)fMK̇r,qα,λℚpn=supk0∈ℤp−k0λ∑k=−∞k0pkαrfχkLqℚpnr1/r.It is eminent thatMK̇r,qα,0ℚpn=K̇qα,rℚpn and MK̇q,qα/q,0ℚpn=Lqxpα.Definition 3 [31].
Let1≤q<∞. The space CMOqℚpn is defined to be the space of all measurable function f∈Llocqℚpn if
(23)fCMOqℚpn=supγ∈ℤ1BγH∫Bγfx−fBγqdx1/q<∞,where fBγ=1/BγH∫Bγfxdx.Definition 4 (see [40]).
Supposeδ is a positive real number. The Lipschitz space Λδℚpn is defined to be the space of all measurable function f on ℚpn such that
(24)fΛδℚpn=supx,h∈ℚpn,h≠0fx+h−fxhpδ<∞.


## 2. CMO Estimates forHβ,b→p,2

In the following section, we acquire the boundedness of commutators of bilinear fractionalp-adic Hardy operator on homogeneous p-adic Herz spaces and Morrey-Herz spaces by considering the symbol function from CMO spaces. We start the section with few lemmas that are helpful to prove the main results.Lemma 5 (see [30]).
Letb be a CMO function and 1≤q<r<∞, then
(25)CMOrℚpn⊂CMOqℚpnandbCMOqℚpn≤bCMOrℚpn.Lemma 6 (see [30]).
Letb be a CMO function, i,k∈ℤ, then
(26)bt−bBk≤bt−bbi+pni−kbCMO1ℚpn.Now, we proceed to state our key results for this section.Theorem 7.
Letα,α1,α2 be arbitrary real numbers, 1<p,p1,p2,q,q1,q2<∞,0≤β<n,α1+α2=α,1/p1+1/p2=1/p, and β/n=1/q1+1/q2−1/q. If for i=1,2 with n/qi′>αi, then Hβ,b→p,2 is bounded from K̇q1α1,p1ℚpn×K̇q2α2,p2ℚpn to K̇qα,pℚpn, where b→=b1,b2,b1,b2∈CMOmaxq,q1.Theorem 8.
Letα,α1,α2 be arbitrary real numbers, 1<p,p1,p2,q,q1,q2<∞,0≤β<n,α1+α2=α,λ1+λ2=λ,0≤λ1,λ2<∞,1/p1+1/p2=1/p, and β/n=1/q1+1/q2−1/q. If for i=1,2 with n/qi′+λi>αi, then Hβ,b→p,2 is bounded from MK̇p1,q1α1,λ1ℚpn×MK̇p2,q2α2,λ2ℚpn to MK̇p,qα,λℚpn, where b→=b1,b2,b1,b2∈CMOmaxq,q1.Note that Theorem7 is a special case of Theorem 8. So, we only prove Theorem 8.Proof.
LetbiBk denotes the average of bi on the ball Bk for i=1,2 and k∈ℤ. By definition, we have
(27)Hβ,b1p,2f1,f2χkLqℚpnq=∫Skxp−q2n−β∫t1,t2p≤xpf1t1f2t2b1x−b1t1dt1dt2qdx≤∫Skp−kq2n−β∫t1,t2p≤pkf1t1f2t2b1x−b1t1dt1dt2qdx≤Cp−kq2n−β∫Sk∫t1,t2p≤pkf1t1f2t2b1x−b1Bkdt1dt2qdx+Cp−kq2n−β∫Sk∫t1,t2p≤pkf1t1f2t2b1t1−b1Bkdt1dt2qdx=I+II.To evaluateI, we use 1/q1+1/q1′=1,1/q2+1/q2′=1, and β/n=1/q1+1/q2−1/q. Applying Hölder’s inequality to get
(28)I=Cp−kq2n−β∫Skb1x−b1Bkqdx∫Bk∫Bkf1t1f2t2dt1dt2q≤Cp−kq2n−βBkH1BkH∫Bkb1x−b1Bkqdx∑j=−∞k∫Sjf1t1dt1×∑i=−∞k∫Sif2t2dt2q≤Cpknq−1/q2′−1/q1′b1CMOqℚpnq∑j=−∞k∫Sjf1t1q1dt11/q1∫Sjdt11/q1′×∑i=−∞k∫Sif2t2q2dt21/q2∫Sidt21/q2′q≤Cb1CMOqℚpnq∑j=−∞kpj−kn/q1′f1χjLq1ℚpn.∑i=−∞kpi−kn/q2′f2χiLq2ℚpnq.Now, we turn our attention towards estimatingII:(29)II=Cp−kq2n−β∫Sk∫Bk∫Bkf1t1f2t2b1t1−b1Bkdt1dt2qdx≤Cpknq−1/q2′−1/q1′∑j=−∞k∫Sjf1t1b1t1−b1Bjdt1∑i=−∞k∫Sif2t2dt2q+Cpknq−1/q2′−1/q1′∑j=−∞k∫Sjf1t1b1Bk−b1Bjdt1∑i=−∞k∫Sif2t2dt2q=II1+II2.An easy application of Hölder’s inequality simplifies the expression ofII1, that is:
(30)II1≤Cpknq−1/q2′−1/q1′∑j=−∞k∫Sjf1t1q1dt11/q1∫Sjb1t1−b1Bjq1′dt11/q1′×∑i=−∞k∫Sif2t2q2dt21/q2∫Sidt21/q2′q≤Cpknq−1/q2′−1/q1′∑j=−∞kf1χjLq1ℚpnpjn/q1′1BjH∫Bjb1t1−b1Bjq1′dt11/q1′×∑i=−∞kf2χiLq2ℚpnpin/q2′q≤Cb1CMOq1′ℚpnq∑j=−∞kpj−kn/q1′f1χjLq1ℚpn.∑i=−∞kpi−kn/q2′f2χiLq2ℚpnq.To estimateII2, we use Lemma 6 along with the Hölder’s inequality to have
(31)II2≤Cpknq−1/q2′−1/q1′b1CMO1ℚpnq2∑j=−∞kk−j∫Sjf1t1dt1×∑i=−∞k∫Sif2t2dt2q≤Cpknq−1/q2′−1/q1′b1CMO1ℚpnq∑j=−∞kk−j∫Sjf1t1q1dt11/q1×∫Sjdt11/q1′∑i=−∞k∫Sif2t2q2dt21/q2∫Sidt21/q2′q≤Cb1CMO1ℚpnq×∑j=−∞kk−jpj−kn/q1′f1χjLq1ℚpn.∑i=−∞kpi−kn/q2′f2χiLq2ℚpnq.Sinceα=α1+α2,λ=λ1+λ2, and 1/p=1/p1+1/p2, by the definition of p-adic Morrey-Herz space along with Lemma 5 and Hölder’s inequality, we are down to
(32)Hβ,b1p,2f1,f2MK̇p,qα,λℚpn=supk0∈ℤp−k0λ∑k=−∞k0pkαpHβ,bp,2f1,f2χkLqℚpnp1/p≤Cb1CMOmaxq,q1′ℚpnsupk0∈ℤp−k0λ∑k=−∞k0pkαp×∑j=−∞kk−jpj−kn/q1′f1χjLq1ℚpn∑i=−∞kpi−kn/q2′f2χiLq2ℚpnp1/p≤Cb1CMOmaxq,q1′ℚpn×supk0∈ℤp−k0λ1∑k=−∞k0pkα1p1∑j=−∞kk−jpj−kn/q1′f1χjLq1ℚpnp11/p1×supk0∈ℤp−k0λ2∑k=−∞k0pkα2p2∑i=−∞kpi−kn/q2′f2χiLq2ℚpnp21/p2=Cb1CMOmaxq,q1′ℚpnE1×E2,where
(33)E1=supk0∈ℤp−k0λ1∑k=−∞k0pkα1p1∑j=−∞kk−jpj−kn/q1′f1χjLq1ℚpnp11/p1,E2=supk0∈ℤp−k0λ2∑k=−∞k0pkα2p2∑i=−∞kpi−kn/q2′f2χiLq2ℚpnp21/p2.To proceed further, we consider:(34)f1χjLq1ℚpn=pjα1p−jα1f1χjLq1ℚpnp11/p1≤p−jα1∑l=−∞jplα1p1f1χjLq1ℚpnp11/p1≤pjλ1−α1f1MK̇p1,q1α1,λ1ℚpn.Sincen/q1′+λ1>α1 and λ1>0, ultimately, we have the following representation of E1.(35)E1≤supk0∈ℤp−k0λ1∑k=−∞k0pkα1p1∑j=−∞kk−jpj−kn/q1′pjλ1−α1p11/p1f1MK̇p1,q1α1,λ1ℚpn≤supk0∈ℤp−k0λ1∑k=−∞k0pkλ1p1∑j=−∞kk−jpk−jα1−n/q1′−λ1p11/p1f1MK̇p1,q1α1,λ1ℚpn=Cf1MK̇p1,q1α1,λ1ℚpn.In a similar fashion,E2 can be estimated as
(36)E2≤Cf2MK̇p2,q2α2,λ2ℚpn.Making use of above inequalities in (32), we get
(37)Hβ,b1p,2f1,f2MK̇p,qα,λℚpn≤Cb1CMOmaxq,q1′ℚpnf1MK̇p1,q1α1,λ1ℚpnf2MK̇p2,q2α2,λ2ℚpn.Similarly, we have(38)Hβ,b2p,2f1,f2MK̇p,qα,λℚpn≤Cb2CMOmaxq,q1′ℚpnf1MK̇p1,q1α1,λ1ℚpnf2MK̇p2,q2α2,λ2ℚpn.Therefore, we complete the proof of Theorem8.


## 3. Lipschitz Estimates forHβ,b→p,2 on p-Adic Herz Type Spaces

In the present section, we establish Lipschitz estimates for commutators of the bilinear fractional Hardy operator on homogeneousp-adic Herz spaces and Morrey-Herz spaces. The main results of the section are as follows.Theorem 9.
Letα,α1,α2 be arbitrary real numbers, 1<p,p1,p2,q,q1,q2<∞,0≤β<n,α1+α2=α,1/p1+1/p2=1/p, and β+γ/n=1/q1+1/q2−1/q. If for i=1,2 with n/qi′>αi, then Hβ,b→p,2 is bounded from K̇q1α1,p1ℚpn×K̇q2α2,p2ℚpn to K̇qα,pℚpn, where b→=b1,b2,b1,b2∈Λγℚpn.Theorem 10.
Letα,α1,α2 be any arbitrary real numbers, 1<p,p1,p2,q,q1,q2<∞,0≤β<n,α1+α2=α,λ1+λ2=λ,0≤λ1,λ2<∞,1/p1+1/p2=1/p, and β+γ/n=1/q1+1/q2−1/q. If for i=1,2 with n/qi′+λi>αi, then Hβ,b→p,2 is bounded from MK̇p1,q1α1,λ1ℚpn×MK̇p2,q2α2,λ2ℚpn to MK̇p,qα,λℚpn, where b→=b1,b2,b1,b2∈Λγℚpn.Since Theorem9 can easily be deduced from Theorem 10, so we opt for proof of later theorem.Proof .
Sinceb1∈Λγℚpn, therefore, we have
(39)b1x−b1t1≤x−t1pγb1ℚpn.Next, consider(40)Hβ,b1p,2f1,f2χkLq2ℚpnq=∫Skxp−q2n−β∫t1,t2p≤xpf1t1f2t2b1x−b1t1dt1dt2qdx≤∫Skp−kq2n−β∫t1,t2p≤pkf1t1f2t2b1x−b1t1dt1dt2qdx≤Cb1Λγℚpnp−kq2n−β∫Sk∫t1,t2p≤pkx−t1pγf1t1f2t2dt1dt2qdx≤Cb1Λγℚpnp−kq2n−β∫Sk∫t1,t2p≤pkxpγf1t1f2t2dt1dt2qdx=Cb1Λγℚpnp−kq2n−β−γ∫Sk∫t1,t2p≤pkf1t1f2t2dt1dt2qdx=ITo evaluateI, we use 1/q1+1/q1′=1,1/q2+1/q2′=1, and β+γ/n=1/q1+1/q2−1/q. Applying Hölder’s inequality to get
(41)I=Cb1Λγℚpnp−kq2n−β−γ+kn∫Bk∫Bkf1t1f2t2dt1dt2q=Cb1Λγℚpnp−kq2n−β−γ+kn∑j=−∞k∫Sjf1t1dt1.∑i=−∞k∫Sif2t2dt2q≤Cb1Λγℚpnpknq−1/q2′−1/q1′∑j=−∞k∫Sjf1t1q1dt11/q1∫Sjdt11/q1′×∑i=−∞k∫Sif2t2q2dt21/q2∫Sidt21/q2′q≤Cb1Λγℚpn∑j=−∞kpj−kn/q1′f1χjLq1ℚpn.∑i=−∞kpi−kn/q2′f2χiLq2ℚpnq.By the definition ofp-adic Morrey-Herz space, we have
(42)Hβ,b1p,2f1,f2MK̇p,qα,λℚpn=supk0∈ℤp−k0λ∑k=−∞k0pkαpHβ,bp,2f1,f2χkLqℚpnp1/p≤Cb1Λγℚpnsupk0∈ℤp−k0λ∑k=−∞k0pkαp∑j=−∞kpj−kn/q1′f1χjLq1ℚpn×∑i=−∞kpi−kn/q2′f2χiLq2ℚpnp1/p.The rest of the proof follows from Theorem8. So, we conclude the theorem.


## 4. Conclusion

Here, we obtained the CMO and Lipschitz estimates for the commutators of the bilinear fractionalp-adic Hardy operator on p-adic Herz-type spaces.



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