# Boundedness for Commutators of Roughp-Adic Hardy Operator on p-Adic Central Morrey Spaces

Authors: Naqash Sarfraz; Muhammad Aslam; Fahd Jarad
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In the present article we obtain the boundedness for commutators of roughp-adic Hardy operator on p-adic central Morrey spaces. Furthermore, we also acquire the boundedness of rough p-adic Hardy operator on Lebesgue spaces.


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## Body

## 1. Introduction

The classical Hardy operator for a non-negative functionf:ℝ+⟶ℝ+is given as
(1)Hfx=1x∫0xftdt,x>0.In [1], Hardy defined the above operator which satisfies
(2)HfLrℝ+≤rr−1fLrℝ+,1<r<∞.The constantr/r−1 in (2) is sharp. In [2], Faris extended the Hardy operator in ℝn by
(3)Hfx=1xn∫B0,xftdt.In this day and age, the Hardy operator has received a relentless consideration, see for example [3–7]. Moreover, the publications [8–12] and the references therein will do world of good to comprehend the Hardy type operators.The past few years has seen an immense attention towards mathematical physics [13, 14] along with harmonic analysis in the p-adic field [15–23]. Furthermore, the applications of p-adic analysis are seen mainly in string theory [24], quantum gravity [25, 26], quantum mechanics [14] and spring glass theory [27, 28].Supposep is a prime number, r∈ℚ, we introduce the p-adic norm rp by a rule
(4)0p=0,rp=p−α,where the integer α=αr is defined by the following notation
(5)r=pαm/n,integers m,n and p are coprime to each other. ·p has many properties of a real norm together with
(6)r+sp≤maxrp,sp.We denote the completion ofℚ in the norm ·p by ℚp. Any nonzero p-adic number can be written in series form as (see [14]):
(7)r=pα∑i=0∞γipi,where γi,α∈ℤ,γi∈ℤ/pℤp,γ0≠0. The series (7) is convergent as pαγipip=p−α−i.The spaceℚpn contains all n-tuples of ℚp. The norm on this space is
(8)rp=max1≤k≤nrkp.Represent byBαa the ball with radius pα and center at a and Sαa its sphere:
(9)Bαa=r∈ℚpn:r−ap≤pα,Sαa=r∈ℚpn:r−ap=pα.Sinceℚpn is a locally compact Hausdorff space, then there exists the Haar measure dx on additive group ℚpn and is normalized by
(10)∫B00dx=B00H=1,where EH denotes the Haar measure of a measurable subset E of ℚpn. Moreover, it is not hard to see that BγaH=pnγ and SγaH=pnγ1−p−n, for any a∈ℚpn.SupposeLsℚpn1≤s<∞ is the space of all complex-valued functions f on ℚpn such that
(11)∥f∥Lsℚpn=∫ℚpnfxsdx1/s<∞.In what follows author in [29] introduced the Hardy operator in the p-adic field as for f∈Llocℚpn, we have
(12)Hpfx=1xpn∫B0,xpftdt.For better understanding of Hardy type operators in thep-adic field we refer the publications [12, 29–32] and the references therein. From here on, we discuss the rough kernel version of an operator which is also considered an important topic in analysis, see for instance [20, 33–37]. In [10], Fu et al. studied the roughness of Hardy operator in the real field. In the p-adic setting, the rough Hardy operator and its commutator are defined and studied in [20]. Suppose f:ℚpn⟶ℝ, b:ℚpn⟶ℝ and Ω:S0⟶ℝ are measurable mappings, then
(13)HΩpfx=1xpn∫B0,xpΩtptftdtHΩp,bfx=1xpn∫B0,xpbx−btΩtptftdt,respectively, whenever
(14)∫B0,xpΩtptftdt<∞∫B0,xpbtΩtptftdt<∞.In [20], authors showed the weighted estimates of HΩp,b on two weighted Herz-Morrey spaces. In the present article, we acquire the λ−central bounded mean oscillations CṀOr,λℚpn estimate of HΩp,b on p-adic central Morrey spaces. In addition, we open up our results with a lemma which shows the boundedness of rough p-adic Hardy operator on Lebesgue spaces. Throughout this paper, we have no intention to obtain the best constants in the inequalities. The occurrence of a letter C does not mean a same constant, its value may vary at different positions.Definition 1 [32].
Supposeλ∈ℝ and 1<r<∞. The p-adic space Ḃr,λℚpn is defined as follows
(15)fḂr,λℚpn=supγ∈ℤ1BγH1+λr∫Bγfxrdx1/r<∞,where Bγ=Bγ0. Interestingly Ḃr,λℚpn reduces to 0 for −1/r>λ.Definition 2 [32].
Supposeλ<1/n and 1<r<∞. The p-adic space CṀOr,λℚpn is given by
(16)fCṀOr,λℚpn=supγ∈ℤ1BγH1+λr∫Bγfx−fBγrdx1/r<∞,where fBγ=1/BγH∫Bγfxdx, BγH is the Haar measure of Bγ.Remark 3.
Ifλ=0, then CṀOr,λℚpn is reduced to CMOrℚpn (see [29]).


## 2. Boundedness for Commutators of Roughp-Adic Hardy Operator on Central Morrey Spaces

In the present sectionCṀOr,λℚpn estimates of HΩp,b on central Morrey spaces in the p-adic field are obtained. However, to prove the result we need few lemmas.Lemma 4 [32].
Letb∈CṀOr,λℚpn and i,j∈ℤ, λ≥0. Then
(17)bBi−bBj≤pni−jbCṀOr,λℚpnmaxBiHλ,BjHλ.Lemma 5.
Suppose1<s<∞ and 1/s+1/s′=1. Then the inequality
(18)HΩpfLsℚpn≤CfLsℚpnholds for all f∈Llocsℚpn and Ω∈LsS0.Proof.
Firstly, we set(19)f~x=11−pn∫ξp=1fxp−1ξdξ,x∈ℚpn.
Obviouslyf~x=f~xp−1. In what follows we take this function a radial function on p-adic Lebesgue space. It is not hard to see that
(20)HΩpf~x=HΩpfx.In [29], it is shown that f~Lsℚpn≤fLsℚpn. Therefore,
(21)HΩpfLsℚpnfLsℚpn≤HΩpf~Lsℚpnf~Lsℚpn.This implies thatf~=f providing f is a radial function. Consequently, the norm of an operator HΩp along with its restriction to the function f^ have the same operator norm. So, we assume f to be a radial function in the rest of the proof.By the change ofp-adic variables t=xp−1y, we have
(22)HΩpfLsℚpn=∫ℚpn1xpn∫B0,∣xpΩtptftdtsdx1/s=∫ℚpn∫B0,1Ω∣y∣pyf∣x∣p−1ydysdx1/s.Now by using Minkowski’s inequality and Hölder’s inequality1/s+1/s′=1, we get
(23)HΩpfLsℚpn≤∫B0,1Ωypy∫ℚpnf∣y∣p−1xsdx1/sdy≤∫B0,1Ωypyyp−n/sdyfLsℚpn=∑j=−∞0∫SjΩpjyp−nj/sdyfLsℚpn≤∑j=−∞0p−jn/s∫SjΩpjysdy1/s∫Sjdy1/s′fLsℚpn.We handle the first part of sum as follows(24)∫SjΩpjysdy=∫zp=1Ωzspjndz=Cpjn.Hence inequality (23) takes the following form
(25)HΩpfLsℚpn≤CfLsℚpn,which completes the proof of a lemma.Now, we turn towards our key result.Theorem 6.
Suppose1<r1<∞, r1′<r2<∞, n1/r2−1/r1<n/r1, 1/r1+1/r2=1/r, −1/r1<λ1<0, λ=λ1+λ2 and 0≤λ2<1/n. If r1′<s<∞, then the below inequality
(26)HΩp,bfḂr,λℚpn≤CfḂr,λ1ℚpn,holds for b∈CMOmaxr2,sr1′/s−r1′,λ2ℚpn and Ω∈LsS0.Proof.
We supposef∈Ḃr1,λ1ℚpn. We also take γ∈ℤ and without any brevity we consider bCMOmaxr2,sr1′/s−r1′,λ2ℚpn=1. Applying Minkowski’s inequality to have
(27)1BγH1+λr∫BγHΩp,bfxrdx1/r=1BγH1+λr∫Bγ1xpn∫B0,∣xpΩ∣t∣ptftbx−btdtrdx1/r≤1BγH1+λr∫Bγ1xpn∫B0,∣xpΩ∣t∣ptftbx−bBγdtrdx1/r+1BγH1+λr∫Bγ1xpn∫B0,∣xpΩ∣t∣ptftbt−bBγdtrdx1/r=I+II.For the evaluation ofI, we make use of Lemma (5) which shows that HΩp is bounded from Lrℚpn to Lrℚpn,1<r<∞. By Hölder’s inequality 1=r/r1+r/r2, we have
(28)I≤BγH−1/r−λ∫Bγbx−bBγr2dx1/r2∫BγHΩpfxr1dx1/r1≤BγH−1/r−λ∫Bγbx−bBγr2dx1/r2∫Bγfxr1dx1/r1=CfḂr1,λ1ℚpn.In order to estimateII, we proceed as follows
(29)IIr≤1BγH1+λr∫Bγ1xpn∫B0,∣xpΩ∣t∣ptftbt−bBγdtrdx≤1BγH1+λr∑k=−∞γ∫Skp−knr∫B0,pk∣Ωtptftbt−bBγdt∣rdx=CBγH1+λr∑k=−∞γpkn1−r∑j=−∞k∫Sj∣Ωpjtftbt−bBγdt∣r≤CBγH1+λr∑k=−∞γpkn1−r∑j=−∞k∫Sj∣Ωpjtftbt−bBjdt∣r+CBγH1+λr∑k=−∞γpkn1−r∑j=−∞k∫Sj∣ΩpjtftbBj−bBγdt∣r=II1+II2.Forj,k∈ℤ with j≤k, we have
(30)∫SjΩpjtsdt=∫zp=1Ωzspjndz≤Cpkn.To evaluateII1, we apply Hölder’s inequality together with (30) to get
(31)II1≤CBγH1+λr∑k=−∞γpkn1−r∑j=−∞k∫SjΩpjtsdt1/s×∫Sjftr1dt1/r1∫Sjbt−bBjr2dt1/r2r≤CBγH1+λrfḂr,λ1ℚpn∑k=−∞γpkn1−r+r/s∑j=−∞kBj1/r1+λ1+1/r2+λ2r≤CBγH1+λrfḂr,λ1ℚpn∑k=−∞γpkn1+λr=CBγH1+λrfḂr,λ1ℚpnpγn1+λr=CfḂr,λ1ℚpn.The convergence of above series is eminent fromλ1+λ2+1/r1+1/r2≥λ1+1−1/s>−1/r+1−1/s=1/r1′−1/s>0.ForII2, we use Lemma 4, inequality (30) and Hölder’s inequality to obtain
(32)II2=CBγH1+λr∑k=−∞γpkn1−r∑j=−∞k∫Sj∣ΩpjtftbBj−bBγdt∣r≤CBγH1+λr∑k=−∞γpkn1−r∑j=−∞k∫SjΩpjtftγ−jBγHλ2dtr=CBγH1+λr∑k=−∞γpkn1−r∑j=−∞kγ−j∫SjΩpjtftdtr≤CBγH1+λ1r∑k=−∞γpkn1−r∑j=−∞kγ−j∫SjΩpjtsdt1/s×∫Sjftr1dt1/r1∫Sjdt1/r1′−1/sr≤CBγH1+λ1rfḂr,λ1ℚpn∑k=−∞γpkn1−r+r/s∑j=−∞kγ−jBjλ1+1−1/sr≤CBγH1+λ1rfḂr,λ1ℚpn∑k=−∞γpkn1−r+r/sγ−krBkλ1+1−1/sr=CBγH1+λ1rfḂr,λ1ℚpn∑k=−∞γγ−krpknr1/r+λ1=CBγH1+λ1rfḂr,λ1ℚpnpγnr1/r+λ1=CfḂr,λ1ℚpn,where we notice that 0<λ1+1−1/s together with λ1+1/r1+1/r2>1/r2>0=λ1+1/r. From (28), (31) and (32), we get
(33)HΩp,bfḂr,λ1ℚpn≤CfḂr,λ1ℚpn.


## 3. Conclusion

We mainly focused on the boundedness for commutators of roughp-adic Hardy operator on p-adic central Morrey spaces. Besides, we also obtained the boundedness of rough p-adic Hardy operator on p-adic Lebesgue spaces.



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