# Weighted Estimates for Commutator of Roughp-Adic Fractional Hardy Operator on Weighted p-Adic Herz–Morrey Spaces

Authors: Naqash Sarfraz; Doaa Filali; Amjad Hussain; Fahd Jarad
Journal: Journal of Mathematics
 (2021)

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## Abstract

The current article investigates the boundedness criteria for the commutator of roughp-adic fractional Hardy operator on weighted p-adic Lebesgue and Herz-type spaces with the symbol function from weighted p-adic bounded mean oscillations and weighted p-adic Lipschitz spaces.


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## Body

## 1. Introduction

For a fixed primep, it is always possible to write a nonzero rational number x in the form x=pγm/n, where p is not divisible by m,n∈ℤ and γ is an integer. The p-adic norm is defined as xp=p−γ∪0:γ∈ℤ. The p-adic norm ⋅p fulfills all the properties of a real norm along with a stronger inequality:(1)x+yp≤maxxp,yp.The completion of the field of rational number with respect to⋅p leads to the field of p-adic numbers ℚp. In [1], it can be seen that any x∈ℚp\0 can be represented in the formal power series form as(2)x=pγ∑j=0∞βjpj,where βj,γ∈ℤ,βj∈ℤ/pℤp,β0≠0. The convergence of series (2) is followed from pγβkpjp=p−γ−j.Then-dimensional vector space ℚpn=ℚp×⋯×ℚp consists of tuples x=x1,x2,…,xn, where xi∈ℚp,i=1,2,…,n, with the following norm:(3)xp=max1≤i≤nxip.The ballBγa and the corresponding sphere Sγa with center at a∈ℚpn and radius pγ in non-Archimedean geometry are given by(4)Bγa=x∈ℚpn:x−ap≤pγ,Sγa=x∈ℚpn:x−ap=pγ.Whena=0, we write Bγ0=Bγ,Sγ0=Sγ.Since the spaceℚpn is locally compact commutative group under addition, it cements the fact from the standard analysis that there exists a translation invariant Haar measure dx. Also, the measure is normalized by(5)∫B0dx=B0H=1,where EH represents the Haar measure of a measurable subset E of ℚpn. Furthermore, one can easily show that BγaH=pnγ, SγaH=pnγ1−p−n, for any a∈ℚpn.The last several decades have seen a growing interest in thep-adic models appearing in various branches of science. The p-adic analysis has cemented its role in the field of mathematical physics (see, for example, [2–4]). Many researchers have also paid relentless attention to harmonic analysis in the p-adic fields [5–11]. The present paper can be considered as an extension of investigation of Hardy-type operators started in [6, 7, 12–16].The one-dimensional Hardy operator(6)Hfx=1x∫0xfydy,x>0,was introduced by Hardy in [17] for measurable functions f:ℝ+⟶ℝ+ which satisfies the inequality(7)HfLqℝ+≤qq−1fLqℝ+,1<q<∞,where the constant q/q−1 is sharp. In [18], Faris proposed an extension of an operator H on higher dimensional space ℝn by(8)Hfx=1xn∫y≤xfydy,where x=∑i=1nxi21/2 for x=x1,…,xn. In addition, Christ and Grafakos [23] obtained the exact value of the norm of an operator H defined by (8). Over the years, Hardy operator has gained a significant amount of attention due to its boundedness properties [19–22]. For complete understanding of Hardy-type operators, we refer the interested readers to study [12, 23–29] and the references therein.In what follows, then-dimensional p-adic fractional Hardy operator(9)Hαpfx=1xpn−α∫yp≤xpfydy,was defined and studied for f∈L1locℚpn and 0≤α<n in [15]. When α=0, the operator Hαp transfers to the p-adic Hardy operator (see [30] for more details). Fu et al. in [30] acquired the optimal bounds of p-adic Hardy operator on Lqℚpn. On the central Morrey spaces, the p-adic Hardy-type operators and their commutators are discussed in [16]. In this link, see also [6, 7, 14, 27].From now on, we turn our attention towards the rough kernel version of an operator which recently received a substantial attention in analysis (see for instance [11, 31–37]). The roughness of Hardy operator was first time studied by Fu et al. in [12]. Motivated from the results of rough Hardy-type operators in Euclidean space, we define a special kind of rough fractional Hardy operator and its commutator in the p-adic field.Letf:ℚpn⟶ℝ, b:ℚpn⟶ℝ and Ω:S0⟶ℝ be measurable functions and let 0<α<n. Then, for x∈ℚpn∖0, we define a rough p-adic fractional Hardy operator HΩ,αp and its commutator HΩ,αp,b as(10)HΩ,αpfx=1xpn−α∫yp≤xpΩypyfydy,and(11)HΩ,αp,bfx=1xpn−α∫yp≤xpbx−byΩypyfydy,whenever(12)∫yp≤xpΩypyfydy<∞,(13)∫yp≤xpbyΩypyfydy<∞.Remark 1.
Obviously(14)yp:y∈ℚpn=pγ:γ∈ℤ∪0,holds for every integer n≥1 and prime p≥2. Since the inclusion(15)0∪pγ:γ∈ℤ⊆ℚp,holds and ℚpn is a linear space over field ℚp, the product ypy is correctly defined. Moreover, if a nonzero y∈ℚpn has a form y=y1,…,yn and(16)yi=pγiβ0,i+β1,ip+β2,ip2+⋯,i=1,…,n,(see (2)), then there is i0∈1,…,n such that(17)yi0p=p−γi0≥p−γi=yip,whenever yi≠0. Using (3), we obtain yp=p−γi0. Now from (16) and (17), it follows that(18)ypyp=max1≤i≤nyi≠0pγi−γi0p=max1≤i≤nyi≠0pγi0−γi=pγi0−γi0=1.
Thus, for every nonzeroy∈ℚpn, the vector ypy belongs to the sphere(19)S00=y∈ℚpn:|y|p=1.
From (12), it directly follows that HΩ,αp∈ℝ for every nonzero x∈ℚpn, and using (12) and (13), we have(20)HΩ,αp,bfx≤bxxpn−α∫yp≤xpΩypyfydy+1xpn−α∫yp≤xpbyΩypyfydy<∞,for every x∈ℚpn∖0. Consequently, the operators HΩ,αp and HΩ,αp,b are correctly defined.
The aim of the present paper is to study the weighted central mean oscillationsCMO and weighted p-adic Lipschitz estimates of HΩ,αp,b on weighted p-adic function spaces like weighted p-adic Lebesgue spaces, weighted p-adic Herz spaces and p-adic Herz–Morrey spaces. Throughout this article, the letter C represents a constant whose value may differ at all of its occurrence. Before turning to our key results, let us define and denote the relevant p-adic function spaces.


## 2. Notations and Definitions

Supposewx is a weight function on ℚpn, which is non-negative and locally integrable function on ℚpn. The weighted measure of E is denoted and defined as wE=∫Ewxdx. Let Lqw,ℚpn,0<q<∞ be the space of all complex-valued functions f on ℚpn such that(21)fLqw,ℚpn=∫ℚpnfxqwxdx1/q<∞.Definition 1.
Suppose1≤q<∞ and w is a weight function. The p-adic space CMOqw,ℚpn is defined as follows:(22)fCMOqw,ℚpn=supγ∈Z1wBγ∫Bγfx−fBγqwx1−qdx1/q,where(23)fBγ=1Bγ∫Bγfxdx.Definition 2.
(see [5]). Suppose α∈ℝ, 0<p,q<∞ and w1 and w2 are weight functions. Then, the weighted p-adic Herz space Kqα,pw1,w2 is defined by(24)Kqα,pw1,w2=f∈Llocqw2,ℚpn\0:fKqα,pw1,w2<∞,where(25)fKqα,pw1,w2=∑k=−∞∞w1Bkαp/nfχkLqw2,ℚpnp1/pand χk is the characteristic function of the sphere Sk=Bk∖Bk−1.Remark 2.
ObviouslyKq0,qw1,w2=Lqw2,ℚpn.Definition 3.
(see [5]). Let α∈ℝ, 0<p,q<∞, w1 and w2 be weight functions and λ be a non-negative real number. Then, the weighted p-adic Herz–Morrey space MKp,qα,λw1,w2 is defined as follows:(26)MKp,qα,λw1,w2=f∈Llocqw2,ℚpn\0:fMKp,qα,λw1,w2<∞,where(27)fMKp,qα,λw1,w2=supk0∈ℤw1Bk0−λ/n∑k=−∞k0w1Bkαp/nfχkLqw2,ℚpnp1/p.Remark 3.
It is evident thatMKp,qα,0w1,w2=Kqα,pw1,w2.
Now, we define the weightedp-adic Lipschitz space.Definition 4.
Suppose1≤q<∞, 0<γ<1 and w is a weight function. The p-adic space Lipγw,ℚpn is defined as(28)fLipγw,ℚpn=supB⊂ℚpn1wBγ/n1wB∫Bfx−fBqwx1−qdx1/q,where(29)fB=1B∫Bfxdx.
Muckenhoupt introduced the theory ofAq weights on ℝn in [38]. Let us define the Aq weights in the p-adic field.Definition 5.
A weight functionw∈Aq1≤q<∞, if there exists a constant C free from choice of B⊂ℚpn such that(30)1B∫Bwxdx1B∫Bwx−1/q−1dx1/q≤C.
For the caseq=1,w∈A1, we have(31)1B∫Bwxdx≤Cessinfx∈Bwx,for every B⊂ℚpn.Remark 4.
A weight functionw∈A∞ if it undergoes the stipulation of Aq1≤q<∞ weights.


## 3. WeightedCMO Estimates of HΩ,αp,b on Weighted p-Adic Herz-Type Spaces

The present section discusses the boundedness ofHΩ,αp,b on weighted p-adic Lebesgue spaces as well as on the weighted p-adic Herz-type spaces. We begin the section with some useful lemmas to prove our main results.Lemma 1.
(see [39]). Supposew∈A1; then, there exists constantsC1,C2and0<μ<1such that(32)C1AB≤wAwB≤C2ABμ,for measurable subset A of a ball B.Remark 5.
Ifw∈A1, then it follows from Lemma 1 that there exists a constant C and μ0<μ<1 such that wBk/wBi≤Cpk−in as i<k and wBk/wBi≤Cpk−inμ as i≥k.Lemma 2.
Supposew∈A1andb∈CMOqw,ℚpn; then, there is a constantCsuch that fori,k∈ℤ,(33)bBi−bBk≤Ci−kbCMOqw,ℚpnwBkBk.Proof.
Firstly, we consider(34)b2Bγ−bBγ≤1Bγ∫Bγbx−b2Bγdx≤1Bγ∫2Bγbx−b2Bγdx≤CBγbCMOqw,ℚpnw2Bγ.
We assume without loss of generality thati>k; then, using Lemma 1, we are down to(35)bBi−bBk≤bBi−bBi−1+⋅⋅⋅+bBk+1−bBk≤1Bi−1∫Bibx−bBidx+⋅⋅⋅+1Bk∫Bk+1bx−bBk+1dx≤CbCMOqw,ℚpnwBiBi−1+⋅⋅⋅+wBk+1Bk≤Ci−kbCMOqw,ℚpnwBkBk.Lemma 3.
Supposew∈A1; thenfor1<q<∞,(36)∫Bwx1−q′dx≤CBq′wB1−q′,where 1/q+1/q′=1.Proof.
SinceA1⊂Aqq>1, w satisfies the Aq conditions(37)1B∫Bwxdx1B∫Bwx−1/q−1dxq−1dx≤C,for every B⊂ℚpn.
From here, we easily get(38)∫Bwx1−q′dx≤CBq′wB1−q′.Theorem 1.
Let1≤p,q<∞,w∈A1,α/n+1=1/s′; then(39)HΩ,αp,bfLqw1−q,ℚpn≤CbCMOpmaxq,q′w,ℚpnfLqw,ℚpn,holds for all b∈CMOpmaxq,q′w,ℚpn, Ω∈LsS00, 1<s<∞ and f∈Llocℚpn.Now we state the results about the boundedness of commutator of roughp-adic fractional Hardy operator on weighted p-adic Herz-type spaces.Theorem 2.
Let0<p1≤p2<∞,1≤p,q<∞and letw∈A1,α/n+1=1/s′.Ifβ<nμ/q′, then the inequality(40)HΩ,αp,bfK˙qβ,p2w,w1−q≤CbCMOpmaxq,q′w,ℚpnfK˙qβ,p1w,wholds for all b∈CMOpmaxq,q′w,ℚpn, Ω∈LsS00, 1<s<∞ and f∈Llocℚpn.Remark 6.
Ifβ=0,p1=p2=q, then Theorem 1 becomes a special case of Theorem 2.Theorem 3.
Let0<p1≤p2<∞,1≤p,q<∞and letw∈A1,α/n+1=1/s′andλ>0. Ifβ<nμ/q′+λ, then(41)HΩ,αp,bfMK˙p2,qβ,λw,w1−q≤CbCMOpmaxq,q′w,ℚpnfMK˙p1,qβ,λw,w,holds for allb∈CMOpmaxq,q′w,ℚpn,Ω∈LsS00,1<s<∞andf∈Llocℚpn.Proof.
of Theorem2. By definition, we firstly have(42)HΩ,αp,bfχkLqw1−q,ℚpnq=∫Skxp−qn−α∫yp≤xpΩypyfybx−bydyqwx1−qdx≤Cp−kqn−α∫Sk∫yp≤pkΩypyfybx−bydyqwx1−qdx≤Cp−kqn−α∫Sk∑j=−∞k∫SjfyΩpjybx−bBkdyqwx1−qdx+Cp−kqn−α∫Sk∑j=−∞k∫SjfyΩpjyby−bBkdyqwx1−qdx=I+II.
Forj,k∈ℤ with j≤k, we get(43)∫SjΩpjysdy=∫zp=1Ωzspjndz≤Cpkn.
Also, sincew∈A1⊂Aq, by the application of Hölder’s inequality 1/q+1/q′=1 together with Lemma 3, we have(44)∫Sjfydy≤∫Sjfyqwydy1/q∫Sjwy−q′/qdy1/q′≤CfχjLqw,ℚpnBjwBj−1/q.
To estimateI, we make use of Hölder’s inequality, Remark 5, and α/n+1=1/s′ along with (43) and (44) to have(45)I≤Cp−kqn−α∫Bkbx−bBkq×∑j=−∞k∫Sjfys′dy1/s′∫SjΩpjysdy1/sqwx1−qdx≤Cpkqn−αbCMOqw,ℚpnqwBk∑j=−∞kpkn/s∫Sjfydyq≤Cp−kqn1−α/n−1/sbCMOqw,ℚpnqwBk∑j=−∞kfχjLqw,ℚpnBjwBj−1/qq≤Cp−knqbCMOqw,ℚpnq∑j=−∞kfχjLqw,ℚpnBjwBkwBj1/qq≤CbCMOqw,ℚpnqpk−jn/q′fχjLqw,ℚpnq.
Now, we turn our attention towards estimatingII.(46)II≤Cp−kqn−α∫Sk∑j=−∞k∫SjfyΩpjyby−bBjdyqwx1−qdx+Cp−kqn−α∫Sk∑j=−∞k∫SjfyΩpjybBk−bBjdyqwx1−qdx=II1+II2.
In order to evaluateII1, we need the following preparation. Apply Hölder’s inequality at the outset to deduce(47)∫Sjfyby−bBjdy≤∫Sjfyqwydy1/q∫Sjby−bBjq′wy−q′/qdy1/q′≤wBj−1/q′fχjLqw,ℚpnbCMOq′w,ℚpn.
We imply Hölder’s inequality, inequality (47), Lemma 3, and Remark 5 to estimate II1.(48)II1≤Cp−kqn−α∫Sk∑j=−∞k∫Sjfyby−bBjs′dy1/s′×∫SjΩpjysdy1/sqwx1−qdx≤Cp−kqn1−α/n−1/s∫Skwx1−qdx∑j=−∞k∫Sjfyby−bBjdyq≤Cp−kqn1−α/n−1/sBkqwBk1−qbCMOq′w,ℚpnq∑j=−∞kfχjLqw,ℚpnwBj1/q′q≤CbCMOq′w,ℚpnq∑j=−∞kwBjwBk1−1/qfχjLqw,ℚpnq≤CbCMOq′w,ℚpnq∑j=−∞kpj−knμ/q′fχjLqw,ℚpnq.
In a similar fashion, we can estimateII2. Using Hölder’s inequality, Lemmas 2 and 3, Remark 5, and inequality (44), we get(49)II2≤Cp−kqn−α×∫Sk∑j=−∞k∫SjfyΩpjyj−kbCMOpw,ℚpnwBjBjdyqwx1−qdx≤Cp−kqn1−α/n−1/sbCMOpw,ℚpnqBkqwBk1−q×∑j=−∞kk−jwBjBj∫Sjfys′dy1/s′∫SjΩpjysdy1/sq≤CbCMOpw,ℚpnqwBk1−q×∑j=−∞kk−jwBjBj∫Sjfydyq≤CbCMOpw,ℚpnq×∑j=−∞kk−jwBjwBk1−1/qfχjLqw,ℚpnq≤CbCMOpw,ℚpnq×∑j=−∞kk−jpj−knμ/q′fχjLqw,ℚpnq.
From (45), (48), and (49) together with Jensen inequality, we have(50)HΩ,αp,bfK˙qβ,p2w,w1−q=∑k=−∞∞wBkβp2/nHΩ,αp,bfχkLqw1−q,ℚpnp21/p2≤∑k=−∞∞wBkβp1/nHΩ,αp,bfχkLqw1−q,ℚpnp11/p1≤CbCMOqw,ℚpn∑k=−∞∞wBkβp1/n∑j=−∞kpj−kn/q′fχjLqw,ℚpnp11/p1+CbCMOq′w,ℚpn∑k=−∞∞wBkβp1/n∑j=−∞kpj−knμ/q′fχjLqw,ℚpnp11/p1+CbCMOpw,ℚpn∑k=−∞∞wBkβp1/n∑j=−∞kk−jpj−knμ/q′fχjLqw,ℚpnp11/p1=S.
Consequently,(51)Sp1≤CbCMOpmaxq,q′w,ℚpnp1×∑k=−∞∞wBkβp1/n∑j=−∞kk−jpj−knμ/q′fχjLqw,ℚpnp1≤CbCMOpmaxq,q′w,ℚpnp1×∑k=−∞∞∑j=−∞kk−jpj−knμ/q′−βfχjLqw,ℚpnp1.
From here on in the proof we consider couple of cases,0<p1≤1 and p1>1.Case 1.
When0<p1≤1, noticing that β<nμ/q′, we proceed as follows.(52)Sp1≤CbCMOpmaxq,q′w,ℚpnp1×∑k=−∞∞∑j=−∞kk−jp1wBjβp1/npj−knμ/q′−βp1fχjLqw,ℚpnp1=CbCMOpmaxq,q′w,ℚpnp1×∑k=−∞∞wBjβp1/nfχjLqw,ℚpnp1∑k=j∞k−jp1pj−knμ/q′−βp1=CbCMOpmaxq,q′w,ℚpnp1fK˙qβ,p1w,wp1.Case 2.
Whenp1>1, applying Hölder’s inequality with β<nμ/q′, we get(53)Sp1≤CbCMOpmaxq,q′w,ℚpnp1∑k=−∞∞∑j=−∞kwBjβp1/nfχjLq0w,ℚpnp1pj−knμ/q′−βp1/2×∑j=−∞kk−jp1′pj−knμ/q′−βp1′/2p1/p1′=CbCMOpmaxq,q′w,ℚpnp1∑k=−∞∞wBjβp1/nfχjLqw,ℚpnp1∑k=j∞pj−knμ/q′−βp1/2=CbCMOpmaxq,q′w,ℚpnp1fK˙q1β,p1w,wp1.
Therefore, the proof of theorem is completed.Proof.
of Theorem3. From Theorem 2, we have(54)HΩ,αp,bfχkLqw1−q,ℚpn≤CbCMOpmaxq,q′w,ℚpnp1∑j=−∞kk−jpj−knμ/q′fχjLqw,ℚpn.
By definition of weightedp-adic Herz–Morrey space and Jensen inequality together with β<nμ/q′+λ, λ>0 and 1<p1<∞, it follows that(55)HΩ,αp,bfMK˙p2,qβ,λw,w1−q=supk0∈ℤwBk0−λ/n∑k=−∞k0wBkβp2/nHΩ,αp,bfχkLqw1−q,ℚpnp21/p2≤supk0∈ℤwBk0−λ/n∑k=−∞k0wBkβp1/nHΩ,αp,bfχkLqw1−q,ℚpnp11/p1≤CbCMOpmaxq,q′w,ℚpnsupk0∈ℤwBk0−λ/n×∑k=−∞k0wBkλp1/n∑j=−∞kk−jpj−knμ/q′wBkwBjλp1/n×wBj−λ/n∑l=−∞jwBlβp1/nfχjLqw,ℚpnp11/p1p11/p1≤CbCMOpmaxq,q′w,ℚpnsupk0∈ℤwBk0−λ/n×∑k=−∞k0wBkλp1/n∑j=−∞kk−jpj−knμ/q′−β+λfMK˙p1,qβ,λw,wp11/p1≤CbCMOpmaxq,q′w,ℚpnsupk0∈ℤwBk0−λ/n×∑k=−∞k0wBkλp1/n1/p1fMK˙p1,qβ,λw,w≤CbCMOpmaxq,q′w,ℚpnfMK˙p1,qβ,λw,w.


## 4. Weighted Lipschitz Estimates for the Commutator of Roughp-Adic Fractional Hardy Operator on Herz–Morrey Spaces

In this section, we obtain the weightedp-adic Lipschitz estimates for the commutator of rough p-adic fractional Hardy operator on p-adic Lebesgue spaces and p-adic Herz-type spaces. We begin the section with a useful lemma which can be proved in the similar lines as Lemma 2.Lemma 5.
Supposew∈A1andb∈Lipγw,ℚpn; then, there is a constantCsuch that fori,k∈ℤ,(56)bBi−bBk≤Ci−kbLipγw,ℚpnwBiγ/nwBkBk.Theorem 4.
Let1≤p,q<∞,1/q1−1/q2=γ/n,w∈A1,α/n+1=1/s′; then,(57)HΩ,αp,bfLqw1−q,ℚpn≤CbLipγw,ℚpnfLqw,ℚpn,holds for allb∈Lipγw,ℚpn,Ω∈LsS00,1<s<∞, andf∈Llocℚpn.Now we state the results about the boundedness of commutator of roughp-adic fractional Hardy operator on weighted p-adic Herz-type spaces.Theorem 5.
Let0<p1≤p2<∞,1≤q1,q2<∞,1/q1−1/q2=γ/nand letw∈A1,α/n+1=1/s′.Ifβ<nμ/q1′, then the inequality(58)HΩ,αp,bfK˙q2β,p2w,w1−q2≤CbLipγw,ℚpnfK˙q1β,p1w,w,holds for allb∈Lipγw,ℚpn,Ω∈LsS00,1<s<∞, andf∈Llocℚpn.Remark 7.
Ifβ=0,p1=q1=p and p2=q2=q, then Theorem 4 can easily be obtained from Theorem 5.Theorem 6.
Let0<p1≤p2<∞,1≤q1,q2<∞,1/q1−1/q2=γ/nand letw∈A1,α/n+1=1/s′, andλ>0.Ifβ<nμ/q1′+λ, then(59)HΩ,αp,bfMK˙p2,q2β,λw,w1−q2≤CbLipγw,ℚpnfMK˙p1,q1β,λw,w,holds for allb∈Lipγw,ℚpn,Ω∈LsS00,1<s<∞, andf∈Llocℚpn.Proof.
of Theorem5. Following the same pattern of Theorem 2, we have(60)HΩ,αp,bfχkLq2w1−q2,ℚpnq2≤Cp−kq2n−α∫Sk∑j=−∞k∫SjfyΩpjybx−bBk|dyq2wx1−q2dx+Cp−kq2n−α∫Sk∑j=−∞k∫SjfyΩpjyby−bBk|dyq2wx1−q2dx=J+JJ.
To estimateJ, we make use of Hölder’s inequality, Remark 5, α/n+1=1/s′, γ/n=1/q1−1/q2, and w∈A1⊂Aq1 along with (43) and (44) to have(61)J≤Cp−kq2n1−α/n−1/sbLipγw,ℚpnq2wBk1+γq2/n∑j=−∞kfχjLq1w,ℚpnBjwBj−1/q1q2≤Cp−knq2bLipγw,ℚpnq2∑j=−∞kfχjLq1w,ℚpnBjwBkwBj1/q1q2≤CbLipγw,ℚpnq2pk−jn/q1′fχjLq1w,ℚpnq2.
For the estimation ofJJ, we need to decompose it as(62)JJ≤Cp−kq2n−α∫Sk∑j=−∞k∫SjfyΩpjyby−bBjdyq2wx1−q2dx+Cp−kq2n−α∫Sk∑j=−∞k∫SjfyΩpjybBk−bBjdyq2wx1−q2dx=JJ1+JJ2.
We need the following preparation to estimateJJ1. Apply Hölder’s inequality to get(63)∫Sjfyby−bBjdy≤∫Sjfyq1wydy1/q1∫Sjby−bBjq1′wy−q1′/q1dy1/q1′≤wBj−1/q1′+γ/nfχjLq1w,ℚpnbLipγw,ℚpn.
We imply Hölder’s inequality, inequality (63), Lemma 3, and Remark 5 to estimate JJ1.(64)JJ1≤Cp−kq2n1−α/n−1/s∫Skwx1−q2dx∑j=−∞kfχjLq1w,ℚpnwBj1/q1′+γ/nbLipγw,ℚpnq2≤Cp−kq2n1−α/n−1/sBkq2wBk1−q2bLipγw,ℚpnq2∑j=−∞kfχjLq1w,ℚpnwBj1/q1′+γ/nq2≤CbLipγw,ℚpnq2∑j=−∞kwBjwBk1−1/q2fχjLq1w,ℚpnq2≤CbLipγw,ℚpnq2∑j=−∞kpj−knμ/q2′fχjLq1w,ℚpnq2.
Now we turn towardsJJ2. Using once again Hölder’s inequality, Lemmas 5 and 3, Remark 5, and inequality (44), we get(65)JJ2≤Cp−kq2n1−α/n−1/sbLipγw,ℚpnq2Bkq2wBk1−q2×∑j=−∞kk−jwBkγ/nwBjBjBjwBj−1/q1fχjLq1w,ℚpnq2=CbLipγw,ℚpnq2×∑j=−∞kk−jwBjwBk1−1/q1fχjLq1w,ℚpnq2≤CbLipγw,ℚpnq2×∑j=−∞kk−jpj−knμ/q1′fχjLq1w,ℚpnq2.
Rest of the proof is similar to the proof of Theorem2. Thus, we come to an end of proof.Proof.
of Theorem6. Let β<nμ/q1′+λ. By the definition of weighted p-adic Herz–Morrey spaces along with inequalities (61), (64), and (65), we are down to(66)HΩ,αp,bfMK˙p2,q2β,λw,w1−q2≤CbLipγw,ℚpnsupk0∈ℤwBk0−λ/n∑k=−∞∞wBkβp2/n∑j=−∞kpj−kn/q1′fχjLq1w,ℚpnp21/p2+CbLipγw,ℚpnsupk0∈ℤwBk0−λ/n∑k=−∞∞wBkβp2/n∑j=−∞kpj−knμ/q2′fχjLq1w,ℚpnp21/p2+CbLipγw,ℚpnsupk0∈ℤwBk0−λ/n∑k=−∞∞wBkβp2/n∑j=−∞kk−jpj−knμ/q1′fχjLq1w,ℚpnp21/p2=L1+L2+L3.
Next by applying the similar arguments as in Theorem3, we get(67)L1≤CbLipγw,ℚpnfMK˙p1,qβ,λw,w,β<nq1′+λ,L2≤CbLipγw,ℚpnfMK˙p1,qβ,λw,w,β<nμq2′+λ,L3≤CbLipγw,ℚpnfMK˙p1,qβ,λw,w,β<nμq1′+λ.
Therefore, we conclude the proof.



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*Source: 5559815-2021-05-28.xml*