# Characterization of Lipschitz Spaces via Commutators of Maximal Function on thep-Adic Vector Space

Authors: Qianjun He; Xiang Li
Journal: Journal of Mathematics
 (2022)

---

## Abstract

In this paper, we give characterization of ap-adic version of Lipschitz spaces in terms of the boundedness of commutators of maximal function in the context of the p-adic version of Lebesgue spaces and Morrey spaces, where the symbols of the commutators belong to the Lipschitz spaces. A useful tool is a Lipschitz norm involving the John-Nirenberg-type inequality for homogeneous Lipschitz functions, which is new in the p-adic field context.


---

## Body

## 1. Introduction and Statement of Main Results

For a prime numberp, let ℚp be the field of p-adic numbers. It is defined as the completion of the field of rational numbers ℚ with respect to the non-Archimedean p-adic norm ⋅p. This norm is defined as follows: 0p=0. If any nonzero rational number x is represented as x=pγm/n, where m and n are integers which are not divisible by p, and γ is an integer, then xp=p−γ. It is not difficult to show that the norm satisfies the following properties:(1)xyp=xpyp,x+yp≤maxxp,yp.It follows from the second property that whenxp≠yp, then x+yp=maxxp,yp. From the standard p-adic analysis [1], we see that any nonzero p-adic number x∈ℚp can be uniquely represented in the canonical series(2)x=pγ∑j=0∞ajpj,γ=γx∈ℤ,where aj are integers, 0≤aj≤p−1, a0≠0. Series (2) converges in the p-adic norm because ajpjp=p−j.The spaceℚpd consists of points x=x1,x2,…,xd, where xj∈ℚp, j=1,2,…,d. The p-adic norm on ℚpd is xp≔max1≤j≤dxjp for x∈ℚpd. Denote by Bγa=x∈ℚpd:x−ap≤pγ, the ball with center at a∈ℚpd and radius pγ, and by Sγa≔x∈ℚpdx−ap=pγ the sphere with center at a∈ℚpd and radius pγ, γ∈ℤ. It is clear that Sγa=Bγa/Bγ−1a and Bγa=∪k≤γSka.Sinceℚpd is a locally compact commutative group under addition, it follows from the standard analysis that there exists a unique Harr measure dx on ℚpd (up to positive constant multiple) which is translation invariant. We normalize the measure dx so that(3)∫B00dx=B00H=1,where EH denotes the Harr measure of a measurable subset E of ℚpd. From this integral theory, it is easy to obtain that BγaH=pγd and SγaH=pγd1−p−d for any a∈ℚpd.In what follows, we say that a (real-valued) measurable functionf defined on ℚpd is in Lqℚpd, 1≤q≤∞, if it satisfies(4)fLqℚpd≔∫ℚpdfxqdx1/q<∞,1≤q<∞,fL∞ℚpd≔infα:x∈ℚpd:fx>αH=0<∞.Here the integral in equation (4) is defined as follows:(5)∫ℚpdfxqdx=limγ⟶∞∫Bγ0fxqdx=limγ⟶∞∑−∞<k≤γ∫Sk0fxqdx.If the limit exists. We now mention some of the previous works on harmonic analysis on thep-adic field, see [2–8] and the references therein.For a functionf∈Lloc1ℚpd, we defined the Hardy-Littlewood maximal function of f on ℚpd by the following equation:(6)ℳpfx=supγ∈ℤ1BγxH∫Bγxfydy.In [9, 10], Kim proved Lq boundedness of the version of maximal function ℳp and gave some properties similar to the Euclidean setting.The maximal commutator ofℳp with a locally integrable function b is defined by the following equation:(7)ℳbpfx=supγ∈ℤ1BγxH∫Bγxbx−byfydy.The first part of this paper is to study the boundedness ofℳbp when the symbol belongs to a Lipschitz space (see in Section 2). Some characterizations of the Lipschitz space via such commutator are given. Our first result can be stated as follows:Theorem 1.
Letb be a locally integrable function and 0<β<1, then the following statements are equivalent.(1)
b∈Λβℚpd;(2)
ℳbp is bounded from Lqℚpd to Lrℚpd for all q,r with 1<q<d/β and 1/r=1/q−β/d;(3)
ℳbp is bounded from Lqℚpd to Lrℚpd for some q,r with 1<q<d/β and 1/r=1/q−β/d;(4)
ℳbp satisfies the weak-type 1,d/d−β estimates, namely, there exists a positive constant C such that for any λ>0,(8)x∈ℚpd:ℳbpfx>λH≤CfL1ℚpdλd/d−β,(5)
ℳbp is bounded from Ld/βℚpd to L∞ℚpd.We remark that the boundedness of the commutators of maximal function were pretty much unknown on thep-adic vector space. However, in Euclidean case, the mapping properties of the maximal commutator have been studied intensively by many authors, see [11–21].It is well-known that the Morrey space introduced by Morrey in 1938, is connected to certain problem in elliptic PDE [22]. Later the Morrey spaces were found to have many important applications to the Navier-Stokes equations, the Schrödinger equations, the elliptic equations with discontinuous coefficients and the potential analysis, see [23–27] and so on. Next, we introduce the p-adic version of Morrey space on p-adic vector space.For1≤q<∞ and 0≤λ≤d. The p-adic version of Morrey space is defined by the following equation:(9)Lq,λℚpd=f∈Llocqℚpd:fLq,λℚpd<∞,where(10)fLq,λℚpd≔supx∈ℚpdsupγ∈ℤ1BγxHλ/d∫Bγxfxqdx1/q.It is well known that if1≤q<∞, then Lq,0ℚpd=Lqℚpd and Lq,dℚpd=L∞ℚpd.Theorem 2.
Letb be a locally integrable function and 0<β<1. Assume that 1<q<d/β, 0<λ<d−qβ and 1/r=1/p−β/d−λ. Then, b∈Λβℚpd if and only if ℳbp is bounded from Lq,λℚpd to Lr,λℚpd.Theorem 3.
Letb be a locally integrable function and 0<β<1. Assume that 1<q<d/β, 0<λ<d−qβ, 1/r=1/p−β/d, and ν/r=λ/q. Then, b∈Λβℚpd if and only if ℳbp is bounded from Lq,λℚpd to Lr,νℚpd.On the other hand, the classical commutator of thep-adic version of maximal function ℳp with a locally integrable function b can be defined by the following equation:(11)b,ℳpfx=bxℳpfx−ℳpbfx.In the Euclidean setting, the commutator of maximal function as in (11) has attracted much more attention. For examples, Milman and Schonbek [28] established a commutator result by real interpolation techniques. As an application, they obtained the Lq-boundedness of the commutator of maximal function b,M when b∈BMOℝd and b≥0. This operator can be used in studying the product of a function in H1 and a function in BMO (see [29] for instance). Bastero et al. [30] studied the necessary and sufficient conditions for the boundedness of b,M on Lq spaces when 1<q<∞. Zhang and Wu [31] extended their results to commutators of the fractional maximal function. The results in [30, 31] were extended to variable Lebesgue spaces in [32, 33]. Recently, Zhang [20] studied the commutator b,M when b belongs to Lipschitz spaces. Some necessary and sufficient conditions for the boundedness of b,M on Lebesgue and Morrey spaces are given. Some of the results were extended to variable Lebesgue spaces in [21]. For more information about the characterization of the commutator of maximal operator, see also [19, 20] and the references therein.We would like to remark that operatorsℳbp and b,ℳp essentially differ from each other. For example, ℳbp is positive and sublinear, but b,ℳp is neither positive nor sublinear. Motivated by the papers mentioned above, in this paper, the second part of this paper aims to study the mapping properties of the commutator b,ℳp when b belongs to some Lipschitz spaces. More precisely, we will give some new necessary and sufficient conditions for the boundedness of b,ℳp on p-adic vector spaces, by which some new characterizations for certain subclasses of Lipschitz spaces are obtained.Theorem 4.
Letb be a locally integrable function and 0<β<1. Assume that 1<q<d/β and 1/r=1/p−β/d. Then, the following statements are equivalent.(1)
b∈Λβℚpd and b≥0;(2)
b,ℳp is bounded from Lqℚpd to Lrℚpd;(3)
There exists a constantC>0 such that(12)supx∈ℚpdsupγ∈ℤ1|Bγx|Hβ/d1BγxH∫Bγxby−ℳBγxpbyrdy1/r≤C.WhereℳBγxp denote the maximal operator with respect to a p-adic ball which is defined by the following equation:(13)ℳBγxpfy=supBγx⊇Bγ0y1Bγ0yH∫Bγ0yfzdz.Here, the supremum is taken over all thep-adic Bγ0y with Bγ0y⊆Bγx for a fixed p-adic ball Bγx.Theorem 5.
Letb≥0 be a locally integrable function, 0<β<1 and b∈Λβℚpd. Then, there exist a positive C such that for any λ>0,(14)x∈ℚpd:b,ℳpfx>λH≤CfL1ℚpdλd/d−β.Theorem 6.
Letb be a locally integrable function and 0<β<1. Assume that 1<q<d/β, 0<λ<d−qβ, and 1/r=1/p−β/d−λ. Then, the following statements are equivalent(1)
b∈Λβℚpd and b≥0;(2)
b,ℳp is bounded from Lq,λℚpd to Lr,λℚpd.Theorem 7.
Letb be a locally integrable function and 0<β<1. Assume that 1<q<d/β, 0<λ<d−qβ, 1/r=1/p−β/d, and ν/r=λ/q. Then, the following statements are equivalent.(1)
b∈Λβℚpd and b≥0;(2)
b,ℳp is bounded from Lq,λℚpd to Lr,νℚpd.Remark 1.
Theorem 6 is essentially the Adams-type result and Theorem 7 is the Spanne-type result. Also, Theorem 7 can be immediately proved via Theorem 6.
The rest of the present paper is organized as follows: in Section 2, we gave some definitions and lemmas. The Proof of Theorems1–3 are presented in Section 3. In Section 4, we gave the proof of Theorems 4–7. By A≲B we mean that A≤cB with some positive constant c independent of appropriate quantities. The positive constants C varies from one occurrence to another. For a real number q, 1<q<∞, q′ is the conjugate number of q, that is, 1/q+1/q′=1.


## 2. Preliminary Definitions and Lemmas

To prove our main results, we need the following definitions and lemmas.Definition 1.
Assume that0<β<1. The p-adic version of homogeneous Lipschitz spaces Λβℚpd (see [34]) is the space of all measurable functions f on ℚpd such that(15)fΛβℚpd=supx,y∈ℚpd,x≠yfx−fyx−ypβ<∞.
For1≤q<∞, the p-adic version of Lipschitz spaces Lipγqℚpd is the space of all measurable functions f on ℚpd such that(16)fLipβqℚpd=supx∈ℚpdsupγ∈ℤ1BγxHβ/d1BγxH∫Bγxfy−fBγxqdy1/q<∞,
wherefBγxdenotes the average of f over Bγx, i.e., fBγx=1/BγxH∫Bγxfydy. When q=1, we use Lipβℚpd as Lipβ1ℚpd.
We shall give thep-adic version of John-Nirenberg inequality for homogeneous Lipschitz functions in the following lemma.Lemma 1.
Let0<β<1, then there exist some c1,c2>0 depending on p and d such that(17)x∈B:BH−β/dfx−fB>λH≤c1exp−c2λfLipβℚpdBH,
for anyf∈Lipβℚpd and for any p-adic B∈ℱp, where ℱp denotes the family of all the p-adic balls which is defined as follows:(18)ℱp=Bγx:γ∈ℤ,x∈ℚpd.Kim in [9] observed several interesting properties on the family ℱp.Lemma 2.
The familyℱp has the following properties:(a)
Ifγ≤γ′, then either Bγx∩Bγ′y=∅ or Bγx⊂Bγ′y.(b)
Bγx=Bγy if and only if y∈Bγx.Proof of Lemma 1.
LetB∈ℱp be a fixed p-adic ball and λ0 be some positive real number which will be determined later. Applying the p-adic version of the Calderón-Zygmund decomposition (see Corollary 3.4 in [9], or see also Theorem 5.16 in [9]) of B for BH−β/dfx−fB relative to λ0 to obtain a pair-wise disjoint family of p-adic balls Bjj=1∞⊂ℱp (by Lemma 2) which satisfies the following equation:(19)x∈B∪j=1∞Bj:BH−β/dfx−fB>λ0H=0,(20)λ0<1BjH∫BjBH−β/dfx−fBdx≤pdλ0,for anyj.
and(21)∑j=1∞BjH≤1λ0∫BBH−β/dfx−fBdx.
Combiningβ>0 and equation (20) gives the following equation:(22)BH−β/dfB−fBj≤1BjH∫BjBH−β/dfx−fBdx+1BjH∫BjBjH−β/dfx−fBjdx≤pdλ0+fLipβℚpd.
We denotet=pdλ0+fLipβℚpd>λ0. For any x∈Bj, we have(23)BH−β/dfx−fB≤BH−β/dfB−fBj+BH−β/dfx−fBj≤t+BjH−β/dfx−fBj.
By using equation (19), we can infer that for any λ>0(24)x∈B:BH−β/dfx−fB>λ+tH≤x∈∪j=1∞Bj:BH−β/dfx−fB>λ+tH+x∈B∪j=1∞Bj:BH−β/dfx−fB>λ+tH≤x∈∪j=1∞Bj:BH−β/dfx−fB>λ+tH+x∈B∪j=1∞Bj:BH−β/dfx−fB>λ0H=∑j∞x∈Bj:BH−β/dfx−fB>λ+tH.
Thus, equations (23) and (24) now implies.(25)x∈B:BH−β/dfx−fB>λ+tH≤∑j=1∞x∈Bj:BH−β/dfx−fB>λ+tH≤∑j=1∞x∈Bj:BjH−β/dfx−fBj>λH.
For anyλ>0, set(26)Gfλ=supB∈ℱp1BHx∈B:BH−β/dfx−fB>λH.
Obviously,Gfλ is a decreasing function 0,∞ and Gf0≤1. Thus, we have the following equation:(27)1BHx∈B:BH−β/dfx−fB>λ+tH≤Gfλ1BH∑j=1∞BjH≤Gfλ1λ0BH∫BBH−β/dfx−fBdx.
Hence, for anyλ>0, we conclude that(28)Gfλ+t≤GfλfLipβℚpdλ0.
Takingλ0=efLipβℚpd, then t is also a fixed positive number and for any λ≥0,(29)Gfλ+t≤Gfλe.
Using induction argument for anyk≥1, we obtain the following equation:(30)Gfk+1t≤e−kGft.
Thus, forλ∈kt,k+1t, we have the following equation:(31)Gfλ≤Gfkt≤e−kGf0≤ee−λ/t.
Notice that this inequality is also true forλ∈0,t, due to Gfλ≤Gf0=1≤ee−λ/t. Therefore, for any λ≥0, we conclude that(32)1BHx∈B:BH−β/dfx−fB>λH≤ee−λ/t.
This finishes the proof of the lemma.
Kim in [9] gave the property of Lqℚpd as in the Euclidean case.Lemma 3.
Forα>0, the distribution function ωHα of f on ℚpd defined by the following equation:(33)ωHα=x∈ℚpd:fx>αH.Iff∈Lqℚpd for q>0, then we have the following equation:(34)∫ℚpdfxqdx=−∫0∞αqdωHα=q∫0∞αq−1ωHαdα.We also need thep-adic version of Lebesgue differential theorem, which is due to Kim [10].Lemma 4.
Iff∈L1ℚpd, then we have that for a.e. x∈ℚpd(35)limγ⟶−∞1BγxH∫Bγxfydy=fx.The following lemma is about some properties of Lipschitz spaceLipβqℚpd.Lemma 5.
Iff∈Lipβℚpd is given, then we have the following properties:(a)
Forq>1, then there exists a constant cpq>0 such that fLipβqℚpd≤cpqfLipβℚpd.(b)
The normfLipβqℚpd is equivalent to the norm fLipβℚpd on Lipschitz space.(c)
For anyλ with 0<λ<c2/fLipβℚpd, where c2 is the constant given in Lemma 1,(36)supx∈ℚpdsupγ∈ℤ1BγxH∫BγxexpλBγxH−β/dfy−fBγxdy<∞.Proof.
.(a)
Forq>1, using Lemmas 1 and 3 we give the following equation:(37)∫BγxBγxH−β/dfy−fBγxqdy=∫0∞qλq−1y∈Bγx:BγxH−β/dfy−fBγx>λHdλ≤c1∫0∞qλq−1exp−c2λfLipβℚpddλ⋅BγxH=c1qfLipβℚpdc2q∫0∞λq−1e−λdλ⋅BγxH=c1qΓqc2−qfLipβℚpdq⋅BγxH=cpqqfLipβℚpdq⋅BγxH,wherecpq=c1qΓqc2−q1/q. Hence, we conclude that fLipβqℚpd≤cpqfLipβℚpd.(b)
Forq>1, by Hölder’s inequality it is obvious that(38)1BγxH1+β/d∫Bγxfy−fBγxqdy≤1BγxHβ/d1BγxH∫Bγxfy−fBγxqdy1/q.for anyγ,x∈ℤ×ℚpd. Thus, we have that fLipβℚpd≤fLipβqℚpd. Hence, by (a) we can obtain the equivalence of those two norms.(c)
We first observe that(39)∫0BγxH−β/dfy−fBγxddteλtdt=expλBγxH−β/dfy−fBγx−1,y∈Bγx.
For anyγ,x∈ℤ×ℚpd. Then, it follows from Lemma 1 and the p-adic version [1] of changing the order of integration that(40)∫BγxexpλBγxH−β/dfy−fBγxdy=∫Bγx∫0BγxH−β/dfy−fBγxλeλtdt+1dy=∫0∞λeλty∈Bγx:BγxH−β/dfy−fBγx>tHdt+BγxH≤c1λ∫0∞expλt−c2λfLipβℚpddt⋅BγxH+BγxH=c1λc2fLipβℚpd−λ−1+1⋅BγxH.
for anyγ,x∈ℤ×ℚpd, provided that 0<λ<c2/fLipβℚpd. Therefore, we have the following equation:(41)supx∈ℚpdsupγ∈ℤ1BγxH∫BγxexpλBγxH−β/dfy−fBγxdy≤c1λc2fLipβℚpd−λ−1+1<∞.
This completes the proof of the lemma.
To prove the theorems, we need the following key lemma. In Euclidean case, the Lipschitz spaceΛβℝd coincides with some Morrey-Compana to space [35] and can be characterized by mean oscillation which is due to DeVore and Sharpley [36], Janson et al. [35] and Harboure et al. [37] (see also Paluszyński [38]).Lemma 6.
For1≤q<∞ and 0<β<1, then the homogeneous Lipschitz space Λβℚpd coincides with the space Lipβqℚpd.Proof.
Using the property (b) of Lemma5, we only need to prove that for some constant C>0,(42)fLipβℚpd≤fΛβℚpd≤CfLipβℚpd.
Now, we give the proof of the right side of inequality (42). Assume that f∈Lipβℚpd. Given x,y∈ℚpd and x≠y, set B=Bx,x−yp and B′=y,x−yp. Then we have the following equation:(43)fx−fy≤fx−fB+fy−fB′+fB−fB′.
We estimate only the first term of the right side, since the others follow similar lines. LettingBi=Bx,p−ix−yp for i≥1 and B0=B, then by Lemma 4, we show that(44)fx−fB≤limγ⟶∞fx−fBγ+∑i=0γ−1fBi+1−fBi≤C∑i=1∞1BiH∫Bifz−fBidz≤CfLipβℚpd∑i=1∞BiHβ/d≤CfLipβℚpd∑i=1∞p−βi+logpx−ypβ≤Cx−ypβfLipβℚpd.
Thus, we obtainfΛβℚpd≤CfLipβℚpd.
On the other hand, fory,z∈Bγx and y≠z, we have the following equation:(45)1BγxH1+β/d∫Bγxfy−fBγxdy≤1BγxH2+β/d∫Bγx∫Bγxfy−fzdydz.
Sincey,z∈Bγx and (b) of Lemma 2 implies that y−zp≤BγxH1/d. Hence, by using above inequality, we can infer that(46)1Bγx|H1+β/d∫Bγxfy−fBγxdy≤fΛβℚpdBγxH2+β/d∫Bγx∫Bγxy−zpβdydz≤fΛβℚpdBγxH2+β/d∫Bγx∫BγxBγxHβ/ddydz=fΛβℚpd.for any γ,x∈ℤ×ℚpd. Thus, we have fLipβℚpd≤fΛβℚpd. This concludes the proof of the lemma.
In [34], Taibleson introduced and studied the Riesz potentials on local fields. Let us recall the definition of Riesz potentials on p-adic vector space as follows:(47)Iαpfx=1Γdα∫ℚpdfyx−ypn−αdy,
whereΓdα=1−pα−n/1−p−α with α∈ℂ/0.
Let us recall the Hardy-Littlewood-Sobolev inequality for the Riesz potentialsIαp on p-adic vector space. More details, the interested reader may refer to the book [34].Lemma 7.
Letα be a complex number with 0<Reα<d and 1≤q<∞ such that 1/r=1/q−Reα/d. Then, the following statements are held.(a)
Ifq>1, then Iαp is bounded from Lqℚpd to Lrℚpd.(b)
Ifq=1, then Iαp is bounded from L1ℚpd to Lr,∞ℚpd.
Letα be a complex number with 0<Reα<d and f be locally integrable function on ℚpd, the p-adic version of fractional maximal function of f is given by the following equation:(48)ℳαpfx=supγ∈ℤ1BγxH1−α/d∫Bγxfydy.
Using the definition of thep-adic version of Riesz potential Iαp in equation (47), we give the following inequality:(49)Iαpfx=1Γdα∫ℚpdfyx−ypn−αdy≥1Γdα∫x−yp≤pγfyx−ypn−Reαdy≥1Γdα1pγn−Reα∫x−yp≤pγfydy.
So inequality (49) gives the following equation:(50)ℳαpfx≤CIαpfx.
Using inequality (50) and Lemma 7, we give the Lq boundedness of p-adic version of fractional maximal function ℳαp on the p-adic vector space.Lemma 8.
Letα be a complex number with 0<Reα<d and 1≤q<∞ such that 1/r=1/q−Reα/d. Then, the following statement are held.(a)
Ifq>1, then ℳαp is bounded from Lqℚpd to Lrℚpd.(b)
Ifq=1, then ℳαp is bounded from L1ℚpd to Lr,∞ℚpd.
We now state the boundedness ofIαp on Morrey space which will be useful in proving our results. This lemma is similar to the results of ones from [39–41]. In Euclidean setting, Adams [23] and Spanne [42] studied the corresponding results, and see also [43].Lemma 9.
Let0<Reα<d,1<q<d/Reα, and 0<λ<d−qReα.(1)
If1/r=1/q−Reα/d−λ, then Iαp is bounded from Lq,λℚpd to Lr,λℚpd.(2)
If1/r=1/q−Reα/d and λ/q=ν/r, then Iαp is bounded from Lq,λℚpd to Lr,νℚpd.Proof.
(1) Letx∈ℚpd and γ∈ℤ. Then, we split the following equation:(51)ΓdαIαpfx=∫x−yp≤pγfyx−ypd−αdy+∫x−yp>pγfyx−ypd−αdy=:I1x+I2x.
First, we estimateI1x. Using the definition of ℳp, we obtain the following equation:(52)I1x≤∑k=−∞γ∫Sγxfyx−ypd−Reαdy,=∑k=−∞γpkReα−d∫Sγxfydy≤CpγReαℳpfx.
On the other hand, by Hölder’s inequality, we have thatReα−d−λ/q<0(53)I2x≤∑k=γ+1∞pkReα−d∫Sγxfydy≤∑k=γ+1∞pkReα−d−λ/qfLq,λℚpd≤CpγReα−d−λ/qfLq,λℚpd.
Combining equations (52) and (53) gives the following equation:(54)Iαpfx≲pγReαℳpfx+pγReα−d−λ/qfLq,λℚpd.
Taking(55)pγ=fLq,λℚpdℳpfxq/d−λ.
then equation (54) becomes as follows:(56)Iαpfx≲fLq,λℚpdqReαd−λℳpfx1−qReα/d−λ.
Since1/r=1/q−Reα/d−λ, we have d−λ−qReαr/d−λ=q. Thus, by the Lq boundedness of the p-adic version of maximal operator ℳp, we can infer that(57)1BγxHλ/d∫BγxIαpfyrdy≲fLq,λℚpdqrReα/d−λ1BγxHλ/d∫Bγxℳpfyqdy≲fLq,λℚpdr−q1BγxHλ/d∫Bγxfyqdy≤fLq,λℚpdr,for any γ,x∈ℤ×ℚpd. Hence, we conclude that IαpfLr,λℚpd≤CfLq,λℚpd.
Now, we start to prove (2). Assume thatr1,q,α,λ,d satisfy the condition of (1), then 1/r1=1/q−Reα/d−λ. It follows that(58)1r=1q−Reαd≥1q−Reαd−λ=1r1.
that is equivalent tor≤r1. Thus, by using Hölder’s inequality, we have the following equation:(59)1BγxHν/d∫BγxIαpfyrdy1/r,≤1BγxHν/dr−1/r+1/r1−λ/dr11BγxHλ/d∫BγxIαpfyr1dy1/r1.
Using conditions1/r1=1/q−Reα/d−λ, 1/r=1/q−Reα/d, and λ/q=ν/r, we show that(60)νdr−1r+1r1−λdr1=λdq−1q+Reαd+n−λd⋅1q−Reαd−λ=0.
Hence, by using the conclusion of equation (1), we obtain the following equation:(61)IαpfLr,νℚpd≤CfLq,λℚpd.
This finishes the proof of the lemma.
Using inequality (50) and Lemma 9, we immediately obtain the following lemma:Lemma 10.
Let0<Reα<d,1<q<d/Reα, and 0<λ<d−qReα.(1)
If1/r=1/q−Reα/d−λ, then ℳαp is bounded from Lq,λℚpd to Lr,λℚpd.(2)
If1/r=1/q−Reα/d and λ/q=ν/r, then ℳαp is bounded from Lq,λℚpd to Lr,νℚpd.
Considering the characteristic functionχBγx, we have the following property.Lemma 11.
Let1≤q<∞ and 0<λ<d, then there exist a constant C>0 such that(62)χBγxLq,λℚpd=BγxHd−λ/dq.Proof.
Using the definition ofLq,λℚpd and Lemma 2, it is easy to compute that for 0<λ<d(63)χBγxLq,λℚpd=pγd−λ/q=BγxHd−λ/dq,
and the lemma is proved.


## 3. The Proof of Theorems 1.1–1.3

Proof of Theorem 1.
Ifb∈Λβℚpd, then(64)ℳbpfx=supγ∈ℤ1BγxH∫Bγxbx−byfydy≤bΛβℚpdsupγ∈ℤ1BγxH1−β/d∫Bγxfydy=bΛβℚpdℳβpfx.
Obviously, equations (2)–(5) follow from Lemma 8 and (64).
(3)⇒ (1): Assume that ℳbp is bounded from Lqℚpd to Lqℚrd for some q,r with 1<q<d/β and 1/r=1/q−β/n. For any p-adic ball Bγx⊂ℚpd, by Hölder’s inequality and noting that 1/q+1/r′=1+β/d, one gives that(65)1BγxH1+β/d∫Bγxby−bBγxdy≤1BγxH1+β/d∫Bγx1BγxH∫Bγxby−bzdzdy=1BγxH1+β/d∫Bγx1BγxH∫Bγxby−bzχBγxzdzdy≤1BγxH1+β/d∫BγxMbpχBγxydy≤1BγxH1+β/d∫BγxMBPχBγxyrdy1/r∫BγxχBγxydy1/r′≤CBγxH1+β/dMbpLqℚpd⟶LrℚpdXBγxLqQPdXBγxLr′Qpd≤CMbpLqℚPD⟶LrℚPd.
This together with Lemma6 implies that b∈Λβℚpd.
(4)⇒ (1): We assume that (8) is true and will verify b∈Λβℚpd. For any p-adic Bγ0x⊂ℚpd, since for any y∈Bγ0x(66)by−bBγ0x≤1Bγ0xH∫Bγ0xby−bzdz,then, for all y∈Bγ0x(67)ℳbpχBγ0xy=supγ∈ℤ1BγyH∫Bγyby−bzχBγ0xzdz,≥1Bγ0xH∫Bγ0xby−bzχBγ0xzdz=1Bγ0xH∫Bγ0xby−bzdz,≥by−bBγ0x.
This together with equation (8) gives the following equation:(68)y∈Bγ0x:by−bBγ0x>λH≤y∈Bγ0x:ℳbpχBγ0xy>λH≤Cλ−1χBγ0xL1ℚpdd/d−β=Cλ−1Bγ0xHd/d−β.
Lett>0 be a constant to be determined later, then by using Lemma 3, we have the following equation:(69)∫Bγ0xby−bBγ0xdy=∫0∞y∈Bγ0x:by−bBγ0x>λHdλ=∫0ty∈Bγ0x:by−bBγ0x>λHdλ+∫t∞y∈Bγ0x:by−bBγ0x>λHdλ≤tBγ0xH+C∫t∞λ−1Bγ0xHd/d−βdλ≤tBγ0xH+CBγ0xHd/d−β∫t∞λ−d/d−βdλ≤CtBγ0xH+t1−d/d−βBγ0xHd/d−β.
Takingt=Bγ0xHβ/d in the above estimate, we have the following equation:(70)∫Bγ0xby−bBγ0xdy≤CBγ0xH1+β/d.
It follows from Lemma6 that b∈Λβℚpd since Bγ0x is an arbitrary p-adic ball in ℚpd.
(5)⇒ (1): If ℳbp is bounded from Ld/βℚpd to L∞ℚpd, then for any p-adic ball Bγx⊂ℚpd(71)1BγxH1+β/d∫Bγxby−bBγxdy≤1BγxH1+β/d∫Bγx1BγxH∫Bγxby−bzχBγxzdzdy≤1BγxH1+β/d∫BγxℳbpχBγxydy≤1BγxHβ/dℳbpχBγxL∞ℚpd≤CBγxHβ/dℳbpLd/βℚpd⟶L∞ℚpdχBγxLd/βℚpd≤CℳbpLd/βℚpd⟶L∞ℚpd.
This together with Lemma6 implies that b∈Λβℚpd.
The Proofof Theorem 1 is completed since (2) ⇒ (1) follows from (3) ⇒ (1).Proof of Theorem 2.
Assume thatb∈Λβℚpd. By using equations (64) and (1) of Lemma 9, we have the following equation:(72)ℳbpfLr,λℚpd≤bΛβℚpdℳβpLr,λℚpd≤CbΛβℚpdfLq,λℚpd.
Conversely, ifℳbp is bounded from Lq,λℚpd to Lr,λℚpd, then for any p-adic ball Bγx⊂ℚpd(73)1BγxHβ/d1BγxH∫Bγxby−bBγxqdy1/q≤1BγxHβ/d1BγxH∫Bγx1BγxH∫Bγxby−bzχBγxzdzqdy1/q≤1BγxHβ/d1BγxH∫BγxℳbpχBγxyqdy1/q=1BγxHβ/dBγxHλ/dBγxH1/q1BγxHλ/d∫BγxℳbpχBγxyqdy1/q≤BγxH−β/d−1/q+λ/dqℳbpLq,λℚpd⟶Lr,λℚpdχBγxLq,λℚpd≤CℳbpLq,λℚpd⟶Lr,λℚpd,where in the last step we have used 1/r=1/q−β/d−λ and Lemma 10.
It follows from Lemma6 that b∈Λβℚpd. This finishes the Proof of Theorem 2.Proof of Theorem 3.
By using similar proof to the one of Theorem2, we can obtain Theorem 3. Hence, we omit the details. □


## 4. The Proofof Theorems 4–7

Proof of Theorem 4.
(2)⇒ (1): For any fixed x∈ℚpd such that ℳpfx<∞, since b≥0 then we have the following equation:(74)b,ℳpfx=bxℳpfx−ℳpbfx|=supγ∈ℤ1BγxH∫Bγxbxfydy−supγ∈ℤ1BγxH∫Bγxbyfydy≤supγ∈ℤ1BγxH∫Bγxbx−byfy|dy=ℳbpx.
It follows from Theorem1 that b,ℳp is bounded from Lqℚpd to Lrℚpd since b∈Λβℚpd.
(2)⇒ (3): For any fixed p-adic ball Bγx⊂ℚpd and all y∈Bγx, we claim that(75)ℳpχBγxy=χBγxx,ℳpbχBγxy=ℳBγxpby.
In fact, similar to the proof of Lemma2.12, it is easy to obtain the claim. Then, by using the above claim, we have the following equation:(76)1BγxHβ/d1BγxH∫Bγxby−ℳBγxpbyrdy1/r=1BγxHβ/d1BγxH∫BγxbyℳpχBγxy−ℳpbχBγxyrdy1/r=1BγxHβ/d1BγxH∫Bγxb,ℳpχBγxyrdy1/r≤1BγxHβ/d+1/rb,ℳpχBγxLrℚpd≤CBγxHβ/d+1/rχBγxLqℚpd≤C.
which gives that (3) since the p-adic ball Bγx⊂ℚpd is arbitrary.
(3)⇒ (1): To prove b∈Λβℚpd, by using Lemma 6, it suffices to verify that there is a constant C>0 such that for any p-adic ball Bγx⊂ℚpd,(77)1BγxH∫Bγxby−bBγxdy≤C.
For any fixedp-adic ball Bγx, let E=y∈Bγx:by≤bBγx and Bγx, let E=y∈Bγx:by<bBγx. The following equality is trivially true.(78)∫Eby−bBγxdy=∫Fby−bBγxdy.
Since for anyy∈E we have by≤bBγx≤ℳBγxpby, then for any y∈E,(79)by−bBγx≤by−ℳBγxpby.
Thus, we can conclude that(80)1BγxH1+β/d∫Bγxby−bBγxdy=1BγxH1+β/d∫E∪Fby−bBγxdy=2BγxH1+β/d∫Eby−bBγxdy=2BγxH1+β/d∫Eby−ℳBγxpbydy=2BγxH1+β/d∫Eby−ℳBγxpbydy=2BγxH1+β/d∫Bγxby−ℳBγxpbydy.
On the other hand, it follows from Hölder’s inequality and equation (12) that(81)1BγxH1+β/d∫Bγxby−ℳBγxpbydy≤1BγxH1+β/d∫Bγxby−ℳBγxpbyrdy1/rBγxH1/r′≤1BγxHβ/d1BγxH∫Bγxby−ℳBγxpbyrdy1/r≤C.
Combining equations (76) with (80) it follows that b∈Λβℚpd.
In order to proveb≥0, it suffices to show b−=0, where b−=−minb,0. Let b+=b−b−, then b=b+−b−. For any fixed p-adic ball Bγx, observe that(82)0≤b+y≤bx≤ℳBγxpby,for y∈Bγx and therefore we have that for y∈Bγx,(83)0≤b−y≤ℳBγxpby−b+y+b−y=ℳBγxpby−by.
Then, it follows from equation (12) that for any p-adic ball y∈Bγx,(84)1BγxH∫Bγxb−ydy≤1BγxH∫BγxℳBγxpby−bydy≤1BγxH∫BγxℳBγxpby−byrdy1/r=BγxHβ/d1BγxHβ/d1BγxH∫BγxℳBγxpby−byrdy1/r≤CBγxHβ/d.
Thus,b−=0 follows from Lemma 4. Hence, the Proof of Theorems 4 is completed.Proof of Theorem 5.
Obviously, Theorem5 follows from equation (74) and Theorem 1. Hence, we omit the details.Proof of Theorem 6.
(1)⇒ (2): Assume that b≥0 and b∈Λβℚpd, then by using equation (74) and Theorem 2, we show that b,ℳp is bounded from Lq,λℚpd to Lr,λℚpd.
(2)⇒ (1): Assume that b,ℳp is bounded from Lq,λℚpd to Lr,λℚpd. Similar to estimate for equation (76), we have that for any p-adic ball Bγx⊂ℚpd,(85)1BγxHβ/d1BγxH∫Bγxby−ℳBγxpbyrdy1/r=1BγxHβ/d1BγxHβ/d∫BγxbyℳpχBγxy−ℳpbχBγxyrdy1/r=1BγxHβ/d1BγxH∫Bγxb,ℳpχBγxyrdy1/r≤BγxHλ/dqBγxHβ/d+1/rb,ℳpχBγxLr,λℚpd≤CBγxHλ/dqBγxHβ/d+1/rχBγxLq,λℚpd≤C,where in the last step we have used the condition 1/r=1/q−β/d−λ and Lemma 10. Thus, by using Theorem 4, we give that b∈Λβℚpd and b≥0. This finish the Proof of Theorem 6.Proof of Theorem 7.
By using the same way of the Proofof Theorem 6, Theorem 2.2 can be proven. Hence, we omit the details.



---

*Source: 7430272-2022-12-05.xml*