# p-Adic Riesz Potential and Its Commutators on Morrey-Herz Spaces

Authors: Yanlong Shi; Yafeng Shi; Shenbao Chen
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we establish the boundedness ofp-adic Riesz potential on Morrey-Herz spaces, as well as the λ-central BMO estimates for multilinear commutators of p-adic Riesz potential on Morrey-Herz spaces.


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## Body

## 1. Introduction

For a prime numberp, the field of p-adic numbers ℚp is defined as the completion of the field of rational numbers ℚ with respect to the non-Archimedean p-adic normal ·p, which is defined as follows: if x=0, 0p=0; if x≠0 is an arbitrary rational number with the unique representation x=pγm/n, where m, n are not divisible by p, γ=γx∈ℤ, then, xp=p−γ. This norm satisfies the following properties:
(i)
xp≥0, ∀x∈ℚp, and xp=0⇔x=0(ii)
xyp=xpyp, ∀x,y∈ℚp(iii)
x+yp≤maxxp,yp, ∀x,y∈ℚp and when xp≠yp, we have x+yp=maxxp,ypThe spaceℚpn=ℚp×⋯×ℚp consists of all points x=x1,⋯,xn, where xj∈ℚp,j=1,2,⋯,n,n≥1. The p-adic norm of ℚpn is defined by
(1)xp=max1≤j≤nxjp,x∈ℚpn.Denote by(2)Bγa=x∈ℚpn:x−ap≤pγ,the ball of radius pγ with center at a∈ℚpn and by
(3)Sγa=Bγa\Bγ−1a=x∈ℚpn:x−ap=pγ,the sphere of radius pγ with center at a∈ℚpn, where γ∈ℤ. For convenience, we denote Bγ=Bγ0 and Sγ=Sγ0, it is easy to see that the equalities x0+Bγ=Bγx0, x0+Sγ=Sγx0, and Bγa=⋃k≤γSka hold for any x0∈ℚpn.Since the spaceℚpn is a locally compact commutative group under addition, there exists the Haar measure dx on the additive group of ℚpn normalized by
(4)∫B0dx=B0=1,whereE denotes the Haar measure of a measurable set E⊂ℚpn. Then, by a simple calculation, we can obtain that
(5)Bγa=pnγ,Sγa=pnγ1−p−n,∀a∈ℚpn,for any a∈ℚpn. We should mention that the Haar measure takes value in ℝ, there also exist p-adic valued measures [1, 2]. For a complete introduction to the p-adic analysis, one can refer to [3] or [4] and references therein.In the last several decades, there has been 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, [4, 5]). Many researchers have also paid relentless attention to harmonic analysis on the p-adic fields [6–18]. It is well known that the Riesz potential operator in harmonic analysis is one of the most important operators and plays an important role in many areas such as Sobolev spaces, potential theory, and PDE, to name a few.On thep-adic field, let Γnα=1−pα−n/1−p−α with α∈ℂ\0, then, the Riesz potential Iα is defined by
(6)Iαfx=1Γnα∫ℚpnfyx−ypn−αdy.Whenn=1, Haran [6, 7] obtained the explicit formula of the Riesz potential Iα on ℚp. Taibleson [3] gave the fundamental analytic properties of Iα on local fields. In particular, Kim [8] gave the p-adic Hardy-Littlewood-Sobolev theorem on ℚpn, which implies Iα is bounded from Lqℚpn to Lrℚpn when q>1 and Iα is of weak type q,r when q=1, where 0<Reα<n and 1/r=1/q−Reα/n>0. Later on, Volosivets [11, 12] extended the theorem to p-adic generalized Morrey spaces and showed analogs of the results in Euclidean space ℝn given by Nakai [19]. Also, Wu and Fu [9] established counterparts of the theorem on p-adic central Morrey spaces. Additionally, Lu and Xu [20] extended Morrey spaces and introduced Morrey-Herz spaces in the setting of ℝn, and they also gave the Hardy-Littlewood-Sobolev inequalities on the spaces, which led many scholars to study the boundedness of Iα on Morrey-Herz spaces, for details, see [21, 22]. The first aim of the current paper is to study the boundedness of the Riesz potential Iα on Morrey-Herz spaces over the p-adic fields.Theorem 1.
Letα be a complex number with 0<Reα<n, 1/r=1/q−Reα/n with 1≤q<r<∞, λ≥0, λ−n/q+Reα<σ<n1−1/q, 0<l≤s≤∞. If q>1, then, there exists a constant C>0 such that
(7)IαfMK̇s,rσ,λℚpn≤CfMK̇l,qσ,λℚpn.
Moreover, ifq=1, then, there exists a constant C>0 such that
(8)IαfWMK̇s,rσ,λℚpn≤CfMK̇l,qσ,λℚpn.Remark 2.
The definitions of the homogeneous Morrey-Herz spacesMK̇s,rσ,λℚpn and weak homogeneous Morrey-Herz spaces WMK̇s,rσ,λℚpn will be given in Section 2, here, we point out that MK̇r,r0,0ℚpn=Lrℚpn for 1≤r<∞. We observe that Theorem 1 is counterparts of famous Hardy-Littlewood-Sobolev theorem on p-adic Morrey-Herz spaces. In addition, Theorem 1 coordinates with those on Morrey-Herz spaces in the setting of ℝn, which was proved by Lu and Xu [20].
Moreover, letb=b1,b2,⋯,bm, where bi∈Llocℚpn for 1≤i≤m, m∈ℕ. Then, multilinear commutator Iαb generated by b and Iα can be defined as follows:
(9)Iαbfx=1Γnα∫ℚpnfyx−ypn−α∏i=1mbix−biydy.
Whenm=1, Wu and Fu [9] obtained the λ-central BMO estimates for commutator Iαb on p-adic central Morrey spaces. Furthermore, Mo et al. [13] considered multilinear case for m≥1 and established the boundedness of Iαb with symbols in Campanato spaces on p-adic generalized Morrey spaces. Motivated by the works, our second goal is to show the λ-central BMO estimates for the multilinear commutator Iαb on Morrey-Herz spaces over the p-adic fields.Theorem 3.
Letα be a complex number with 0<Reα<n, 1/r=1/q1+1/q2+⋯+1/qm+1/q−Reα/n, where r>n/n−Reα and 1<q,q1,⋯,qm<∞. λ≥0, 0<l≤s≤∞, ν=ν1+ν2+⋯+νm with 0≤ν1,ν2,⋯,νm<1/n, nν+λ−n/q+Reα<σ1<n1−1/q+λ, σ2=σ1−n1/q1+1/q2+⋯+1/qm+ν. If bi∈CBMOqi,νiℚpn, then, there exists a constant C>0 such that
(10)IαbfMK̇s,rσ2,λℚpn≤C∏i=1mbiCBMOqi,νifMK̇l,qσ1,λℚpn.
We notice that the important particular case ofλ-central BMO spaces CBMOq,λℚpn is CBMOqℚpn defined in [14] when λ=0. Hence, if let ν1=ν2=⋯=νm=0 in Theorem 3, we obtain the following conclusion.Theorem 4.
Letα be a complex number with 0<Reα<n, 1/r=1/q1+1/q2+⋯+1/qm+1/q−Reα/n, where r>n/n−Reα and 1<q,q1,⋯,qm<∞. λ≥0, 0<l≤s≤∞, λ−n/q+Reα<σ1<n1−1/q+λ, σ2=σ1−n1/q1+1/q2+⋯+1/qm. If bi∈CBMOqiℚpn, then, there exists a constant C>0 such that
(11)IαbfMK̇s,rσ2,λℚpn≤C∏i=1mbiCBMOqifMK̇l,qσ1,λℚpn.
On the other hand, it is eminent thatMK̇s,rσ,0ℚpn=K̇rσ,sℚpn and WMK̇s,rσ,0ℚpn=WK̇rσ,sℚpn. Therefore, by letting λ=0 in Theorem 1, 3, and 4, respectively, we will get counterparts on p-adic Herz spaces.Corollary 5.
Letα be a complex number with 0<Reα<n, 1/r=1/q−Reα/n with 1≤q<r<∞, Reα−n/q<σ<n1−1/q, 0<l≤s≤∞. If q>1, then, there exists a constant C>0 such that
(12)IαfK̇rσ,sℚpn≤CfK̇qσ,lℚpn.
Moreover, ifq=1, then, there exists a constant C>0 such that
(13)IαfWK̇rσ,sℚpn≤CfK̇qσ,lℚpn.Corollary 6.
Letα be a complex number with 0<Reα<n, 1/r=1/q1+1/q2+⋯+1/qm+1/q−Reα/n, where r>n/n−Reα and 1<q,q1,⋯,qm<∞. 0<l≤s≤∞, ν=ν1+ν2+⋯+νm with 0≤ν1,ν2,⋯,νm<1/n, nν−n/q+Reα<σ1<n1−1/q, σ2=σ1−n1/q1+1/q2+⋯+1/qm+ν. If bi∈CBMOqi,νiℚpn, then, there exists a constant C>0 such that
(14)IαbfK̇rσ2,sℚpn≤C∏i=1mbiCBMOqi,νifK̇qσ1,lℚpn.Corollary 7.
Letα be a complex number with 0<Reα<n, 1/r=1/q1+1/q2+⋯+1/qm+1/q−Reα/n, where r>n/n−Reα and 1<q,q1,⋯,qm<∞. 0<l≤s≤∞, Reα−n/q<σ1<n1−1/q, σ2=σ1−n1/q1+1/q2+⋯+1/qm. If bi∈CBMOqiℚpn, then, there exists a constant C>0 such that
(15)IαbfMK̇rσ2,sℚpn≤C∏i=1mbiCBMOqifMK̇qσ1,lℚpn.
Throughout this paper, fort>0 and α∈ℂ, we always assume that tα takes the principal branch, that is
(16)tα=P.V.tα=eαlogt.
Thus, we have(17)tα=tReα.
The letterC will be used to denote various constants, and the various uses of the letter do not, however, denote the same constant.


## 2. Some Notations and Lemmas

We start with some notations and definitions. Here and in what follows, denote byχk the characteristic function of the sphere Sk.Definition 8 (see [15]).
Letσ∈ℝ and 0<s,r≤∞. The homogeneous Herz space K̇rσ,sℚpn is defined by
(18)K̇rσ,sℚpn=f∈Llocrℚpn\0:fK̇rσ,sℚpn<∞,where
(19)fK̇rσ,sℚpn=∑k=−∞∞pkσsfχkLrℚpns1/s,
fors<∞, and the usual modifications should be made when s=∞.Definition 9 (see [15]).
Letσ∈ℝ, 0<s,r≤∞, and λ be a nonnegative real number. Then, the homogeneous Morrey-Herz space MK̇l,qσ,λℚpn is defined by
(20)MK̇s,rσ,λℚpn=f∈Llocrℚpn\0:fMK̇s,rσ,λℚpn<∞,where
(21)fMK̇s,rσ,λℚpn=supk0∈ℤp−k0λ∑k=−∞k0pkσsfχkLrℚpns1/s,for s<∞, and the usual modifications should be made when s=∞.
Here, we introduce thep-adic weak Herz spaces and p-adic weak Morrey-Herz spaces.Definition 10.
Letσ∈ℝ, 0<s,r≤∞. The weak homogeneous Herz space WK̇rσ,sℚpn defined by
(22)WK̇rσ,sℚpn=fisameasurablefunctiononℚpnandfWK̇rσ,sℚpn<∞,where
(23)fWK̇rσ,sℚpn=supμ>0μ∑k=−∞k=∞pkσsx∈Sk:fx>μs/r1/s,for s<∞, and the usual modifications should be made when s=∞.Definition 11.
Letσ∈ℝ, 0<s,r≤∞, and λ be a nonnegative real number. The weak homogeneous Morrey-Herz space WMK̇s,rσ,λℚpn defined by
(24)WMK̇s,rσ,λℚpn=fisameasurablefunctiononℚpnandfWMK̇s,rσ,λℚpn<∞,where
(25)fWMK̇s,rσ,λℚpn=supμ>0μsupk0∈ℤp−k0λ∑k=−∞k0pkσsx∈Sk:fx>μs/r1/s,for s<∞, and the usual modifications should be made when s=∞.Definition 12 (see [9]).
Givenλ<1/n, 1<q<∞, the λ-central bounded mean oscillation space CBMOq,λℚpn is defined as the set of all functions f∈Llocrℚpn\0 such that
(26)fCBMOq,λℚpn=supk∈ℤ1Bk1+qλ∫Bkfx−fBkqdx1/q<∞,where
(27)fBk=1Bk∫Bkfxdx.Lemma 13 (see [10]).
Suppose that0≤λ<1/n, 1<q<∞, b∈CBMOq,λℚpn and j,k∈ℤ. Then
(28)bBk−bBj≤pnj−kbCBMOq,λℚpnmaxBkλ,Bjλ.Lemma 14 (see [8]).
Letα be a complex number with 0<Reα<n, 1/r=1/q−Reα/n with 1≤q<r<∞.If q>1, then, there exists a constant C>0 such that
(29)IαfLrℚpn≤CfLqℚpn.
Moreover, ifq=1, there is a constant C>0 such that
(30)x∈ℚpn:Iαfx>μ≤CfL1ℚpnμr,μ>0.Lemma 15.
Letα be a complex number with 0<Reα<n, 1/r=1/q−Reα/n with 1≤q<r<∞. Then. there exists a constant C>0 such that and
(31)IαfχjχkLrℚpn≤Cp−k−jn/q′fχjLqℚpn,ifj≤k−2,Cpk−jnq−ReαfχjLqℚpn,ifj≥k+2,for any integers j and k.Proof.
Obviously, by the properties of·p, it is not difficult to check that
(32)x−yp=maxxp,yp=maxpk,pj=pmaxk,j,for any x∈Sk, y∈Sj. Thus, by applying the Hölder’s inequality, we have
(33)Iαfχjx≤C∫Sjfyx−ypn−αdy≤C∫Sjp−maxk,jn−Reαfydy≤Cp−maxk,jn−ReαSj1/q′fχjLqℚpn≤Cp−maxk,jn−Reαpjn1−p−n1/q′fχjLqℚpn≤Cp−maxk,jn−Reαpjn/q′fχjLqℚpn,(34)∫SkIαfχjxr1/r≤Cp−maxk,jn−Reαpjn/q′fχjLqℚpnχkLrℚpn≤Cp−maxk,jn−Reαpjn/q′pkn1−p−n1/rfχjLqℚpn,(35)≤Cp−k−jn/q′fχjLqℚpn,ifj≤k−2,Cpk−jn/q−ReαfχjLqℚpn,ifj≥k+2.
Therefore, Lemma15 is completely proved.


## 3. Proof of Theorems

Proof of Theorem 16.
Suppose thatf∈MK̇l,qσ,λℚpn, we decompose f in the following way
(36)fx=∑j=−∞∞fxχjx≡∑j=−∞∞fjx,j∈ℤ.
First, we prove that forq>1. By the decomposition of f above and the fact l≤s, we get
(37)IαfMK̇s,rσ,λℚpn≤Csupk0∈ℤp−k0λ∑k=−∞k0pkσlIαfχkLrℚpnl1/l≤Csupk0∈ℤp−k0λ∑k=−∞k0pkσl∑j=−∞k−2IαfjχkLrℚpnl1/l+Csupk0∈ℤp−k0λ∑k=−∞k0pkσl∑j=k−1k+1IαfjχkLrℚpnl1/l+Csupk0∈ℤp−k0λ∑k=−∞k0pkσl∑j=k+2∞IαfjχkLrℚpnl1/l≡CN1+N2+N3.
Ifk−1≤j≤k+1, by the Lr-boundedness of Iα (see Lemma 14), we can obtain the estimates of N2 by
(38)N2≤Csupk0∈ℤp−k0λ∑k=−∞k0pkσl∑j=k−1k+1fjLqℚpnl1/l≤Csupk0∈ℤp−k0λ∑j=−∞k0+1∑k=j−1j+1pk−jσlpjσlfjLqℚpnl1/l≤Csupk0∈ℤp−k0λ∑j=−∞k0+1pjσlfjLqℚpnl∑k=j−1j+1pk−jσl1/l≤Csupk0∈ℤp−k0λ∑j=−∞k0+1pjσlfjLqℚpnl1/l≤fMK̇l,qσ,λℚpn.
Ifj≤k−2, applying Lemma 15, Jensen’s inequality, Hölder’s inequality, and the fact σ<n1−1/q=n/q′, we can get the estimates for N1 by
(39)N1≤supk0∈ℤp−k0λ∑k=−∞k0pkσl∑j=−∞k−2p−k−jn/q′fjLqℚpnl1/l≤supk0∈ℤp−k0λ∑k=−∞k0∑j=−∞k−2pk−jσ−nq′pjσfjLqℚpnl1/l≤supk0∈ℤp−k0λ∑j=−∞k0−2∑k=j+2k0pk−jlσ−nq′pjσlfjLqℚpnl1/l,if0<l≤1supk0∈ℤp−k0λ∑k=−∞k0∑j=−∞k−2pk−jσ−nq′pjσlfjLqℚpnl×∑j=−∞k−2pk−jσ−nq′l−11/l,if1<l<∞supk0∈ℤp−k0λsupk≤k0supj≤k−2pjσfjLqℚpn∑j=−∞k−2pk−jσ−nq′1/l,ifl=∞≤supk0∈ℤp−k0λ∑j=−∞k0−2pjσlfjLqℚpnl∑k=j+2∞pk−jlσ−nq′1/l,if0<l≤1supk0∈ℤp−k0λ∑j=−∞k0−2pjσlfjLqℚpnl∑k=j+2∞pk−jσ−nq′1/l,if1<l<∞supk0∈ℤp−k0λsupk≤k0supj≤k−2pjσfjLqℚpn∑k=j+2∞pk−jσ−nq′1/l,ifl=∞≤fMK̇l,qσ,λℚpn.
Ifj≥k+2, Lemma 15 shows the estimates for N3 by
(40)N3≤Csupk0∈ℤp−k0λ∑k=−∞k0pkσl∑j=k+2∞pk−jnq−ReαfjLqℚpnl1/l≤Csupk0∈ℤp−k0λ∑k=−∞k0∑j=k+2k0pk−jnq−Reα+σpjσfjLqℚpnl1/l+Csupk0∈ℤp−k0λ∑k=−∞k0∑j=k0+1∞pk−jnq−Reα+σpjσfjLqℚpnl1/l≡J1+J2.
ForJ1, using similar methods as that for N1, by the fact σ>λ−n/q+Reα≥−n/q+Reα, Jensen’s inequality, and Hölder’s inequality, we obtain
(41)J1≤supk0∈ℤp−k0λ∑j=−∞k0∑k=−∞j−2pk−jlnq−Reα+σpjσlfjLqℚpnl1/l,if0<l≤1,supk0∈ℤp−k0λ∑k=−∞k0∑j=k+2k0pk−jnq−Reα+σpjσlfjLqℚpnl×∑k=j+2k0pk−jnq−Reα+σl−11/l,if1<l<∞,supk0∈ℤp−k0λsupk≤k0supj≤k0pjσfjLqℚpn∑j=k+2k0pk−jnq−Reα+σ1/l,ifl=∞,≤supk0∈ℤp−k0λ∑j=−∞k0pjσlfjLqℚpnl∑k=−∞j−2pk−jlnq−Reα+σ1/l,if0<l≤1,supk0∈ℤp−k0λ∑j=−∞k0pjσlfjLqℚpnl∑k=−∞j−2pk−jnq−Reα+σ1/l,if1<l<∞,supk0∈ℤp−k0λsupk≤k0supj≤k0pjσfjLqℚpn∑k=−∞j−2pk−jnq−Reα+σ1/l,ifl=∞,(42)≤fMK̇l,qσ,λℚpn.
ForJ2, one can see from the definition of Morrey-Herz space that
(43)pjσfjLq≤C∑ℓ=−∞jpℓσlfℓLqℚpnl1/l≤CpjλfMK̇l,qσ,λℚpnforq≥1.
Here, by the estimates (43) and the fact σ>λ−n/q+Reα≥−n/q+Reα, we obtain
(44)J2≤CfMK̇l,qσ,λℚpnsupk0∈ℤp−k0λ∑k=−∞k0∑j=k0+1∞pk−jnq−Reα+σpjλl1/l≤CfMK̇l,qσ,λℚpnsupk0∈ℤp−k0λ∑k=−∞k0∑j=k0+1∞pk−jnq−Reα+σpjλl1/l≤CfMK̇l,qσ,λℚpnsupk0∈ℤ∑k=−∞k0pk0−k−nq+Reα−σl×∑j=k0+1∞pj−k0λ−nq+Reα−σl1/l≤CfMK̇l,qσ,λℚpn.
Therefore, by a combination for the estimates ofN1, N2 and N3, we can conclude that IαfMK̇s,rσ,λℚpn≤CfMK̇l,qσ,λℚpn.
Next, forq=1, we will consider the weak type boundedness of Iα. Note that WMK̇l,rσ,λℚpn⊂WMK̇s,rσ,λℚpn for l<s, so we only need to prove that in the case l=s. For any μ>0, the decomposition of f shows that
(45)μsupk0∈ℤp−k0λ∑k=−∞k0pkσlx∈Sk:Iαfx>2μl/r1/l=μsupk0∈ℤp−k0λ∑k=−∞k0pkσlx∈Sk:∑j=−∞k+1Iαfjx>μl/r1/l+μsupk0∈ℤp−k0λ∑k=−∞k0pkσlx∈Sk:∑j=k+2∞Iαfjx>μl/r1/l≡CM1+M2.
ForM1, applying the weak type property 1,r of the operator Iα (see Lemma 14) and the estimates (43). Notice that λ−n+Reα<σ<0 when q=1, λ≥0, so we can conclude that
(46)M1≤supk0∈ℤp−k0λ∑k=−∞k0pkσl∑j=−∞k+1fjL1ℚpnl1/l≤supk0∈ℤp−k0λ∑k=−∞k0∑j=−∞k+1pk−jσlpjσlfjL1ℚpnl1/l≤supk0∈ℤp−k0λ∑k=−∞k0∑j=−∞k+1pk−jσlpjλlfMK̇l,1σ,λℚpnl1/l≤fMK̇l,1σ,λℚpnsupk0∈ℤp−k0λ∑k=−∞k0pkλl∑j=−∞k+1pk−jσ−λl1/l≤fMK̇l,1σ,λℚpnsupk0∈ℤp−k0λ∑k=−∞k0pkλl1/l≤fMK̇l,1σ,λℚpn.
ForM2, by the Chebychev inequality, Lemma 15, the estimates (43), and the fact λ−n+Reα<σ<0, we have
(47)M2=μsupk0∈ℤp−k0λ∑k=−∞k0pkσlx∈Sk:∑j=k+2∞Iαfjx>μl/r1/l=supk0∈ℤp−k0λ∑k=−∞k0pkσl∑j=k+2∞IαfjχkLrℚpnl1/l=supk0∈ℤp−k0λ∑k=−∞k0pkσl∑j=k+2∞pk−jn−ReαfjL1ℚpnl1/l=supk0∈ℤp−k0λ∑k=−∞k0∑j=k+2∞pk−jn−Reα+σpjσfjL1ℚpnl1/l=supk0∈ℤp−k0λ∑k=−∞k0∑j=k+2∞pk−jn−Reα+σpjλfMK̇l,1σ,λℚpnl1/l=fMK̇l,1σ,λℚpnsupk0∈ℤp−k0λ∑k=−∞k0pkλ∑j=k+2∞pk−jn−Reα+σ−λl1/l≤fMK̇l,1σ,λℚpnsupk0∈ℤp−k0λ∑k=−∞k0pkλl1/l≤fMK̇l,1σ,λℚpn.
Hence, by a combination for the estimates ofM1, and M2, we can get the desired inequality IαfWMK̇s,rσ,λℚpn≤CfMK̇l,1σ,λℚpn.
Therefore, we complete the proof of Theorem16.Proof of Theorem 17.
Similarly to the proof of Theorem16, let f∈MK̇l,qσ1,λℚpn and decompose f into
(48)fx=∑j=−∞∞fxχjx≡∑j=−∞∞fjx,j∈ℤ.
Whenm=1, denote Iαbf by Iαb1f, we consider
(49)Iαb1fMK̇s,rσ2,λℚpn≤Csupk0∈ℤp−k0λ∑k=−∞k0pkσ2lIαb1fχkLrℚpnl1/l≤Csupk0∈ℤp−k0λ∑k=−∞k0pkσ2l∑j=−∞k−2Iαb1fjχkLrℚpnl1/l+Csupk0∈ℤp−k0λ∑k=−∞k0pkσ2l∑j=k−1k+1Iαb1fjχkLrℚpnl1/l+Csupk0∈ℤp−k0λ∑k=−∞k0pkσ2l∑j=k+2∞Iαb1fjχkLrℚpnl1/l≡CE1+E2+E3.
Let us first estimateE2, note that
(50)Iαb1fjχk=b1−b1BkIαfjχk−Iαb1−b1Bkfjχk.
Suppose that1/u=1/q−Reα/n, 1/t=1/q1+1/q, then, it is easy to see 1/r=1/u+1/q1=1/t−Reα/n for q>1 and t>1. Applying the fact σ2=σ1−n1/q1+ν1, Hölder’s inequality, Minkowski inequality, the boundedness of Iα from Lqℚpn to Luℚpn and from Ltℚpn to Lrℚpn, we have
(51)Iαb1fjχkLrℚpn≤b1−b1BkIαfjχkLrℚpn+Iαb1−b1BkfjχkLrℚpn=∫Bkb1x−b1BkrIαfjxrdx1/r+∫BkIαb1−b1Bkfjxrdx1/r≤C∫Bkb1x−b1Bkq1dx1/q1∫BkIαfjxudx1/u+C∫Bkb1x−b1Bkfjxtdx1/t≤CBk1/q1+ν1b1CBMOq1,ν1fjLqℚpn+C∫Bkb1x−b1Bkq1dx1/q1∫Bkfjxqdx1/q≤CBk1/q1+ν1b1CBMOq1,ν1fjLqℚpn≤Cpkn1/q1+ν1b1CBMOq1,ν1fjLqℚpn≤Cpkσ1−σ2b1CBMOq1,ν1fjLqℚpn.
Therefore, we get(52)E2≤Cb1CBMOq1,ν1supk0∈ℤp−k0λ∑k=−∞k0pkσ2l∑j=k−1k+1pkσ1−σ2fjLqℚpnl1/l≤Cb1CBMOq1,ν1supk0∈ℤp−k0λ∑k=−∞k0∑j=k−1k+1pkσ1lfjLqℚpnl1/l≤Cb1CBMOq1,ν1supk0∈ℤp−k0λ∑j=−∞k0+1pjσ1lfjLqℚpnl∑k=j−1j+1pk−jσ1l1/l≤Cb1CBMOq1,ν1supk0∈ℤp−k0λ∑j=−∞k0+1pjσ1lfjLqℚpnl1/l≤Cb1CBMOq1,ν1fMK̇l,qσ1,λℚpn.
Now, let us turn to the estimates ofE1 and E3. If x∈Sk, we can easily deduce that
(53)Iαb1fχjx≤C∫Sjb1x−b1yfyx−ypn−αdy≤Cp−maxk,jn−Reα∫Sjb1x−b1Bjfydy+∫Sjb1y−b1Bjfydy.
Then, using the facts1/r=1/q1+1/q−Reα/n, σ2=σ1−n1/q1+ν1, j−k≥2, Lemma 13, and Hölder’s inequality, we can get
(54)Iαb1fjχkLrℚpn≤Cp−maxk,jn−Reα∫Sk∫Sjb1x−b1yfydyrdx1/r≤Cp−maxk,jn−Reα∫Skb1x−b1Bjr∫Sjfydyrdx1/r+Cp−maxk,jn−Reα∫Sk∫Sjb1y−b1Bjfydyrdx1/r≤Cp−maxk,jn−ReαfjL1ℚpn∫Skb1x−b1Bjrdx1/r+Cp−maxk,jn−ReαSk1/r∫Sjb1y−b1Bjfydy≤Cp−maxk,jn−ReαSj1−1/qfjLqℚpnSk1/r−1/q1∫Skb1x−b1Bjq1dx1/q1+Cp−maxk,jn−ReαSk1/rSj1−1/q−1/q1fjLqℚpn∫Sjb1y−b1Bjq1dy1/q1≤Cp−maxk,jn−ReαSj1−1/qfjLqℚpnSk1/r−1/q1×∫Skb1x−b1Bkq1dx1/q1+Bk1/q1b1Bk−b1Bj+Cp−maxk,jn−ReαSk1/rSj1−1/q−1/q1Bj1/q1+ν1fjLqℚpnb1CBMOq1,ν1≤Cp−maxk,jn−ReαSj1−1/qSk1/r−1/q1fjLqℚpnb1CBMOq1,ν1×Bk1/q1+ν1+Bk1/q1pnj−kmaxBkv1,Bjv1+Cp−maxk,jn−ReαSk1/rSj1−1/q−1/q1Bj1/q1+ν1fjLqℚpnb1CBMOq1,ν1≤Cp−maxk,jn−Reαpjn1−1/qpkn1/r−1/q1b1CBMOq1,ν1fjLqℚpn×pkn1/q1+ν1+pkn/q1pnj−kpmaxk,jnv1+Cp−maxk,jn−Reαpjn1−1/qpkn/rpjnν1b1CBMOq1,ν1fjLqℚpn≤Cp−maxk,jn−Reαpjn1−1/qpkn/rb1CBMOq1,ν1fjLqℚpn×pknν1+pnj−kpmaxk,jnv1+Cp−maxk,jn−Reαpjn1−1/qpkn/rpjnν1b1CBMOq1,ν1fjLqℚpn≤Cp−maxk,jn−Reαpjn1−1/qpkn/rpmaxk,jnv1j−kb1CBMOq1,ν1fjLqℚpn≤Ck−jpknv1−1+Reα/n+1/rpjn1−1/qb1CBMOq1,ν1fjLqℚpn,ifj≤k−2Cj−kpjnv1−1+Reα/npjn1−1/qpkn/rb1CBMOq1,ν1fjLqℚpn,ifj≥k+2≤Ck−jpknv1−1+1/q1+1/qpjn1−1/qb1CBMOq1,ν1fjLqℚpn,ifj≤k−2Cj−kpjnv1+1/q1−1/rpkn/rb1CBMOq1,ν1fjLqℚpn,ifj≥k+2≤Ck−jpk−jσ1−n1−1/qpjσ1p−kσ2b1CBMOq1,ν1fjLqℚpn,ifj≤k−2Cj−kpk−jσ1+n/q−Reα−nν1pjσ1p−kσ2b1CBMOq1,ν1fjLqℚpn,ifj≥k+2.
Thus, the factnν1+λ−n/q+Reα<σ1<n1−1/q+λ and (43) imply
(55)E1≤Cb1CBMOq1,ν1×supk0∈ℤp−k0λ∑k=−∞k0pkσ2l∑j=−∞k−2k−jpk−jσ1−n1−1/qpjσ1p−kσ2fjLqℚpnl1/l≤Cb1CBMOq1,ν1×supk0∈ℤp−k0λ∑k=−∞k0∑j=−∞k−2k−jpk−jσ1−n1−1/qpjσ1fjLqℚpnl1/l≤Cb1CBMOq1,ν1×supk0∈ℤp−k0λ∑k=−∞k0∑j=−∞k−2k−jpk−jσ1−n1−1/qpjλfMK̇l,qσ1,λℚpnl1/l≤Cb1CBMOq1,ν1fMK̇l,qσ1,λℚpn×supk0∈ℤp−k0λ∑k=−∞k0pkλl∑j=−∞k−2k−jpk−jσ1−n1−1/q−λl1/l≤Cb1CBMOq1,ν1fMK̇l,qσ1,λℚpnsupk0∈ℤp−k0λ∑k=−∞k0pkλl1/l≤Cb1CBMOq1,ν1fMK̇l,qσ1,λℚpn.(56)E3≤Cb1CBMOq1,ν1×supk0∈ℤp−k0λ∑k=−∞k0∑j=k+2∞j−kpk−jkσ1+n/q−Reα−nν1pjσ1fjLqℚpnl1/l≤Cb1CBMOq1,ν1×supk0∈ℤp−k0λ∑k=−∞k0∑j=k+2∞j−kpk−jkσ1+n/q−Reα−nν1pjλfMK̇l,qσ1,λℚpnl1/l≤Cb1CBMOq1,ν1fMK̇l,qσ1,λℚpn×supk0∈ℤp−k0λ∑k=−∞k0pkλl∑j=k+2∞j−kpk−jkσ1+n/q−Reα−nν1−λl1/l≤Cb1CBMOq1,ν1fMK̇l,qσ1,λℚpnsupk0∈ℤp−k0λ∑k=−∞k0pkλl1/l≤Cb1CBMOq1,ν1fMK̇l,qσ1,λℚpn.
Combining the estimates forE1, E2 and E3, which completes the proof for Theorem 3 of the case m=1.
Now, we consider the casem≥2. In order to simplify the proof, for positive integer m and 1≤i≤m, we denote by Cim the family of all finite subsets θ=θ1,θ2,⋯,θm of 1,2,⋯,m of i different elements, let θc=1,2,⋯,m\θ for any θ∈Cim. For b=b1,b2,⋯,bm, let bθ=bθ1,bθ2,⋯,bθi, bθ=bθ1bθ2⋯bθi and biBk denote the integral average of the function bi over the set Bk, then
(57)bx−bBkθ=bθ1x−bθ1Bk⋯bθix−bθiBk,(58)bBj−bBkθ=bθ1Bj−bθ1Bk⋯bθiBj−bθiBk,(59)bθCBMOq¯,ν¯=bθ1CBMOq¯1,ν¯1⋯bθiCBMOq¯m,ν¯m,where 1/q=1/q¯1+⋯+1/q¯m and ν¯=ν¯1+⋯+ν¯m.
We write(60)IαbfMK̇s,rσ2,λℚpn≤Csupk0∈ℤp−k0λ∑k=−∞k0pkσ2lIαbfχkLrℚpnl1/l≤Csupk0∈ℤp−k0λ∑k=−∞k0pkσ2l∑j=−∞k−2IαbfjχkLrℚpnl1/l+Csupk0∈ℤp−k0λ∑k=−∞k0pkσ2l∑j=k−1k+1IαbfjχkLrℚpnl1/l+Csupk0∈ℤp−k0λ∑k=−∞k0pkσ2l∑j=k+2∞IαbfjχkLrℚpnl1/l≡CG1+G2+G3.
Let us first estimateG2, we can obtain
(61)Iαbfjx=1Γnα∫ℚpnfjyx−ypn−α∏i=1mbix−biBk+biy−biBkdy=∏i=1mbix−biBkIαfjx+−1mIα∏i=1mbi−biBkfjx+∑i=1m−1∑θ∈Cim−1m−ibx−bBkθ1Γnα∫ℚpnfjyx−ypn−α∏i=1mby−bBkθcdy=∏i=1mbix−biBkIαfjx+−1mIα∏i=1mbi−biBkfjx+∑i=1m−1∑θ∈Cim−1m−ibx−bBkθIαb−bBkθcfjx≡H1+H2+H3.
ForH1, taking 1/s=1/q−Reα/n, then, 1/r=1/q1+1/q2+⋯+1/qm+1/s. Applying Hölder’s inequality, the boundedness of Iα from Lqℚpn to Lsℚpn and the fact σ2=σ1−n1/q1+⋯+1/qm+ν1+⋯+νm, we obtain
(62)∏i=1mbi−biBkIαfjχkLrℚpn≤C∫Bk∏i=1mbix−biBkIαfjxrdx1/r≤C∏i=1m∫Bkbix−biBkqidx1/qi∫BkIαfjxsdx1/s≤C∏i=1mBk1/qi+νibiCBMOqi,νifjLqℚpn≤C∏i=1mpkn1/qi+νibiCBMOqi,νifjLqℚpn≤Cpkσ1−σ2∏i=1mbiCBMOqi,νifjLqℚpn.
ForH2, taking 1/h=1/q1+⋯+1/qm+1/q, then, 1/r=1/h−Reα/n for h>1. By Hölder’s inequality, the boundedness of Iα from Lhℚpn to Lrℚpn and the fact σ2=σ1−n1/q1+⋯+1/qm+ν1+⋯+νm, we get
(63)−1mIα∏i=1mbi−biBkfjχkLrℚpn≤C∫Bk∏i=1mbix−biBkfjxhdx1/h≤C∏i=1m∫Bkbix−biBkqidx1/qi∫Bkfjxqdx1/q≤C∏i=1mBk1/qi+νibiCBMOqi,νifjLqℚpn≤C∏i=1mpkn1/qi+νibiCBMOqi,νifjLqℚpn≤Cpkσ1−σ2∏i=1mbiCBMOqi,νifjLqℚpn.
ForH3, we denote by
(64)1h=∑θi∈θ1qi,1z=∑θi∈θc1qi,1ν′=∑θi∈θ1νi,1ν″=∑θi∈θ1νi,and 1/ω=1/z+1/q−Reα/n, then 1/r=1/h+1/ω. Using Hölder’s inequality and the Lω-boundedness of Iα, we have
(65)∑i=1m−1∑θ∈Cim−1m−ib−bBkθIαb−bBkθcfjχkLrℚpn≤C∑i=1m−1∑θ∈Cim∫Bkbx−bBkθhdx1/h∫BkIαb−bBkθcfjxωdx1/ω≤C∑i=1m−1∑θ∈Cim∫Bkbx−bBkθhdx1/h∫Bkbx−bBkθczdx1/z×∫Bkfjxqdx1/q≤CBk1/h+ν′bθCBMOh,ν′Bk1/z+ν′′bθcCBMOz,ν′′fjLqℚpn≤C∏i=1mBk1/qi+νibiCBMOqi,νifjLqℚpn≤C∏i=1mpkn1qi+νibiCBMOqi,νifjLqℚpn≤Cpkσ1−σ2∏i=1mbiCBMOqi,νi∥fj∥Lqℚpn.
Then, similarly to the method estimating forE2, it is not difficult for us to get
(66)G2≤C∏i=1m∥bi∥CBMOqi,νisupk0∈ℤp−k0λ∑k=−∞k0pkσ2l∑j=k−1k+1pkσ1−σ2∥fj∥Lqℚpnl1/l≤C∏i=1m∥bi∥CBMOqi,νi∥f∥MK̇l,qσ1,λℚpn.
Next, we will estimateG1 and G3. Let
(67)1τ=∑θi∈θ1qi,1τ′=∑θi∈θc1/qi1ϑ=∑θi∈θ1νi,1ϑ′=∑θi∈θ1νi.
Therefore, by Minkowski’s inequality, Hölder’s inequality, and Lemma13, it follows that
(68)IαbfjχkLrℚpn≤Cp−maxk,jn−Reα∫Sk∫Sj∏i=1mbix−biyfydyrdx1/r≤Cp−maxk,jn−Reα∫Sj∫Sk∏i=1mbix−biyrdx1/rfydy≤Cp−maxk,jn−Reα∑i=0m∑θ∈Cim∫Skbx−bBkθrdx1/r×∫Sjby−bBkθcfydy≤Cp−maxk,jn−Reα∑i=0m∑θ∈Cim∫Skbx−bBkθτdx1/τBk1/r−1/τ×∫Sjby−bBkθcτ′dy1/τ′∫Sjfyq1/qBj1−1/τ′−1/q≤Cp−maxk,jn−Reα∥fj∥Lqℚpn∑i=0m∑θ∈Cim∫Skbx−bBkθτdx1/τBk1/r−1/τ×∫Sjby−bBjθcτ′dy1/τ′+Bj1/τ′bBk−bBjθcBj1−1/τ′−1/q≤Cp−maxk,jn−Reαpkn/rpjn1−1/q∥fj∥Lqℚpn∑i=0m∑θ∈Cimpknϑ∥bθ∥CBMOτ,ϑ×∥bθc∥CBMOτ′,ϑ′pjnϑ′+pmnj−kmpmaxk,jnϑ′≤Cp−maxk,jn−Reαpkn/rpjn1−1/q∥fj∥Lqℚpn×j−kmpmaxk,jnϑ∥bθ∥CBMOτ,ϑpmaxk,jnϑ′∥bθc∥CBMOτ′,ϑ′≤Cp−maxk,jn−Reαpkn/rpjn1−1/q∥fj∥Lqℚpnj−km∏i=1mpmaxk,jnνi∥bi∥CBMOqi,νi≤Cpmaxk,jnν−1+Reα/npkn/rpjn1−1/qj−km∏i=1m∥bi∥CBMOqi,νi∥fj∥Lqℚpn≤Ck−jmpk−jσ1−n1−1/qpjσ1p−kσ2∏i=1m∥bi∥CBMOqi,νi∥fj∥Lqℚpn,ifj≤k−2;Cj−kmpk−jσ1+n/q−Reα−nνpjσ1p−kσ2∏i=1m∥bi∥CBMOqi,νi∥fj∥Lqℚpn,ifj≥k+2.
Thus, similarly to the method estimating forE1 and E3, we get
(69)G1≤C∏i=1m∥bi∥CBMOqi,νi×supk0∈ℤp−k0λ∑k=−∞k0∑j=−∞k−2k−jmpk−jσ1−n1−1/qpjσ1∥fj∥Lqℚpnl1/l≤C∏i=1m∥bi∥CBMOqi,νifMK̇l,qσ1,λℚpn×supk0∈ℤp−k0λ∑k=−∞k0pkλl∑j=−∞k−2k−jmpk−jσ1−n1−1/q−λl1/l≤C∏i=1m∥bi∥CBMOqi,νifMK̇l,qσ1,λℚpn.(70)G3≤C∏i=1m∥bi∥CBMOqi,νi×supk0∈ℤp−k0λ∑k=−∞k0∑j=k+2∞j−kmpk−jσ1+n/q−Reα−nνpjσ1∥fj∥Lqℚpnl1/l≤C∏i=1m∥bi∥CBMOqi,νifMK̇l,qσ1,λℚpn×supk0∈ℤp−k0λ∑k=−∞k0pkλl∑j=k+2∞j−kmpk−jσ1+n/q−Reα−nν−λl1/l≤C∏i=1m∥bi∥CBMOqi,νifMK̇l,qσ1,λℚpn.
So far, by a combination for the estimates ofG1, G2, and G3, we will finish the proof for Theorem 3 of the case m≥2. This completes the proof of Theorem 17.



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