# Sharp Bounds for Generalizedm-Linear n-Dimensional p-Adic Hardy-Littlewood-Pólya Operator

Authors: Xiang Li; Xiran Zhang; Qianjun He
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we study the sharp bounds for the generalizedm-linear n-dimensional p-adic Hardy-Littlewood-Pólya operator on central and noncentral p-adic Morrey spaces with power weight. As an application, the sharp bounds for the p-adic Hardy-Littlewood-Pólya operator on corresponding Morrey spaces are obtained. These results generalize substantially some well-known results.


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## Body

## 1. Introduction and Main Results

In recent years,p-adic analysis has received a lot of attention due to its application in Mathematical Physics (cf. [1–6]). And harmonic analysis on the p-adic field has been drawing more and more concern (cf. [7–11] and references therein).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 [6], 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. The series (2) converges in the p-adic norm because ajpjp=p−γ.The spaceℚpn consists of points x=x1,x2,⋯,xn, where xj∈ℚp, j=1,2⋯,n. The p-adic norm on ℚpn is
(3)xp≔max1≤j≤nxjp,x∈ℚpn.Denote byBγa=x∈ℚpn:x−ap≤pγ, the ball with center at a∈ℚpn and radius pγ, and by Sγa≔x∈ℚpn:x−ap=pγ the sphere with center at a∈ℚpn and radius pγ, γ∈ℤ. It is clear that Sγa=Bγa\Bγ−1a, and
(4)Bγa=⋃k≤γSka.We setBγ0=Bγ and Sγ0=Sγ.Sinceℚpn is a locally compact commutative group under addition, it follows from the standard analysis that there exists a unique Harr measure dx on ℚpn (up to positive constant multiple) which is translation invariant. We normalize the measure dx so that
(5)∫B00dx=B00H=1,where EH denotes the Harr measure of a measurable subset E of ℚpn. From this integral theory, it is easy to obtain that BγaH=pγn and SγaH=pγn1−p−n for any a∈ℚpn. For a more complete introduction to the p-adic field, see [6, 12].Hardy-Littlewood-Pólya’s linear operator [13] is defined by
(6)Pfx=∫0+∞fymaxx,ydy.In [13], Bényi and Oh obtained that the norm of Hardy-Littlewood-Pólya’s operator on Lqℝ+ (see also P. 254 in [14]), 1<q<∞, is
(7)PLqℝ+⟶Lqℝ+=q2q−1.As a multilinear generalization of the Calderón operator, them-linear n-dimensional Hardy-Littlewood-Pólya operator is defined by
(8)Pf1,⋯,fmx=∫ℝnmf1y1⋯fmymmaxxn,y1n,⋯,ynnmdy1⋯dym.Computation of the operator norm of integral operators is a challenging work in harmonic analysis. In 2017, Batblod and Sawano [15] studied one-dimensional m-linear Hilbert-type operators that include the Hardy-Littlewood-Pólya operator on weighted Morrey spaces, and they obtained the sharp bounds. Later, He et al. [16] extended the results in [15] and obtain the sharp bound for the generalized Hardy-Littlewood-Pólya operator on power weighted central and noncentral homogeneous Morrey space. In 2011, Wu and Fu [17] got the best estimate of m-linear p-adic Hardy-Littlewood-Pólya operator on Lebesgue spaces with power weights. Recently, Deng et al. [18] improved the results in [17] on Morrey spaces, and they consider the following operator with n=1:
(9)Ppf1,⋯,fm=∫ℚpn⋯∫ℚpnf1y1⋯fmymmaxxpn,y1pn,⋯,ympnmdy1⋯ym.Inspired by [16, 17], we study a more general operator which includes the p-adic Hardy-Littlewood-Pólya operator as a special case and consider its operator norm on two power weighted p-adic Morrey spaces and their central version. Our results concern the sharp bound for the p-adic Hardy-Littlewood-Pólya operator on the product of weighted p-adic Morrey spaces, which generalizes the results in [17, 18]. For more information about the boundedness of some operators on function spaces (including p-adic field), see also [19–27].We present the definition of the weightedp-adic Morrey space Lq,λℚpn,w1,w2 and the weighted p-adic central Morrey spaces Bq,λℚpn,w1,w2 on ℚpn, where w1,w2: Qpn⟶0,+∞ are positive measurable functions.Definition 1.
Let1≤q<∞ and 1/q<λ<0, then the weighted p-adic Morrey space Lq,λℚpn,w1,w2 is the set of all f∈Llocqℚpn for which the norm
(10)fLq,λℚpn,w1,w2=supγ∈ℤ,a∈ℚpn∫Bγaw1xdx−λ+1/q∫Bγafxqw2xdx1/q<+∞.IfBγ replace with Bγa in the above definition, Lq,λℚpn,w1,w2 becomes the weighted p-adic central Morrey spaces Bq,λℚpn,w1,w2. In this paper, we are particularly interested in the norm
(11)fLq,λℚpn,xpα,xpβ=supγ∈ℤ,a∈ℚpn∫Bγaxpαdx−λ+1/q∫Bγafxqxpβdx1/q,fBq,λℚpn,xpα,xpβ=supγ∈ℤ,a∈ℚpn∫Bγxpαdx−λ+1/q∫Bγfxqxpβdx1/q.It is easy to verify the following scaling law.Lemma 2.
Let1≤q<∞, −1/q≤λ<0 and α,β∈ℝ. Ift∈ℚpn and f∈Lq,λℚpn,xpα,xpγorf∈Bq,λℚpn,xpα,xpγ, then we have
(12)ftp−1·Lq,λℚpn,xpα,xpβ=tpnλ−β/q+αλ+1/qfLq,λℚpn,xpα,xpβ,ftp−1·Bq,λℚpn,xpα,xpβ=tpnλ−β/q+αλ+1/qfBq,λℚpn,xpα,xpβ.Definition 3.
The generalizedm-linear p-adic operator is defined by
(13)Tpf1,⋯,fmx=∫ℚpn⋯∫ℚpnKy1,⋯,ym∏i=1mfixp−1yidy1⋯dym,for x∈ℚpn,∗ and fi is a measurable function on ℚpn with i=1,⋯,m. Note that Tp is in fact an integral operator having a homogeneous radial kernel K~ of degree −mn,
(14)Tpf1,⋯,fmx=∫ℚpn⋯∫ℚpnK~x,y1,⋯,ym∏i=1mfiyidy1⋯dym,where K~x,y1,⋯,ym=xp−mnKxpy1,⋯,xpym.In this paper, placing ourselves mainly in the setting ofLq,λℚpn,xpα,xpβ and Bq,λℚpn,xpα,xpβ, we obtain the operator norm of the m-linear p-adic operator as well as the one for the Hardy-Littlewood-Pólya operator Pp. We shall prove the following results.Theorem 4.
Let1<q<qi<∞, 1/q=1/q1+⋯+1/qm, β=β1+⋯+βm, λ=λ1+⋯+λmand−1/qi<λi<0 with i=1,⋯,m.
(i)
If(15)Cm=∫ℚpn⋯∫ℚpnKy1,⋯,ym∏i=1myip−nλi+βi/q−αλi+1/qidy1⋯dym<∞,then Tp is bounded from Bq1,λ1ℚpn,xpα,xpq1β1/q×⋯×Bqm,λmℚpn,xpα,xpqmβm/q to Bq,λℚpn,xpα,xpβ with its operator norm not more than Cm(ii)
Assume thatλq=λ1q1=⋯=λmqm and α≠−n. In this case, condition (15) is also necessary for the boundedness of Tp:Bq1,λ1ℚpn,xpα,xpq1β1/q×⋯×Bqm,λmℚpn,xpα,xpqmβm/q⟶Bq,λℚpn,xpα,xpβ. Moreover,(16)TpBq1,λ1ℚpn,xpα,xpq1β1/q×⋯×Bqm,λmℚpn,xpα,xpqmβm/q⟶Bq,λℚpn,xpα,xpβ=Cm.We remark Theorem4 when α=β=βj=0j=1,⋯,m goes back to [18], Theorem 2.2.A corresponding conclusion ofPp is also true.Corollary 5.
With the same assumptions as in Theorem4, the operator Pp maps Bq1,λ1ℚpn,xpα,xpq1β1/q×⋯×Bqm,λmℚpn,xpα,xpqmβm/q to Bq,λℚpn,xpα,xpβ with its operator norm not more than
(17)1−p−nm1−p−mn1−p−nλ+β/q−αλ+1/q∏i=1m1−pαλi+1/qi+nλi−n−βi/q.In particular, whenλq=λ1q1=⋯=λmqm and α≠−n, then the operator norm of Pp is equal to the above quantity.Next, we give sharp estimates of the generalizedm-linear p-adic operator on the product of p-adic Morrey spaces.Theorem 6.
Let1<q<qi<∞, 1/q=1/q1+⋯+1/qm, β=β1+⋯+βm, λ=λ1+⋯+λmand−1/qi<λi<0 with i=1,⋯,m. Suppose that fi∈Bqj,λiℚpn,xpα,xpqiβi/q satisfies that fix=fixp−1.
(i)
If condition (15) holds, then Tp is bounded from Lq1,λ1ℚpn,xpα,xpq1β1/q×⋯×Lqm,λmℚpn,xpα,xpqmβm/q to Lq,λℚpn,xpα,xpβ with its operator norm not more than Cm(ii)
Assume thatλq=λ1q1=⋯=λmqm and α≠−n. In this case, condition (15) is also necessary for the boundedness of Tp:Lq1,λ1ℚpn,xpα,xpq1β1/q×⋯×Lqm,λmℚpn,xpα,xpqmβm/q⟶Lq,λℚpn,xpα,xpβ. Moreover,(18)TpLq1,λ1ℚpn,xpα,xpq1β1/q×⋯×Lqm,λmℚpn,xpα,xpqmβm/q⟶Lq,λℚpn,xpα,xpβ=CmA corresponding conclusion ofPp is also true.Corollary 7.
With the same assumptions as in Theorem6, the operator Pp maps Lq1,λ1ℚpn,xpα,xpq1β1/q×⋯×Lqm,λmℚpn,xpα,xpqmβm/q to Lq,λℚpn,xpα,xpβ with its operator norm not more than
(19)1−p−nm1−p−mn1−p−nλ+β/q−αλ+1/q∏i=1m1−pαλi+1/qi+nλi−n−βi/q.In particular, whenλq=λ1q1=⋯=λmqm and α≠−n, then the operator norm of Pp is equal to the above quantity.


## 2. Proof of the Main Results

First, we shall show the scaling law in the weightedp-adic Morrey space. We prove Lemma 2.Proof of Lemma 2.
We only prove the scaling inLq,λℚpn,xpα,xpβ, and the other one is similar. We compute that
(20)ftp−1·Lq,λℚpn,xpα,xpβ=supa∈ℚpn,γ∈ℤ∫Bγaxpαdx−λ+1/q∫Bγaftp−1xqxpβdx1/q=supa∈ℚpn,γ∈ℤ∫Bγatp−1xpα∣tp−αdx−λ+1/q∫Bγaftp−1xqtp−1xpβtp−βdx1/q=tpnλ−β/q+αλ+1/qsupa∈ℚpn,γ∈ℤ∫Bγ+logptpa/tpxpαdx−λ+1/q∫Bγ+logptpa/tpfxqxpβdx1/q=tpnλ−β/q+αλ+1/qfLq,λℚpn,xpα,xpβ.This finishes the Proof of Lemma2.Next, we will give the proof of Theorems4 and 6.Proof of Theorem 4.
Taking(21)gjx=11−p−n∫ξp=1fjxp−1ξdξ,x∈ℚpn,j=1,⋯,m.
Obviously,gj satisfies gjx=gjxp−1 and Tpg1,⋯,gmx is equal to
(22)∫ℚpn⋯∫ℚpnKy1,⋯,ymg1xp−1y1⋯gmxp−1ymdy1⋯dym=∫ℚpn⋯∫ℚpnKy1,⋯,ym∏j=1m11−p−n∫ξjp=1fjxp−1yjp−1ξjdξjdy1⋯dym=11−p−nm∫ℚpn⋯∫ℚpnKy1,⋯,ym∏j=1m∫ξjp=1fjxp−1yjp−1ξjdξjdy1⋯dym=11−p−nm∫ℚpn⋯∫ℚpnKy1,⋯,ym∏j=1m∫ξjp=yjpfxp−1zjyjp−ndzjdy1⋯dym=11−p−nm∫ℚpn⋯∫ℚpnKy1,⋯,ym∏j=1m∫yjp=zjpyjp−ndyjf1xp−1z1⋯fmxp−1zmdz1⋯dzm=∫ℚpn⋯∫ℚpnKy1,⋯,ymf1xp−1z1⋯fmxp−1zmdz1⋯dzm=Tpf1,⋯,fmx.Using generalized Minkowski’s inequality and Hölder’s inequality, forj=1,⋯,m, we conclude that
(23)gjBqj,λjℚpn,xpα,xpqjβj/q≤11−p−nsupγ∈ℤ∫Bγxpαdx−λ−1/q∫ξjp=1∫Bγfjxp−1ξjqxpqjβj/qdx1/qdξj≤supγ∈ℤ∫Bγxpαdx−λ−1/q11−p−n∫ξjp=1∫Bγfjxp−1ξjqxpqjβj/qdxdξj1/q=supγ∈ℤ∫Bγxpαdx−λ−1/q11−p−n∫Bγ∫zjp=xpfzjqxpqjβj/qxp−ndzjdx1/q=fjBqj,λjℚpn,xpα,xpqjβj/q.Therefore, one has(24)Tpf1,⋯,fmBq,λℚpn,xpα,xpβ∏j=1mfjBqj,λjℚpn,xpα,xpqjβj/q≤Tpg1,⋯,gmBq,λℚpn,xpα,xpβ∏j=1mgjBqj,λjℚpn,xpα,xpqjβj/q.This implies that the operatorTp and its restriction to the functions g satisfying gx=gxp−1 have the same operator norm in Bq,λℚpn,xpα,xpβ. So, without loss of generality, we assume that fj∈Bqj,λjℚpn,xpα,xpqjβj/q, j=1,⋯,m, satisfying that fjx=fjxp−1 in the rest of the proof.By Minkowski’s inequality and Hölder’s inequality, we have(25)Tpf1,⋯,fmBq,λℚpn,xpα,xpβ≤∫ℚpn⋯∫ℚpnKy1,⋯,ym∏j=1mfjyjp−1·Bq,λℚpn,xpα,xpβdy1⋯dym≤∫ℚpn⋯∫ℚpnKy1,⋯,ym∏j=1mfyjp−1·Bqj,λjℚpn,xpα,xpqjβj/qdy1⋯dym.Using Lemma2, we can deduce that
(26)Tpf1,⋯,fmBq,λℚpn,xpα,xpβ≤∫ℚpn⋯∫ℚpnKy1,⋯,ymyip−nλ1+βi/q−αλi+1/qidy1⋯dym∏j=1mfjBqj,λjℚpn,xpα,xpqjβj/q=Cm∏j=1mfjBqj,λjℚpn,xpα,xpqjβj/q.This means that(27)TpBq1,λ1ℚpn,xpα,xpq1β1/q×⋯×Bqm,λmℚpn,xpα,xpqmβm/q⟶Bq,λℚpn,xpα,xpβ≤Cm.For the necessity whenλq=λ1q1=⋯=λmqm and α≠−n, let fjx=xpnλj−βj/q+αλj+1/qj for all x∈ℚpn\0 and fj0≔0 (j=1,⋯,m). Then, for any Bγ=B0,pγ, we have that for λj>−1/qj,
(28)fjBqj,λjℚpn,xpα,xpqjβj/q=supγ∈ℤ∫Bγxpαdx−λj−1/qj∫Bγxpnλjqj−qjβj/q+αλjqj+1xpqjβj/qdx1/qj=supγ∈ℤ∑k=−∞γpαk∫Skdx−λj−1/qj∑k=−∞γpλjqjα+n+αk∫Skdx1/qj=supγ∈ℤ1−p−n−λj∑k=−∞γpα+nk−λj−1/qj∑k=−∞γpλjqj+1α+nk1/qj.Whenα+n>0, we have
(29)∑k=−∞γpα+nk=pα+nγ1−p−α+n,∑k=−∞γpλjqj+1α+nk=pλjqj+1α+nγ1−p−λjqj+1α+n.In this case, we obtain(30)fjBqj,λjℚpn,xpα,xpqjβj/q=1−p−n−λj1−p−α+n−λj+1/qj1−p−λjqj+1α+n1/qj.Whenα+n<0, then we calculate that
(31)supγ∈ℤ∑k=−∞γpα+nk−λj−1/qj∑k=−∞γpλjqj+1α+nk1/qj=supγ∈ℤlimN⟶+∞∑k=−Nγpkα+n−λj+1/qj∑k=−Nγpkqjλj+1α+n1/qj=limN⟶+∞pγα+n1−p−α+nN+γ1−p−α+n−λ+1/qpγqλ+1α+n1−p−qλ+1α+nN+γ1−p−qjλj+1α+n1/qj=1−p−n−λjp−α+n−1λj+1/qjp−α+nqjλj+1−11/qjlimN⟶+∞supγ∈ℤp−α+nN+γ−1−λj+1/qjp−α+nqλ+1N+γ−11/qj.Using the fact:tab−1≤ta−1b when a>0, 0<b≤1 and t≥1. Then, we have
(32)p−α+nqjλj+1N+γ−1≤p−α+nN+γ−1qjλj+1.Therefore, using the above inequality, we obtain(33)limN⟶+∞supγ∈ℤp−α+nN+γ−1−λj+1/qjp−α+nqjλj+1N+γ−11/qj=1.Hence, we show that forα+n<0,
(34)fjBqj,λjℚpn,xpα,xpqjβj/q=1−p−n−λjp−α+n−1λj+1/qjp−α+nqjλj+1−11/qj.Using (30) and (34), we have that fj∈Bqj,λjℚpn,xpα,xpqjβj/q. Since λ=λ1+⋯+λm, −1/qj<λj<0 and 1<q<qj<∞ with j=1,⋯,m, we have
(35)supγ∈ℤ∫Bγxpαdx−λ+1/q∫BγTpf1,⋯,fmxqxpβdx1/q=Cmsupγ∈ℤ∫Bγxpαdx−λ+1/q∫Bγxpnλq−β+αλq+1xpβdx1/q.Using the above argument forλq=λ1q1=⋯=λmqm, we have
(36)supγ∈ℤ∫Bγxpαdx−λ+1/q∫Bγxpnλq−β+αλq+1xpβdx1/q=∏j=1mfjBqj,λjℚpn,xpα,xpqjβj/q.Therefore, we show that(37)Cm≤TpBq1,λ1ℚpn,xpα,xpq1β1/q×⋯×Bqm,λmℚpn,xpα,xpqmβm/q⟶Bq,λℚpn,xpα,xvpβ<∞.Combining (27) and (37) then concludes the proof. This finishes the Proof of Theorem 4.Proof of Corollary 5.
Taking(38)Ky1,⋯,ym=1max1,y1pn,⋯,ympnm.
Using the result in Theorem4, similar to computing the sharp bound of Theorem 3.1 in [17], we obtain
(39)Cm∗≔∫ℚpn⋯∫ℚpn∏i=1myip−nλi+βi/q−αλi+1/qimax1,y1pn,⋯,ympnmdy1⋯dym=1−p−nm1−p−mn1−p−nλ+β/q−αλ+1/q∏i=1m1−pαλi+1/qi+nλi−n−βi/q.
This completes the Proof of Corollary5.Proof of Theorem 6.
Similar to Proof of Theorem4, it is easy to obtain that
(40)TpLq1,λ1ℚpn,xpα,xpq1β1/q×⋯×Lqm,λmℚpn,xpα,xpqmβm/q⟶Lq,λℚpn,xpα,xpβ≤Cm.
For the necessity whenλq=λ1q1=⋯=λmqm and α≠−n, let fjx=xpnλj−βj/q+αλj+1/qj for all x∈ℚpn\0 and fj0≔0 (j=1,⋯,m). Then, for any Bγa=Ba,pγ, we need to show that fj∈Lqj,λjℚpn,xpα,xpqjβj/q. Consider the following two cases.
(i)
Ifap>pγ and x∈Bγa, then xp=maxx−ap,ap=ap>pγ. Since −1/qj<λj<0, we have(41)fjLqj,λjℚpn,xpα,xpqjβj/q=supa∈ℚpn,γ∈ℤ∫Bγaxpαdx−λj+1/qj∫Bγaxpnλjqj−qjβj/q+αλjqj+1xpqjβj/qdx1/qj=supa∈ℚpn,γ∈ℤapnλjBγaH−λj=supa∈ℚpn,γ∈ℤapnλjp−nλjγ≤1(ii)
Ifap≤pγ and x∈Bγa, then xp=maxx−ap,ap≤pγ. Therefore, x∈Bγa. Recall that two balls in ℚpn are either disjoint or contained in the other (see [28]). So, we have Bγa=Bγ; thus,(42)fjLqj,λjℚpn,xpα,xpqjβj/q=supa∈ℚpn,γ∈ℤ∫Bγaxpαdx−λj+1/qj∫Bγaxpnλjqj−qjβj/q+αλjqj+1xpqjβj/qdx1/qj=supγ∈ℤ∫Bγxpαdx−λj+1/qj∫Bγxpnλjqj+αλjqj+1dx1/qj
By the argument in the Proof of Theorem4, we have that for ap≤pγ and x∈Bγa,
(43)fjLqj,λjℚpn,xpα,xpqjβj/q<∞.
From the previous discussion, we can see thatfj∈Lqj,λjℚpn,xpα,xpqjβj/q. By similar estimates to the method of Theorem 4, we have
(44)Cm≤TpBq1,λ1ℚpn,xpα,xpq1β1/q×⋯×Bqm,λmℚpn,xpα,xpqmβm/q⟶Bq,λℚpn,xpα,xpβ<∞.
Combining (40) and (44) then yields the desired result.Proof of Corollary 7.
This proof is similar to the Proof of Corollary5; we omit the details.



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