# Sharp Estimates ofm-Linear p-Adic Hardy and Hardy-Littlewood-Pólya Operators

Authors: Qingyan Wu; Zunwei  Fu
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

The sharp estimates of them-linear p-adic Hardy and Hardy-Littlewood-Pólya operators on Lebesgue spaces with power weights are obtained in this paper.


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## Body

## 1. Introduction

In recent years,p-adic numbers are widely used in theoretical and mathematical physics (cf. [1–8]), such as string theory, statistical mechanics, turbulence theory, quantum mechanics, and so forth.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: |0|p=0; If any nonzero rational number x is represented as x=pγ(m/n), where m and n are integers which are not divisible by p and γ is an integer, then |x|p=p-γ. It is not difficult to show that the norm satisfies the following properties: (1.1)|xy|p=|x|p|y|p,|x+y|p≤max⁡{|x|p,|y|p}.
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 (1.2)x=pγ∑j=0∞ajpj,γ=γ(x)∈Z,
where aj are integers, 0≤aj≤p-1, a0≠0. The series (1.2) converges in the p-adic norm because |ajpj|p=p-j. Denote by ℚp*=ℚp∖{0} and ℤp={x∈ℚp:|x|p≤1}.The spaceℚpn consists of points x=(x1,x2,…,xn), where xj∈ℚp,j=1,2,…,n. The p-adic norm on ℚpn is (1.3)|x|p:=max⁡1≤j≤n|xj|p,x∈Qpn.
Denote by (1.4)Bγ(a)={x∈Qpn:|x-a|p≤pγ},
the ball with center at a∈ℚpn and radius pγ, and (1.5)Sγ(a):={x∈Qpn:|x-a|p=pγ}=Bγ(a)∖Bγ-1(a).
Since ℚpn is a locally compact commutative group under addition, it follows from the standard analysis that there exists a Haar measure dx on ℚpn, which is unique up to positive constant multiple and is translation invariant. We normalize the measure dx by the equality (1.6)∫B0(0)dx=|B0(0)|H=1,
where |E|H denotes the Haar measure of a measurable subset E of ℚpn. By simple calculation, we can obtain that (1.7)|Bγ(a)|H=pγn,|Sγ(a)|H=pγn(1-p-n),
for any a∈ℚpn. For a more complete introduction to the p-adic field, see [6] or [9].The spaceℚpn×ℚpn×⋯×ℚpn︸m consists of points (y1,y2,…,ym), where yi=(yi1,yi2,…,yin)∈ℚpn, i=1,2,…,m. The p-adic norm of m-tuple (y1,y2,…,ym) is (1.8)|(y1,y2,…,ym)|p:=max⁡1≤i≤m|yi|p.Recently,p-adic analysis has received a lot of attention due to its application in mathematical physics. There are numerous papers on p-adic analysis, such as [10, 11] about Riesz potentials, [12–16] about p-adic pseudodifferential equations, and so forth. The harmonic analysis on p-adic field has been drawing more and more concern (cf. [17–21] and references therein).The well-known Hardy’s integral inequality [22] tells us that for 1<q<∞, (1.9)‖Hf‖Lq(R+)≤qq-1‖f‖Lq(R+),
where the classical Hardy operator is defined by (1.10)Hf(x):=1x∫0xf(t)dt,
for nonnegative integral function f on ℝ+, and the constant q/(q-1) is the best possible. Thus the norm of Hardy operator on Lq(ℝ+) is (1.11)‖H‖Lq(R+)→Lq(R+)=qq-1.Faris [23] introduced the following n-dimensional Hardy operator, for nonnegative function f on ℝn, (1.12)Hf(x):=1Ωn|x|n∫|y|<|x|f(y)dy,x∈Rn∖{0},
where Ωn is the volume of the unit ball in ℝn. Christ and Grafakos [24] obtained that the norm of ℋ on Lq(ℝn) is (1.13)‖H‖Lq(Rn)→Lq(Rn)=qq-1,
which is the same as that of the 1-dimension Hardy operator. In [25], Fu et al. introduced the m-linear Hardy operator, which is defined by (1.14)Hm(f1,…,fm)(x)=1Ωmn|x|mn∫|(y1,…,ym)|<|x|f1(y1)⋯fm(ym)dy1⋯dym,
where x∈ℝn∖{0} and f1,…,fm are nonnegative locally integrable functions on ℝn. And they obtained the precise norms of ℋm on Lebesgue spaces with power weight. The authors of  [26] also got the best constants  of m-linear Hilbert, Hardy and Hardy-Littlewood-Pólya operators on Lebesgue spaces.The study of multilinear averaging operators in Euclidean spaces is a byproduct of the recent interest in multilinear singular integral operator theory. This subject was established by Coifman and Meyer [27] in 1975. In this article, we consider the sharp estimates of m-linear p-adic Hardy and Hardy-Littlewood-Pólya operators. In contrast with [25], we use a new technique in calculations based on the feature of p-adic field, and Theorem 3.1 is also new. They cannot be obtained immediately by [25]. In [28], we defined the p-adic Hardy operator.Definition 1.1.
For a functionf on ℚpn, we define the p-adic Hardy operator as follows
(1.15)Hpf(x)=1|x|pn∫B(0,|x|p)f(t)dt,x∈Qpn∖{0},
where B(0,|x|p) is a ball in ℚpn with center at 0∈ℚpn and radius |x|p.It is obvious that|ℋpf|≤ℳpf, where ℳp is the Hardy-Littlewood maximal operator [17] defined by (1.16)Mpf(x)=sup⁡γ∈Z1|Bγ(x)|H∫Bγ(x)|f(y)|dy,f∈Lloc⁡1(Qpn).
The Hardy-Littlewood maximal operator plays an important role in harmonic analysis. The boundedness of ℳp on Lq(ℚpn) has been solved (see, e.g., [9]). But the best estimate of ℳp on Lq(ℚpn),q>1, even that of Hardy-Littlewood maximal operator on Euclidean spaces ℝn is very difficult to obtain. Instead, we have obtained the sharp estimates of ℋp (and p-adic Hardy-Littlewood-Pólya operator) elsewhere.Definition 1.2.
Letm be a positive integer and f1,…,fm be nonnegative locally integrable functions on ℚpn. The m-linearp-adic Hardy operator is defined by
(1.17)Hmp(f1,…,fm)(x)=1|x|pmn∫|(y1,…,ym)|p≤|x|pf1(y1)⋯fm(ym)dy1⋯dym,
where x∈ℚpn∖{0}.The Hardy-Littlewood-Pólya’s linear operator [26] is defined by (1.18)Tf(x)=∫0∞f(y)max⁡(x,y)dy.
In [26], the authors obtained that the norm of Hardy-Littlewood-Pólya’s operator on Lq(ℝ+) (see also [22, page 254]), 1<q<∞, is (1.19)‖T‖Lq(R+)→Lq(R+)=q2q-1.We define thep-adic Hardy-Littlewood-Pólya operator as (see [28]) (1.20)Tp(x)=∫Qpf(y)max⁡⁡(|x|p,|y|p)dy,x∈Qp*.Definition 1.3.
Letm be a positive integer and f1,…,fm be nonnegative locally integrable functions on ℚpn. The m-linear p-adic Hardy-Littlewood-Pólya operator is defined by
(1.21)Tmp(f1,…,fm)(x)=∫Qp⋯∫Qpf1(y1)⋯fm(ym)[max⁡(|x|p,|y1|p,…,|ym|p)]mdy1⋯dym,x∈Qp*.We obtain the sharp estimates of them-linear p-adic Hardy operator on Lebesgue spaces with power weights in Section 2. In Section 3, we get the best estimate of m-linear p-adic Hardy-Littlewood-Pólya operator on Lebesgue spaces with power weights.In the following sequel, fork∈ℤ, we denote Bk={x∈ℚpn:|x|p≤pk} and Sk={x∈ℚpn:|x|p=pk}.


## 2. Sharp Estimates ofm-Linear p-Adic Hardy Operator

Theorem 2.1.
Letm∈ℤ+,  fi∈Lqi(|x|pαiqi/qdx),  1<qi<∞,  i=1,2,…,m,  1≤q<∞,  1/q=∑i=1m(1/qi),  αi<qn(1-(1/qi)) and α=∑i=1mαi. Then
(2.1)‖Hp(f1,…,fm)‖Lq(|x|pαdx)≤CH‖f1‖Lq1(|x|pα1q1/qdx)⋯‖fm‖Lqm(|x|pαmqm/qdx),
where
(2.2)CH=(1-p-n)m∏i=1m(1-pn/qi+(αi/q)-n)
is the best constant.Whenα=0, we get the sharp estimates of the m-linear p-adic Hardy operator on Lebesgue spaces.Corollary 2.2.
Letm∈ℤ+,  1<qi<∞,  i=1,2,…,m,  1≤q<∞ and 1/q=∑i=1m1/qi.  Then
(2.3)‖Hmp‖Lq1(Qpn)×⋯×Lqm(Qpn)→Lq(Qpn)=(1-p-n)m∏i=1m(1-p(n/qi)-n).Proof of Theorem2.1.
Since the proof of the case whenm=1 is similar to and even simpler than that of the case when m≥2, for simplicity, we will only give the proof of case when m≥2. To make the proof clearer, we will discuss it in two parts.(I) Whenm=2
Firstly, we claim that the operatorℋp and its restriction to the functions g satisfying g(x)=g(|x|p-1) have the same operator norm on Lq(|x|pαdx). In fact, set
(2.4)gi(x)=11-p-n∫|ξi|p=1fi(|x|p-1ξi)dξi,x∈Qpn,i=1,2.
It’s clear that gi(x)=gi(|x|p-1), i=1,2, and
(2.5)H2p(g1,g2)(x)=1|x|p2n∫|(y1,y2)|p≤|x|pg1(y1)g2(y2)dy1dy2=1(1-p-n)21|x|p2n∫|(y1,y2)|p≤|x|p∏i=12(∫|ξi|p=1fi(|yi|p-1ξi)dξi)dy1dy2=1(1-p-n)21|x|p2n∫|(y1,y2)|p≤|x|p∏i=12(∫|zi|p=|yi|pfi(zi)|yi|p-ndzi)dy1dy2=1(1-p-n)21|x|p2n∫|(z1,z2)|p≤|x|p∏i=12(∫|yi|p=|zi|p|yi|p-ndyi)f1(z1)f2(z2)dz1dz2=1|x|p2n∫|(z1,z2)|p≤|x|pf1(z1)f2(z2)dz1dz2=H2p(f1,f2)(x).
By Hölder's inequality, we get(2.6)‖gi‖Lqi(|x|pαiqi/qdx)=(∫Qpn|11-p-n∫|ξi|p=1fi(|x|p-1ξi)dξi|qi|x|pαiqi/qdx)1/qi≤{∫Qpn1(1-p-n)qi(∫|ξi|p=1|fi(|x|p-1ξi)|qidξi)(∫|ξi|p=1dξi)qi-1|x|pαiqi/qdx}1/qi={∫Qpn11-p-n(∫|ξi|p=1|fi(|x|p-1ξi)|qidξi)|x|pαiqi/qdx}1/qi=1(1-p-n)1/qi{∫Qpn(∫|zi|p=|x|p|fi(zi)|qidzi)|x|p(αiqi/q)-ndx}1/qi=1(1-p-n)1/qi{∫Qpn(∫|x|p=|zi|p|x|p(αiqi/q)-ndx)|fi(zi)|qidzi}1/qi=‖fi‖Lqi(|x|pαiqi/qdx),i=1,2.
Therefore,
(2.7)‖H2p(f1,f2)‖Lq(|x|pαdx)‖f1‖Lq1(|x|pα1q1/qdx)‖f2‖Lq2(|x|pα2q2/qdx)≤‖H2p(g1,g2)‖Lq(|x|pαdx)‖g1‖Lq1(|x|pα1q1/qdx)‖g2‖Lq2(|x|pα2q2/qdx),
which implies the claim. In the following, without loss of generality, we may assume that fi∈Lqi(|x|pαiqi/qdx), i=1,2, which satisfy that fi(x)=fi(|x|p-1), i=1,2.
By changing of variablesyi=|x|p-1zi, i=1,2, we have
(2.8)‖H2p(f1,f2)‖Lq(|x|pαdx)=(∫Qpn|1|x|p2n∫|(y1,y2)|p≤|x|pf1(y1)f2(y2)dy1dy2|q|x|pαdx)1/q=(∫Qpn|∫|(z1,z2)|p≤1f1(|x|p-1z1)f2(|x|p-1z2)dz1dz2|q|x|pαdx)1/q=(∫Qpn|∫|(z1,z2)|p≤1f1(|z1|p-1x)f2(|z2|p-1x)dz1dz2|q|x|pαdx)1/q.
Then using Minkowski's integral inequality and Hölder's inequality ((q/q1)+(q/q2)=1), we get
(2.9)‖H2p(f1,f2)‖Lq(|x|pαdx)≤∫|(z1,z2)|p≤1(∫Qpn|f1(|z1|p-1x)f2(|z2|p-1x)|q|x|pαdx)1/qdz1dz2≤∫|(z1,z2)|p≤1∏i=12(∫Qpn|fi(|zi|p-1x)|qi|x|pαiqi/qdx)1/qidz1dz2=(∫|(z1,z2)|p≤1∏i=12|zi|p-(n/qi)-(αi/q)dz1dz2)∏i=12‖fi‖Lqi(|x|pαiqi/qdx).
By calculation, we have(2.10)∫|(z1,z2)|p≤1∏i=12|zi|p-(n/qi)-(αi/q)dz1dz2=∫|z1|p≤1∫|z2|p≤|z1|p∏i=12|zi|p-(n/qi)-(αi/q)dz1dz2+∫|z2|p≤1∫|z1|p<|z2|p∏i=12|zi|p-(n/qi)-(αi/q)dz1dz2=∫|z1|p≤1|z1|p-(n/q1)-(α1/q)(∑k=-∞log⁡p|z1|p∫Sk|z2|p-(n/q2)-(α2/q)dz2)dz1+∫|z2|p≤1|z2|p-(n/q2)-(α2/q)(∑k=-∞log⁡p|z2|p-1∫Sk|z1|p-(n/q1)-(α1/q)dz1)dz2=(1-p-n)1-p(n/q2)+(α2/q)-n∫|z1|p≤1|z1|p-(n/q)-(α/q)+ndz1+(1-p-n)p(n/q1)+(α1/q)-n1-p(n/q1)+(α1/q)-n∫|z2|p≤1|z2|p-(n/q)-(α/q)+ndz2=(1-p-n)2∏i=12(1-p(n/qi)+(αi/q)-n).
Therefore,
(2.11)‖H2p‖Lq1(|x|pα1q1/qdx)×Lq2(|x|pα2q2/qdx)→Lq(|x|pαdx)≤(1-p-n)2∏i=12(1-p(n/qi)+(αi/q)-n).
Now let us prove that our estimate is sharp. For0<ϵ<1 and |ϵ|p>1, we take
(2.12)fiϵ(xi)={0,|xi|p<1,|xi|p-(n/qi)-(αi/q)-(q2ϵ/qi),|xi|p≥1,i=1,2.
Then by calculation, we have
(2.13)‖f1ϵ‖Lq1(|x|pα1q1/qdx)q1=‖f2ϵ‖Lq2(|x|pα2q2/qdx)q2=1-p-n1-p-ϵq2.
It is clear that when|x|p<1, ℋ2p(f1ϵ,f2ϵ)(x)=0. But when |x|p≥1,
(2.14)H2p(f1ϵ,f2ϵ)(x)=|x|p-(n/q)-(α/q)-(q2ϵ/q)∫|(y1,y2)|p≤1,|y1|p≥1/|x|p,|y2|p≥1/|x|p∏i=12|yi|p-(n/qi)-(αi/q)-(q2ϵ/qi)dy1dy2.
Since|ϵ|p>1, we get
(2.15)‖H2p(f1ϵ,f2ϵ)‖Lq(|x|pαdx)={∫|x|p≥1(∏i=12|yi|p-(n/qi)-(αi/q)-q2ϵ/qi|x|p-(n/q)-(α/q)-(q2ϵ/q)×∫|(y1,y2)|p≤1,|y1|p≥1/|x|p,|y2|p≥1/|x|p∏i=12|yi|p-(n/qi)-(αi/q)-(q2ϵ/qi)dy1dy2)q|x|pαdx}1/q≥{∫|x|p≥|ϵ|p(∏i=12|yi|p-(n/qi)-(αi/q)-q2ϵ/qi|x|p-(n/q)-(α/q)-(q2ϵ/q)×∫|(y1,y2)|p≤1,|y1|p≥1/|ϵ|p,|y2|p≥1/|ϵ|p∏i=12|yi|p-(n/qi)-(αi/q)-(q2ϵ/qi)dy1dy2)q|x|pαdx}1/q=(∫|(y1,y2)|p≤1,|y1|p≥1/|ϵ|p,|y2|p≥1/|ϵ|p∏i=12|yi|p-(n/qi)-(αi/q)-(q2ϵ/qi)dy1dy2)(∫|x|p≥|ϵ|p|x|p-n-ϵq2dx)1/q=(∫|(y1,y2)|p≤1,|y1|p≥1/|ϵ|p,|y2|p≥1/|ϵ|p∏i=12|yi|p-(n/qi)-(αi/q)-(q2ϵ/qi)dy1dy2)|ϵ|p-ϵq2/q∏i=12‖fiϵ‖Lqi(|x|pαiqi/qdx).
By the same calculation as that in (2.10), we obtain that
(2.16)∫|(y1,y2)|p≤1,|y1|p≥1/|ϵ|p,|y2|p≥1/|ϵ|p∏i=12|yi|p-(n/qi)-(αi/q)-(q2ϵ/qi)dy1dy2=(1-p-n)2[1-(p|ϵ|p)(n/q)+(α/q)+(q2ϵ/q)-2n]∏i=12(1-p(n/qi)+(αi/q)+(q2ϵ/qi)-n).
Therefore,
(2.17)|ε|p-ϵq2/q(1-p-n)2[1-(p|ϵ|p)(n/q)+(α/q)+(q2ϵ/q)-2n]∏i=12(1-p(n/qi)+(αi/q)+(q2ϵ/qi)-n)≤‖H2p‖Lq1(|x|pα1q1/qdx)×Lq2(|x|pα2q2/qdx)→Lq(|x|pαdx).
Now takeϵ=p-k, k=1,2,3,…. Then |ϵ|p=pk>1. Letting k approach to ∞, then ϵ approaches to 0 and |ϵ|p-ϵq2/q approaches to 1. Since αi<qn(1-(1/qi)), i=1,2, we have
(2.18)(1-p-n)2∏i=12(1-p(n/qi)+(αi/q)-n)≤‖H2p‖Lq1(|x|pα1q1/qdx)×Lq2(|x|pα2q2/qdx)→Lq(|x|pαdx).
Then (2.11) and (2.18) imply that
(2.19)‖H2p‖Lq1(|x|pα1q1/qdx)×Lq2(|x|pα2q2/qdx)→Lq(|x|pαdx)=(1-p-n)2∏i=12(1-p(n/qi)+(αi/q)-n).(II) Whenm≥3
The proof of the upper bound in this case is similar to that of the previous case, and we can obtain that(2.20)‖Hmp(f1,…,fm)‖Lq(|x|pαdx)≤CH‖f1‖Lq1(|x|pα1q1/qdx)⋯‖fm‖Lqm(|x|pαmqm/qdx),
where
(2.21)CH=∫|(z1,…,zm)|p≤1∏k=1m|zk|p-(n/qk)-(αk/q)dz1⋯dzm.
Let(2.22)D1={(z1,…,zm)∈Qpn×⋯×Qpn∣|z1|p≤1,|zk|p≤|z1|p,1<k≤m},Di={(z1,…,zm)∈Qpn×⋯×Qpn∣|zi|p≤1,|zj|p<|zi|p,|zk|p≤|zi|p,1≤j<i<k≤m},Dm={(z1,…,zm)∈Qpn×⋯×Qpn∣|zm|p≤1,|zj|p<|zm|p,1≤j<m}.
It is  clear that
(2.23)⋃j=1mDj={(z1,…,zm)∈Qpn×⋯×Qpn∣|(z1,…,zm)|p≤1},
and Di∩Dj=∅. Then
(2.24)CH=∑j=1m∫Dj∏k=1m|zk|p-(n/qk)-(αk/q)dz1⋯dzm:=∑j=1mIj.
Now let us calculateIj, j=1,2,…,m, respectively,
(2.25)I1=∫D1∏k=1m|zk|p-(n/qk)-(αk/q)dz1⋯dzm=∫|z1|p≤1|z1|p-(n/q1)-(α1/q)(∏k=2m∫|zk|p≤|z1|p|zk|p-(n/qk)-(αk/q)dzk)dz1=∫|z1|p≤1|z1|p-(n/q1)-(α1/q)(∏k=2m(∑j=-∞log⁡p|z1|p∫Sj|zk|p-(n/qk)-(αk/q)dzk))dz1=(1-p-n)m-1∏k=2m(1-p(n/qk)+(αk/q)-n)∫|z1|p≤1|z1|p-(n/q)-(α/q)+(m-1)ndz1=(1-p-n)m(1+p(n/q)+(α/q)-mn)∏k=2m(1-p(n/qk)+(αk/q)-n).
Similarly, fori=2,…,m-1, we have
(2.26)Ii=∫Di∏k=1m|zk|p-(n/qk)-(αk/q)dz1⋯dzm=∫|zi|p≤1|zi|p-(n/qi)-(αi/q)(∏j=1i-1∫|zj|p<|zi|p|zj|p-(n/qj)-(αj/q)dzj)×(∏k=i+1m∫|zk|p≤|zi|p|zk|p-(n/qk)-(αk/q)dzk)dzi=(1-p-n)m-1∏j=1i-1p(n/qj)+(αj/q)-n∏1≤k≤m,k≠i(1-p(n/qk)+(αk/q)-n)∫|zi|p≤1|zi|p-(n/q)-(α/q)+(m-1)ndzi=(1-p-n)m∏j=1i-1p(n/qj)+(αj/q)-n(1+p(n/q)+(α/q)-mn)∏1≤k≤m,k≠i(1-p(n/qk)+(αk/q)-n),Im=∫|zm|p≤1|zm|p-(n/qm)-(αm/q)(∏j=1m-1∫|zj|p<|zm|p|zj|p-(n/qj)-(αj/q)dzj)dzm=(1-p-n)m∏j=1m-1p(n/qj)+(αj/q)-n(1+p(n/q)+(α/q)-mn)∏j=1m-1(1-p(n/qj)+(αj/q)-n).
Therefore,
(2.27)CH=(1-p-n)m(1+p(n/q)+(α/q)-mn)∏k=2m(1-p(n/qk)+(αk/q)-n)+∑i=2m-1(1-p-n)m∏j=1m-1p(n/qj)+(αj/q)-n(1+p(n/q)+(α/q)-mn)∏k=1m-1(1-p(n/qk)+(αk/q)-n)+(1-p-n)m∏j=1m-1p(n/qj)+(αj/q)-n(1+p(n/q)+(α/q)-mn)∏j=1m-1(1-p(n/qj)+(αj/q)-n)=(1-p-n)m∏i=1m(1-p(n/qi)+(nαi/q)-n).
To show thatCℋ is the best constant, we should prove that it is also the lower bound of the norm of ℋmp from Lq1(|x|pα1q1/qdx)×⋯×Lqm(|x|pαmqm/qdx) to Lq(|x|pαdx). For 0<ϵ<1 and |ϵ|p>1, we take
(2.28)fiϵ(xi)={0,|xi|p<1,|xi|p-(n/qi)-(αi/q)-(qmϵ/qi),|xi|p≥1,i=1,2,…,m.
By simple calculation, we have
(2.29)‖f1ϵ‖Lq1(|x|pα1q1/qdx)q1=⋯=‖fmϵ‖Lqm(|x|pαmqm/qdx)qm=1-p-n1-p-ϵqm.
And when |x|p<1, ℋmp(f1ϵ,…,fmϵ)(x)=0. But when |x|p≥1,
(2.30)Hmp(f1ϵ,…,fmϵ)(x)=|x|p-(n/q)-(α/q)-(qmϵ/q)∫|(y1,…,ym)|p≤1,|y1|p≥1/|x|p,…,|ym|p≥1/|x|p∏i=1m|yi|p-(n/qi)-(αi/q)-(q2ϵ/qi)dy1⋯dym.
Then by the similar discussion to that in previous case, we can obtain that
(2.31)‖Hmp‖Lq1(|x|pα1q1/qdx)×⋯×Lqm(|x|pαmqm/qdx)→Lq(|x|pαdx)≥CH.
Theorem 2.1 is established by (2.20), (2.27), and (2.31).


## 3. Sharp Estimate ofm-Linear p-Adic  Hardy-Littlewood-Pólya Operator

We get the following best estimate ofm-linear p-adic Hardy-Littlewood-Pólya operator on Lebesgue spaces with power weights.Theorem 3.1.
Letm∈ℤ+,  fi∈Liq(|x|pαiqi/qdx),  1<qi<∞,  i=1,2,…,m, 1≤q<∞,  1/q=∑i=1m1/qi,  αi<q(1-(1/qi)) and α=∑i=1mαi. Then
(3.1)‖Tmp(f1,…,fm)‖Lq(|x|pαdx)=CT‖f1‖Lq1(|x|pα1q1/qdx)⋯‖fm‖Lqm(|x|pαmqm/qdx),
where
(3.2)CT=(1-p-1)m(1-q-m)(1-p-(1/q)-(α/q))∏i=1m(1-p(1/qi)+(αi/q)-1),
is the best constant.In particular, whenα=0, we obtain the norm of the m-linear p-adic Hardy-Littlewood-Pólya operator on Lebesgue spaces.Corollary 3.2.
Letm∈ℤ+,  1<qi<∞,  i=1,2,…,m,  1≤q<∞ and 1/q=∑i=1m1/qi.  Then
(3.3)‖Tmp‖Lq1(Qp)×⋯×Lqm(Qp)→Lq(Qp)=(1-p-1)m(1-q-m)(1-p-1/q)∏i=1m(1-p(1/qi)-1).Proof of Theorem3.1.
Just as the proof of Theorem2.1, we will only give the proof of case when m≥2.(I) Casem=2
By definition and the change of variablesyi=xzi, i=1,2, we have
(3.4)‖T2p(f1,f2)‖Lq(|x|pαdx)=(∫Qp|∬Qpf1(y1)f2(y2)[max⁡(|x|p,|y1|p,|y2|p)]2dy1dy2|q|x|pαdx)1/q≤(∫Qp(∬Qp|f1(y1)f2(y2)|[max⁡(|x|p,|y1|p,|y2|p)]2dy1dy2)q|x|pαdx)1/q=(∫Qp(∬Qp|f1(xz1)f2(xz2)|[max⁡(1,|z1|p,|z2|p)]2dz1dz2)q|x|pαdx)1/q.
By Minkowski’s integral inequality and Hölder's inequality((q/q1)+(q/q2)=1), we get
(3.5)‖T2p(f1,f2)‖Lq(|x|pαdx)≤∬Qp(∫Qp|f1(xz1)f2(xz2)|q|x|pαdx)1/q1[max⁡(1,|z1|p,|z2|p)]2dz1dz2≤∬Qp∏i=12(∫Qp|fi(xzi)|qi|x|pαiqi/qdx)1/qi1[max⁡(1,|z1|p,|z2|p)]2dz1dz2≤(∬Qp∏i=12|zi|p-(1/qi)-(αi/q)[max⁡(1,|z1|p,|z2|p)]2dz1dz2)∏i=12‖fi‖Lqi(|x|pαiqi/qdx).
By calculation, we have(3.6)∬Qp∏i=12|zi|p-(1/qi)-(αi/q)[max⁡(1,|z1|p,|z2|p)]2dz1dz2=∫|z1|p≤1∫|z2|p≤1∏i=12|zi|p-(1/qi)-(αi/q)dz1dz2+∫|z1|p>1∫|z2|p≤|z1|p|z1|p-(1/q1)-(α1/q)-2|z2|p-(1/q2)-(α2/q)dz1dz2+∫|z2|p>1∫|z1|p<|z2|p|z1|p-(1/q1)-(α1/q)|z2|p-(1/q2)-(α2/q)-2dz1dz2:=L0+L1+L2.
By definition,(3.7)L0=∫|z1|p≤1∫|z2|p≤1∏i=12|zi|p-(1/qi)-(αi/q)dz1dz2=(1-p-1)2(1-p(1/q1)+(α1/q)-1)(1-p(1/q2)+(α2/q)-1),L1=∫|z1|p>1∫|z2|p≤|z1|p|z1|p-(1/q1)-(α1/q)-2|z2|p-(1/q2)-α2/qdz1dz2=(1-p-1)1-p(1/q2)+(α2/q)-1∫|z1|p>1|z1|p-(1/q)-(α/q)-1dz1=(1-p-1)2p-(1/q)-(α/q)(1-p(1/q2)+(α2/q)-1)(1-p-(1/q)-(α/q)).
Similarly,
(3.8)L2=∫|z2|p>1∫|z1|p<|z2|p|z1|p-(1/q1)-(α1/q)|z2|p-(1/q2)-(α2/q)-2dz1dz2=(1-p-1)2p(1/q1)+(α1/q)-1p-(1/q)-(α/q)(1-p(1/q1)+(α1/q)-1)(1-p-(1/q)-(α/q)).
Substituting (3.7) and (3.8) into (3.6), we get
(3.9)∬Qp∏i=12|zi|p-(1/qi)-(αi/q)[max⁡(1,|z1|p,|z2|p)]2dz1dz2=(1-p-1)2(1-q-2)(1-p-(1/q)-(α/q))∏i=12(1-p(1/qi)+(αi/q)-1).
Then (3.5) and (3.9) imply that
(3.10)‖T2p‖Lq1(|x|pα1q1/qdx)×Lq2(|x|pα2q2/qdx)→Lq(|x|pαdx)≤(1-p-1)2(1-p-2)(1-p-(1/q)-(α/q))∏i=12(1-p(1/qi)+(αi/q)-1).
On the other hand, for0<ϵ<1 and |ϵ|p>1, we take
(3.11)fiϵ(xi)={0,|xi|p<1,|xi|p-(1/qi)-(αi/q)-(q2ϵ/qi),|xi|p≥1,i=1,2.
Then
(3.12)‖f1ϵ‖Lq1(|x|pα1q1/qdx)q1=‖f2ϵ‖Lq2(|x|pα2q2/qdx)q2=1-p-11-p-ϵq2.
Since|ϵ|p>1, we  have
(3.13)‖T2p(f1ϵ,f2ϵ)‖Lq(|x|pαdx)=(∫Qp|∬Qpf1ϵ(x1)f2ϵ(x2)[max⁡(|x|p,|x1|p,|x2|p)]2dx1dx2|q|x|pαdx)1/q≥(∫|x|p≥1(∫|x1|p≥1∫|x2|p≥1∏i=12|xi|p-(1/qi)-(αi/q)-(q2ϵ/qi)[max⁡(|x|p,|x1|p,|x2|p)]2dx1dx2)q|x|pαdx)1/q=(∫|x|p≥1(∫|y1|p≥1/|x|p∫|y2|p≥1/|x|p∏i=12|yi|p-(1/qi)-(αi/q)-(q2ϵ/qi)[max⁡(1,|y1|p,|y2|p)]2dy1dy2)q|x|p-1-q2ϵdx)1/q≥(∫|x|p≥|ϵ|p(∫|y1|p≥1/|ϵ|p∫|y2|p≥1/|ϵ|p∏i=12|yi|p-(1/qi)-(αi/q)-(q2ϵ/qi)[max⁡(1,|y1|p,|y2|p)]2dy1dy2)q|x|p-1-q2ϵdx)1/q=∏i=12‖fiϵ‖Lqi(|x|pαiqi/qdx)|ϵ|p-q2ϵ/q∫|y1|p≥1/|ϵ|p∫|y2|p≥1/|ϵ|p∏i=12|yi|p-(1/qi)-(αi/q)-(q2ϵ/qi)[max⁡(1,|y1|p,|y2|p)]2dy1dy2.
Therefore,
(3.14)‖T2p‖Lq1(|x|pα1q1/qdx)×Lq2(|x|pα2q2/qdx)→Lq(|x|pαdx)≥|ϵ|p-q2ϵ/q∫|y1|p≥1/|ϵ|p∫|y2|p≥1/|ϵ|p∏i=12|yi|p-(1/qi)-(αi/q)-(q2ϵ/qi)[max⁡(1,|y1|p,|y2|p)]2dy1dy2.
As the calculation of (3.6)–(3.8), we obtain that
(3.15)∫|y1|p≥1/|ϵ|p∫|y2|p≥1/|ϵ|p∏i=12|yi|p-(1/qi)-(αi/q)-(q2ϵ/qi)[max⁡(1,|y1|p,|y2|p)]2dy1dy2=(1-p-1)2∏i=12[1-(p|ϵ|p)(1/qi)+(αi/q)+(q2ϵ/qi)-1]∏i=12(1-p(1/qi)+(αi/q)+(q2ϵ/qi)-1)+(1-p-1)2p-(1/q)-(α/q)-(q2ϵ/q)(1-p(1/q2)+(α2/q)+ϵ-1)(1-p-(1/q)-(α/q)-(q2ϵ/q))+(1-p-1)2p(1/q1)+(α1/q)+(q2ϵ/q1)-1p-(1/q)-(α/q)-(q2ϵ/q)(1-p(1/q1)+(α1/q)+(q2ϵ/q1)-1)(1-p-(1/q)-(α/q)-(q2ϵ/q)).
Now takeϵ=p-k,k∈ℤ+ and let k approach to ∞, then by (3.9), (3.14), (3.15), and the fact that αi<qn(1-(1/qi)), i=1,2, we have
(3.16)‖T2p‖Lq1(|x|pα1q1/qdx)×Lq2(|x|pα2q2/qdx)→Lq(|x|pαdx)≥(1-p-1)2(1-q-2)(1-p-(1/q)-(α/q))∏i=12(1-p(1/qi)+(αi/q)-1).
Then by (3.10) and (3.16), we get
(3.17)‖T2p‖Lq1(|x|pα1q1/qdx)×Lq2(|x|pα2q2/qdx)→Lq(|x|pαdx)=(1-p-1)2(1-q-2)(1-p-(1/q)-(α/q))∏i=12(1-p(1/qi)+(αi/q)-1).(II) Casem≥3
The upper bound estimate for the norm can be obtained by the same way as that whenm=2, and we can obtain that
(3.18)‖Tmp(f1,…,fm)‖Lq(|x|pαdx)≤CT∏i=1m‖fi‖Lqi(|x|pαiqi/qdx),
where
(3.19)CT=∫|(z1,…,zm)|p≤1∏j=1m|zj|p-(1/qj)-(αj/q)[max⁡(1,|z1|p,…,|zm|p)]mdz1⋯dzm.
Let(3.20)E0=Zp×⋯×Zp,E1={(z1,…,zm)∈Qp×⋯×Qp∣|z1|p>1,|zk|p≤|z1|p,1<k≤m},Ei={(z1,…,zm)∈Qp×⋯×Qp∣|zi|p>1,|zj|p<|zi|p,|zk|p≤|zi|p,1≤j<i<k≤m},Em={(z1,…,zm)∈Qp×⋯×Qp∣|zm|p>1,|zj|p<|zm|p,1≤j<m}.
Obviously,
(3.21)⋃k=1mEk=Qp×⋯×Qp,Ei∩Ej=∅,i≠j,1≤i,j≤m.
Then
(3.22)CT=∑k=0m∫Ek∏j=1m|zj|p-(1/qj)-(αj/q)[max⁡(1,|z1|p,…,|zm|p)]mdz1⋯dzm:=∑k=0mJk.
Now let us calculateJk,k=0,1,…,m, respectively,
(3.23)J0=∫Zp⋯∫Zp∏j=1m|zj|p-(1/qj)-(αj/q)dz1⋯dzm=∏j=1m∫Zp|zj|p-(1/qj)-(αj/q)dzj=∏j=1m(∑k=-∞0∫Sk|zj|p-(1/qj)-(αj/q)dzj)=(1-p-1)m∏j=1m(1-p(1/qj)+(αj/q)-1),J1=∫|z1|p>1∫|z2|p≤|z1|p⋯∫|zm|p≤|z1|p|z1|p-(1/q1)-(α1/q)-m∏j=2m|zj|p-(1/qj)-(αj/q)dz1⋯dzm=∫|z1|p>1|z1|p-(1/q1)-(α1/q)-m(∏j=2m∫|zj|p≤|z1|p|zj|p-(1/qj)-(αj/q)dzj)dz1=∫|z1|p>1|z1|p-(1/q1)-(α1/q)-m∏j=2m((1-p-1)|z1|p-(1/qj)-(αj/q)+11-p(1/qj)+(αj/q)-1)dz1=(1-p-1)m-1∏j=2m(1-p(1/qj)+(αj/q)-1)∫|z1|p>1|z1|p-(1/q)-(α/q)-1dz1=(1-p-1)m-1p-(1/q)-(α/q)∏j=2m(1-p(1/qj)+(αj/q)-1)(1-p-(1/q)-(α/q)).
Similar to J1, for 1<i<m, it is true that
(3.24)Ji=∫|zi|p>1|zi|p-(1/qi)-(αi/q)-m(∏j=1i-1∫|zj|p<|zi|p|zj|p-(1/qj)-(αj/q)dzj)×(∏k=i+1m∫|zk|p≤|zi|p|zk|p-(1/qk)-(αk/q)dzk)dzi=(1-p-1)m-1∏j=1i-1p(1/qj)+(αj/q)-1∏1≤k≤m,k≠i(1-p(1/qk)+(αk/q)-1)∫|zi|p>1|zi|p-(1/q)-(α/q)-1dzi=(1-p-1)m(∏j=1i-1p(1/qj)+(αj/q)-1)p-(1/q)-(α/q)(1-p-(1/q)-(α/q))∏1≤k≤m,k≠i(1-p(1/qk)+(αk/q)-1),Jm=∫|zm|p>1|zi|p-(1/qi)-(αi/q)-m(∏j=1m-1∫|zj|p<|zm|p|zj|p-(1/qj)-(αj/q)dzj)dzm=(1-p-1)m-1∏j=1m-1p(1/qj)+(αj/q)-1∏j=1m-1(1-p(1/qk)+(αk/q)-1)∫|zm|p>1|zm|p-(1/q)-(α/q)-1dzm=(1-p-1)m(∏j=1m-1p(1/qj)+(αj/q)-1)p-(1/q)-(α/q)(1-p-(1/q)-(α/q))∏j=1m-1(1-p(1/qj)+(αj/q)-1).
Consequently, we have
(3.25)CT=∑k=0mJk=(1-p-1)m(1-q-m)(1-p-(1/q)-(α/q))∏i=1m(1-p(1/qi)+(αi/q)-1).
To obtain thatCT is also the lower bound, for 0<ϵ<1 and |ϵ|p>1, we define
(3.26)fiϵ={0,|xi|p<1,|xi|p-(1/qi)-(αi/q)-(q2ϵ/qi),|xi|p≥1,i=1,2,…,m.
By the similar discussion to that in Case m=2, we can also get that
(3.27)‖Tmp‖Lq1(|x|pα1q1/qdx)×⋯×Lqm(|x|pαmqm/qdx)→Lq(|x|pαdx)≥CT.
Combining (3.18) with (3.27), we complete the proof of Theorem 3.1.



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*Source: 472176-2011-07-14.xml*