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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2011, Article ID 472176, 20 pagesdoi:10.1155/2011/472176
Research ArticleSharp Estimates of m-Linear p-Adic Hardy andHardy-Littlewood-Polya Operators
Qingyan Wu and Zunwei Fu
Department of Mathematics, Linyi Uinverstiy, Linyi 276005, China
Correspondence should be addressed to Zunwei Fu, [email protected]
Received 16 April 2011; Accepted 12 May 2011
Academic Editor: Mark A. Petersen
Copyright q 2011 Q. Wu and Z. Fu. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
The sharp estimates of the m-linear p-adic Hardy and Hardy-Littlewood-Polya operators onLebesgue spaces with power weights are obtained in this paper.
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 number p, let Qp be the field of p-adic numbers. It is defined as thecompletion of the field of rational numbers Q with respect to the non-Archimedean p-adicnorm | · |p. This norm is defined as follows: |0|p = 0; If any nonzero rational number x isrepresented as x = pγ(m/n), where m and n are integers which are not divisible by p and γis an integer, then |x|p = p−γ . It is not difficult to show that the norm satisfies the followingproperties:
∣∣xy∣∣p = |x|p
∣∣y∣∣p,
∣∣x + y
∣∣p ≤ max
{
|x|p,∣∣y∣∣p
}
. (1.1)
From the standard p-adic analysis [6], we see that any nonzero p-adic number x ∈ Qp can beuniquely represented in the canonical series
x = pγ∞∑
j=0
ajpj , γ = γ(x) ∈ Z, (1.2)
2 Journal of Applied Mathematics
where aj are integers, 0 ≤ aj ≤ p − 1, a0 /= 0. The series (1.2) converges in the p-adic normbecause |ajp
j |p= p−j . Denote by Q∗
p = Qp \ {0} and Zp = {x ∈ Qp : |x|p ≤ 1}.The space Qn
p consists of points x = (x1, x2, . . . , xn), where xj ∈ Qp, j = 1, 2, . . . , n. Thep-adic norm on Qn
p is
|x|p := max1≤j≤n
∣∣xj
∣∣p, x ∈ Qn
p. (1.3)
Denote by
Bγ(a) ={
x ∈ Qnp : |x − a|p ≤ pγ
}
, (1.4)
the ball with center at a ∈ Qnp and radius pγ , and
Sγ(a) :={
x ∈ Qnp : |x − a|p = pγ
}
= Bγ(a) \ Bγ−1(a). (1.5)
Since Qnp is a locally compact commutative group under addition, it follows from the standard
analysis that there exists a Haar measure dx on Qnp , which is unique up to positive constant
multiple and is translation invariant. We normalize the measure dx by the equality
∫
B0(0)dx = |B0(0)|H = 1, (1.6)
where |E|H denotes the Haar measure of a measurable subset E of Qnp . By simple calculation,
we can obtain that
∣∣Bγ(a)
∣∣H
= pγn,∣∣Sγ(a)
∣∣H
= pγn(
1 − p−n)
, (1.7)
for any a ∈ Qnp . For a more complete introduction to the p-adic field, see [6] or [9].
The space Qnp × Qn
p × · · · × Qnp
︸ ︷︷ ︸
m
consists of points (y1, y2, . . . , ym), where yi = (yi1, yi2,
. . . , yin) ∈ Qnp , i = 1, 2, . . . , m. The p-adic norm of m-tuple (y1, y2, . . . , ym) is
∣∣(
y1, y2, . . . , ym
)∣∣p := max
1≤i≤m
∣∣yi
∣∣p. (1.8)
Recently, p-adic analysis has received a lot of attention due to its application inmathematical 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. Theharmonic analysis on p-adic field has been drawing more and more concern (cf. [17–21] andreferences therein).
The well-known Hardy’s integral inequality [22] tells us that for 1 < q < ∞,
∥∥Hf
∥∥Lq(R+) ≤
q
q − 1∥∥f∥∥Lq(R+), (1.9)
Journal of Applied Mathematics 3
where the classical Hardy operator is defined by
Hf(x) :=1x
∫x
0f(t)dt, (1.10)
for nonnegative integral function f on R+, and the constant q/(q−1) is the best possible. Thusthe norm of Hardy operator on Lq(R+) is
‖H‖Lq(R+)→Lq(R+) =q
q − 1. (1.11)
Faris [23] introduced the following n-dimensional Hardy operator, for nonnegativefunction f on Rn,
Hf(x) :=1
Ωn|x|n∫
|y|<|x|f(
y)
dy, x ∈ Rn \ {0}, (1.12)
where Ωn is the volume of the unit ball in Rn. Christ and Grafakos [24] obtained that thenorm of H on Lq(Rn) is
‖H‖Lq(Rn)→Lq(Rn) =q
q − 1, (1.13)
which is the same as that of the 1-dimension Hardy operator. In [25], Fu et al. introduced them-linear Hardy operator, which is defined by
Hm(f1, . . . , fm)
(x) =1
Ωmn|x|mn
∫
|(y1,...,ym)|< |x|f1(
y1) · · · fm
(
ym
)
dy1 · · ·dym, (1.14)
where x ∈ Rn \ {0} and f1, . . . , fm are nonnegative locally integrable functions on Rn. Andthey obtained the precise norms of Hm on Lebesgue spaces with power weight. The authorsof [26] also got the best constants of m-linear Hilbert, Hardy and Hardy-Littlewood-Polyaoperators on Lebesgue spaces.
The study of multilinear averaging operators in Euclidean spaces is a byproduct of therecent interest in multilinear singular integral operator theory. This subject was establishedby Coifman and Meyer [27] in 1975. In this article, we consider the sharp estimates of m-linear p-adic Hardy and Hardy-Littlewood-Polya operators. In contrast with [25], we use anew technique in calculations based on the feature of p-adic field, and Theorem 3.1 is alsonew. They cannot be obtained immediately by [25]. In [28], we defined the p-adic Hardyoperator.
Definition 1.1. For a function f on Qnp , we define the p-adic Hardy operator as follows
Hpf(x) =1
|x|np
∫
B(0,|x|p)f(t)dt, x ∈ Qn
p \ {0}, (1.15)
where B(0, |x|p) is a ball in Qnp with center at 0 ∈ Qn
p and radius |x|p.
4 Journal of Applied Mathematics
It is obvious that |Hpf | ≤ Mpf , where Mp is the Hardy-Littlewood maximal operator[17] defined by
Mpf(x) = supγ∈Z
1∣∣Bγ(x)
∣∣H
∫
Bγ (x)
∣∣f(
y)∣∣dy, f ∈ L1
loc
(
Qnp
)
. (1.16)
The Hardy-Littlewood maximal operator plays an important role in harmonic analysis. Theboundedness of Mp on Lq(Qn
p) has been solved (see, e.g., [9]). But the best estimate of Mp
on Lq(Qnp), 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-adicHardy-Littlewood-Polya operator) elsewhere.
Definition 1.2. Let m be a positive integer and f1, . . . , fm be nonnegative locally integrablefunctions on Qn
p . Them-linear p-adic Hardy operator is defined by
Hpm
(
f1, . . . , fm)
(x) =1
|x|mnp
∫
|(y1,...,ym)|p≤|x|pf1(
y1) · · · fm
(
ym
)
dy1 · · ·dym, (1.17)
where x ∈ Qnp \ {0}.
The Hardy-Littlewood-Polya’s linear operator [26] is defined by
Tf(x) =∫∞
0
f(
y)
max(
x, y)dy. (1.18)
In [26], the authors obtained that the norm of Hardy-Littlewood-Polya’s operator on Lq(R+)(see also [22, page 254]), 1 < q < ∞, is
‖T‖Lq(R+)→Lq(R+) =q2
q − 1. (1.19)
We define the p-adic Hardy-Littlewood-Polya operator as (see [28])
Tp(x) =∫
Qp
f(
y)
max(
|x|p,∣∣y∣∣p
)dy, x ∈ Q∗p. (1.20)
Definition 1.3. Let m be a positive integer and f1, . . . , fm be nonnegative locally integrablefunctions on Qn
p . Them-linear p-adic Hardy-Littlewood-Polya operator is defined by
Tpm
(
f1, . . . , fm)
(x) =∫
Qp
· · ·∫
Qp
f1(
y1) · · · fm
(
ym
)
[
max(
|x|p,∣∣y1∣∣p, . . . ,
∣∣ym
∣∣p
)]mdy1 · · ·dym, x ∈ Q∗p.
(1.21)
Journal of Applied Mathematics 5
We obtain the sharp estimates of the m-linear p-adic Hardy operator on Lebesguespaces with power weights in Section 2. In Section 3, we get the best estimate of m-linearp-adic Hardy-Littlewood-Polya operator on Lebesgue spaces with power weights.
In the following sequel, for k ∈ Z, we denote Bk = {x ∈ Qnp : |x|p ≤ pk} and Sk = {x ∈
Qnp : |x|p = pk}.
2. Sharp Estimates of m-Linear p-Adic Hardy Operator
Theorem 2.1. Let m ∈ Z+, fi ∈ Lqi(|x|αiqi/qp dx), 1 < qi < ∞, i = 1, 2, . . . , m, 1 ≤ q < ∞,
1/q =∑m
i=1(1/qi), αi < qn(1 − (1/qi)) and α =∑m
i=1 αi. Then
∥∥Hp(f1, . . . , fm
)∥∥Lq(|x|αpdx) ≤ CH
∥∥f1∥∥Lq1 (|x|α1q1/qp dx) · · ·
∥∥fm∥∥Lqm (|x|αmqm/q
p dx), (2.1)
where
CH =
(
1 − p−n)m
∏mi=1(
1 − pn/qi+(αi/q)−n) (2.2)
is the best constant.
When α = 0, we get the sharp estimates of the m-linear p-adic Hardy operator onLebesgue spaces.
Corollary 2.2. Letm ∈ Z+, 1 < qi < ∞, i = 1, 2, . . . , m, 1 ≤ q < ∞ and 1/q =∑m
i=1 1/qi. Then
∥∥∥Hp
m
∥∥∥Lq1 (Qn
p)×···×Lqm (Qnp)→Lq(Qn
p)=
(
1 − p−n)m
∏mi=1(
1 − p(n/qi)−n) . (2.3)
Proof of Theorem 2.1. Since the proof of the case when m = 1 is similar to and even simplerthan that of the case when m ≥ 2, for simplicity, we will only give the proof of case whenm ≥ 2. To make the proof clearer, we will discuss it in two parts.
(I) Whenm = 2
Firstly, we claim that the operator Hp and its restriction to the functions g satisfying g(x) =g(|x|−1p ) have the same operator norm on Lq(|x|αpdx). In fact, set
gi(x) =1
1 − p−n
∫
|ξi|p=1fi(
|x|−1p ξi)
dξi, x ∈ Qnp, i = 1, 2. (2.4)
6 Journal of Applied Mathematics
It’s clear that gi(x) = gi(|x|−1p ), i = 1, 2, and
Hp
2
(
g1, g2)
(x) =1
|x|2np
∫
|(y1,y2)|p≤|x|pg1(
y1)
g2(
y2)
dy1dy2
=1
(
1 − p−n)2
1
|x|2np
∫
|(y1,y2)|p≤|x|p
2∏
i=1
(∫
|ξi|p=1fi(∣∣yi
∣∣−1p ξi)
dξi
)
dy1dy2
=1
(
1 − p−n)2
1
|x|2np
∫
|(y1,y2)|p≤|x|p
2∏
i=1
(∫
|zi|p=|yi|pfi(zi)
∣∣yi
∣∣−np dzi
)
dy1dy2
=1
(
1 − p−n)2
1
|x|2np
∫
|(z1,z2)|p≤|x|p
2∏
i=1
(∫
|yi|p=|zi|p
∣∣yi
∣∣−np dyi
)
f1(z1)f2(z2)dz1dz2
=1
|x|2np
∫
|(z1,z2)|p≤|x|pf1(z1)f2(z2)dz1dz2
= Hp
2
(
f1, f2)
(x).(2.5)
By Holder’s inequality, we get
∥∥gi∥∥Lqi (|x|αiqi/qp dx) =
(∫
Qnp
∣∣∣∣∣
11 − p−n
∫
|ξi|p=1fi(
|x|−1p ξi)
dξi
∣∣∣∣∣
qi
|x|αiqi/qp dx
)1/qi
≤⎧
⎨
⎩
∫
Qnp
1(
1 − p−n)qi
(∫
|ξi|p=1
∣∣∣fi(
|x|−1p ξi)∣∣∣
qidξi
)(∫
|ξi|p=1dξi
)qi−1|x|αiqi/q
p dx
⎫
⎬
⎭
1/qi
=
{∫
Qnp
11 − p−n
(∫
|ξi|p=1
∣∣∣fi(
|x|−1p ξi)∣∣∣
qidξi
)
|x|αiqi/qp dx
}1/qi
=1
(
1 − p−n)1/qi
{∫
Qnp
(∫
|zi|p=|x|p
∣∣fi(zi)
∣∣qidzi
)
|x|(αiqi/q)−np dx
}1/qi
=1
(
1 − p−n)1/qi
{∫
Qnp
(∫
|x|p=|zi|p|x|(αiqi/q)−n
p dx
)
∣∣fi(zi)
∣∣qidzi
}1/qi
=∥∥fi∥∥Lqi (|x|αiqi/qp dx), i = 1, 2.
(2.6)
Therefore,
∥∥∥Hp
2
(
f1, f2)∥∥∥Lq(|x|αpdx)
∥∥f1∥∥Lq1 (|x|α1q1/qp dx)
∥∥f2∥∥Lq2 (|x|α2q2/qp dx)
≤
∥∥∥Hp
2
(
g1, g2)∥∥∥Lq(|x|αpdx)
∥∥g1∥∥Lq1 (|x|α1q1/qp dx)
∥∥g2∥∥Lq2 (|x|α2q2/qp dx)
, (2.7)
Journal of Applied Mathematics 7
which implies the claim. In the following, without loss of generality, we may assume thatfi ∈ Lqi(|x|αiqi/q
p dx), i = 1, 2, which satisfy that fi(x) = fi(|x|−1p ), i = 1, 2.By changing of variables yi = |x|−1p zi, i = 1, 2, we have
∥∥∥Hp
2
(
f1, f2)∥∥∥Lq(|x|αpdx)
=
⎛
⎝
∫
Qnp
∣∣∣∣∣∣
1
|x|2np
∫
|(y1,y2)|p≤|x|pf1(y1)f2(y2)dy1dy2
∣∣∣∣∣∣
q
|x|αpdx⎞
⎠
1/q
=
(∫
Qnp
∣∣∣∣∣
∫
|(z1,z2)|p≤1f1(
|x|−1p z1)
f2(
|x|−1p z2)
dz1dz2
∣∣∣∣∣
q
|x|αpdx)1/q
=
(∫
Qnp
∣∣∣∣∣
∫
|(z1,z2)|p≤1f1(
|z1|−1p x)
f2(
|z2|−1p x)
dz1dz2
∣∣∣∣∣
q
|x|αpdx)1/q
.
(2.8)
Then using Minkowski’s integral inequality and Holder’s inequality ((q/q1) + (q/q2) = 1),we get
∥∥∥Hp
2
(
f1, f2)∥∥∥Lq(|x|αpdx)
≤∫
|(z1,z2)|p≤1
(∫
Qnp
∣∣∣f1(
|z1|−1p x)
f2(
|z2|−1p x)∣∣∣
q|x|αpdx
)1/q
dz1dz2
≤∫
|(z1,z2)|p≤1
2∏
i=1
(∫
Qnp
∣∣∣fi(|zi|−1p x)
∣∣∣
qi |x|αiqi/qp dx
)1/qi
dz1dz2
=
(∫
|(z1,z2)|p≤1
2∏
i=1
|zi|−(n/qi)−(αi/q)p dz1dz2
)2∏
i=1
∥∥fi∥∥Lqi (|x|αiqi/qp dx).
(2.9)
By calculation, we have
∫
|(z1,z2)|p≤1
2∏
i=1
|zi|−(n/qi)−(αi/q)p dz1dz2
=∫
|z1|p≤1
∫
|z2|p≤|z1|p
2∏
i=1
|zi|−(n/qi)−(αi/q)p dz1dz2
+∫
|z2|p≤1
∫
|z1|p<|z2|p
2∏
i=1
|zi|−(n/qi)−(αi/q)p dz1dz2
=∫
|z1|p≤1|z1|−(n/q1)−(α1/q)
p
⎛
⎝
logp |z1|p∑
k=−∞
∫
Sk
|z2|−(n/q2)−(α2/q)p dz2
⎞
⎠dz1
+∫
|z2|p≤1|z2|−(n/q2)−(α2/q)
p
⎛
⎝
logp |z2|p−1∑
k=−∞
∫
Sk
|z1|−(n/q1)−(α1/q)p dz1
⎞
⎠dz2
8 Journal of Applied Mathematics
=
(
1 − p−n)
1 − p(n/q2)+(α2/q)−n
∫
|z1|p≤1|z1|−(n/q)−(α/q)+np dz1
+
(
1 − p−n)
p(n/q1)+(α1/q)−n
1 − p(n/q1)+(α1/q)−n
∫
|z2|p≤1|z2|−(n/q)−(α/q)+np dz2
=
(
1 − p−n)2
∏2i=1(
1 − p(n/qi)+(αi/q)−n).
(2.10)
Therefore,
∥∥∥Hp
2
∥∥∥Lq1 (|x|α1q1/qp dx)×Lq2 (|x|α2q2/qp dx)→Lq(|x|αpdx)
≤(
1 − p−n)2
∏2i=1(
1 − p(n/qi)+(αi/q)−n). (2.11)
Now let us prove that our estimate is sharp. For 0 < ε < 1 and |ε|p > 1, we take
fεi (xi) =
⎧
⎨
⎩
0, |xi|p < 1,
|xi|−(n/qi)−(αi/q)−(q2ε/qi)p , |xi|p ≥ 1,
i = 1, 2. (2.12)
Then by calculation, we have
∥∥fε
1
∥∥q1
Lq1(
|x|α1q1/qp dx) =∥∥fε
2
∥∥q2
Lq2(
|x|α2q2/qp dx) =
1 − p−n
1 − p−εq2. (2.13)
It is clear that when |x|p < 1, Hp
2(fε1 , f
ε2 )(x) = 0. But when |x|p ≥ 1,
Hp
2
(
fε1 , f
ε2
)
(x)
= |x|−(n/q)−(α/q)−(q2ε/q)p
∫
|(y1,y2)|p≤1,|y1|p≥1/|x|p,|y2|p≥1/|x|p
2∏
i=1
∣∣yi
∣∣−(n/qi)−(αi/q)−(q2ε/qi)p dy1dy2.
(2.14)
Journal of Applied Mathematics 9
Since |ε|p > 1, we get
∥∥∥Hp
2
(
fε1 , f
ε2
)∥∥∥Lq(|x|αpdx)
=
{∫
|x|p≥1
(
|x|−(n/q)−(α/q)−(q2ε/q)p
×∫
|(y1,y2)|p≤1,|y1|p≥1/|x|p,|y2|p≥1/|x|p
2∏
i=1
∣∣yi
∣∣−(n/qi)−(αi/q)−(q2ε/qi)p dy1dy2
)q
|x|αpdx}1/q
≥{∫
|x|p≥|ε|p
(
|x|−(n/q)−(α/q)−(q2ε/q)p
×∫
|(y1,y2)|p≤1,|y1|p≥1/|ε|p,|y2|p≥1/|ε|p
2∏
i=1
∣∣yi
∣∣−(n/qi)−(αi/q)−(q2ε/qi)p dy1dy2
)q
|x|αpdx}1/q
=
(∫
|(y1,y2)|p≤1,|y1|p≥1/|ε|p,|y2|p≥1/|ε|p
2∏
i=1
∣∣yi
∣∣−(n/qi)−(αi/q)−(q2ε/qi)p dy1dy2
)(∫
|x|p≥|ε|p|x|−n−εq2p dx
)1/q
=
(∫
|(y1,y2)|p≤1,|y1|p≥1/|ε|p,|y2|p≥1/|ε|p
2∏
i=1
∣∣yi
∣∣−(n/qi)−(αi/q)−(q2ε/qi)p dy1dy2
)
|ε|−εq2/qp
2∏
i=1
∥∥fε
i
∥∥Lqi (|x|αiqi/qp dx).
(2.15)
By the same calculation as that in (2.10), we obtain that
∫
|(y1,y2)|p≤1,|y1|p≥1/|ε|p,|y2|p≥1/|ε|p
2∏
i=1
∣∣yi
∣∣−(n/qi)−(αi/q)−(q2ε/qi)p dy1dy2
=
(
1 − p−n)2[
1 −(
p|ε|p)(n/q)+(α/q)+(q2ε/q)−2n
]
∏2i=1(
1 − p(n/qi)+(αi/q)+(q2ε/qi)−n).
(2.16)
Therefore,
|ε|−εq2/qp
(
1 − p−n)2[
1 −(
p|ε|p)(n/q)+(α/q)+(q2ε/q)−2n
]
∏2i=1(
1 − p(n/qi)+(αi/q)+(q2ε/qi)−n)
≤∥∥∥Hp
2
∥∥∥Lq1 (|x|α1q1/qp dx)×Lq2 (|x|α2q2/qp dx)→Lq(|x|αpdx)
.
(2.17)
10 Journal of Applied Mathematics
Now take ε = p−k, k = 1, 2, 3, . . .. Then |ε|p = pk > 1. Letting k approach to ∞, then ε
approaches to 0 and |ε|−εq2/qp approaches to 1. Since αi < qn(1 − (1/qi)), i = 1, 2, we have
(
1 − p−n)2
∏2i=1(
1 − p(n/qi)+(αi/q)−n)≤∥∥∥Hp
2
∥∥∥Lq1 (|x|α1q1/qp dx)×Lq2 (|x|α2q2/qp dx)→Lq(|x|αpdx)
. (2.18)
Then (2.11) and (2.18) imply that
∥∥∥Hp
2
∥∥∥Lq1 (|x|α1q1/qp dx)×Lq2 (|x|α2q2/qp dx)→Lq(|x|αpdx)
=
(
1 − p−n)2
∏2i=1(
1 − p(n/qi)+(αi/q)−n). (2.19)
(II) Whenm ≥ 3
The proof of the upper bound in this case is similar to that of the previous case, and we canobtain that
∥∥∥Hp
m
(
f1, . . . , fm)∥∥∥Lq(|x|αpdx)
≤ CH∥∥f1∥∥Lq1 (|x|α1q1/qp dx) · · ·
∥∥fm∥∥Lqm (|x|αmqm/q
p dx), (2.20)
where
CH =∫
|(z1,...,zm)|p≤1
m∏
k=1
|zk|−(n/qk)−(αk/q)p dz1 · · ·dzm. (2.21)
Let
D1 ={
(z1, . . . , zm) ∈ Qnp × · · · × Qn
p | |z1|p ≤ 1, |zk|p ≤ |z1|p, 1 < k ≤ m}
,
Di ={
(z1, . . . , zm) ∈ Qnp × · · · × Qn
p | |zi|p ≤ 1,∣∣zj∣∣p< |zi|p, |zk|p ≤ |zi|p, 1 ≤ j < i < k ≤ m
}
,
Dm ={
(z1, . . . , zm) ∈ Qnp × · · · × Qn
p | |zm|p ≤ 1,∣∣zj∣∣p< |zm|p, 1 ≤ j < m
}
.
(2.22)
It is clear that
m⋃
j=1
Dj ={
(z1, . . . , zm) ∈ Qnp × · · · × Qn
p | |(z1, . . . , zm)|p ≤ 1}
, (2.23)
and Di ∩Dj = ∅. Then
CH =m∑
j=1
∫
Dj
m∏
k=1
|zk|−(n/qk)−(αk/q)p dz1 · · ·dzm :=
m∑
j=1
Ij . (2.24)
Journal of Applied Mathematics 11
Now let us calculate Ij , j = 1, 2, . . . , m, respectively,
I1 =∫
D1
m∏
k=1
|zk|−(n/qk)−(αk/q)p dz1 · · ·dzm
=∫
|z1|p≤1|z1|−(n/q1)−(α1/q)
p
(m∏
k=2
∫
|zk |p≤|z1|p|zk|−(n/qk)−(αk/q)
p dzk
)
dz1
=∫
|z1|p≤1|z1|−(n/q1)−(α1/q)
p
⎛
⎝
m∏
k=2
⎛
⎝
logp |z1|p∑
j=−∞
∫
Sj
|zk|−(n/qk)−(αk/q)p dzk
⎞
⎠
⎞
⎠dz1
=
(
1 − p−n)m−1
∏mk=2(
1 − p(n/qk)+(αk/q)−n)∫
|z1|p≤1|z1|−(n/q)−(α/q)+(m−1)n
p dz1
=
(
1 − p−n)m
(
1 + p(n/q)+(α/q)−mn)∏m
k=2(
1 − p(n/qk)+(αk/q)−n) .
(2.25)
Similarly, for i = 2, . . . , m − 1, we have
Ii =∫
Di
m∏
k=1
|zk|−(n/qk)−(αk/q)p dz1 · · ·dzm
=∫
|zi|p≤1|zi|−(n/qi)−(αi/q)
p
⎛
⎝
i−1∏
j=1
∫
|zj |p<|zi|p
∣∣zj∣∣−(n/qj )−(αj/q)p
dzj
⎞
⎠
×(
m∏
k=i+1
∫
|zk |p≤|zi|p|zk|−(n/qk)−(αk/q)
p dzk
)
dzi
=
(
1 − p−n)m−1∏i−1
j=1p(n/qj )+(αj/q)−n
∏
1≤k≤m,k /= i
(
1 − p(n/qk)+(αk/q)−n)∫
|zi|p≤1|zi|−(n/q)−(α/q)+(m−1)n
p dzi
=
(
1 − p−n)m∏i−1
j=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|−(n/qm)−(αm/q)
p
⎛
⎝
m−1∏
j=1
∫
|zj |p<|zm|p
∣∣zj∣∣−(n/qj )−(αj/q)p
dzj
⎞
⎠dzm
=
(
1 − p−n)m∏m−1
j=1 p(n/qj )+(αj/q)−n
(
1 + p(n/q)+(α/q)−mn)∏m−1
j=1(
1 − p(n/qj )+(αj/q)−n).
(2.26)
12 Journal of Applied Mathematics
Therefore,
CH =
(
1 − p−n)m
(
1 + p(n/q)+(α/q)−mn)∏m
k=2(
1 − p(n/qk)+(αk/q)−n)
+m−1∑
i=2
(
1 − p−n)m∏m−1
j=1 p(n/qj )+(αj/q)−n
(
1 + p(n/q)+(α/q)−mn)∏m−1
k=1(
1 − p(n/qk)+(αk/q)−n)
+
(
1 − p−n)m∏m−1
j=1 p(n/qj )+(αj/q)−n
(
1 + p(n/q)+(α/q)−mn)∏m−1
j=1(
1 − p(n/qj )+(αj/q)−n)
=
(
1 − p−n)m
∏mi=1(
1 − p(n/qi)+(nαi/q)−n) .
(2.27)
To show that CH is the best constant, we should prove that it is also the lower boundof the norm of Hp
m from Lq1(|x|α1q1/qp dx) × · · · × Lqm(|x|αmqm/q
p dx) to Lq(|x|αpdx). For 0 < ε < 1and |ε|p > 1, we take
fεi (xi) =
⎧
⎨
⎩
0, |xi|p < 1,
|xi|−(n/qi)−(αi/q)−(qmε/qi)p , |xi|p ≥ 1,
i = 1, 2, . . . , m. (2.28)
By simple calculation, we have
∥∥fε
1
∥∥q1
Lq1(
|x|α1q1/qp dx) = · · · = ∥∥fε
m
∥∥qm
Lqm(
|x|αmqm/qp dx
) =1 − p−n
1 − p−εqm. (2.29)
And when |x|p < 1, Hpm(fε
1 , . . . , fεm)(x) = 0. But when |x|p ≥ 1,
Hpm
(
fε1 , . . . , f
εm
)
(x)
= |x|−(n/q)−(α/q)−(qmε/q)p
∫
|(y1,...,ym)|p≤1,|y1|p≥1/|x|p,...,|ym|p≥1/|x|p
m∏
i=1
∣∣yi
∣∣−(n/qi)−(αi/q)−(q2ε/qi)p dy1 · · ·dym.
(2.30)
Then by the similar discussion to that in previous case, we can obtain that
∥∥∥Hp
m
∥∥∥Lq1 (|x|α1q1/qp dx)×···×Lqm (|x|αmqm/q
p dx)→Lq(|x|αpdx)≥ CH. (2.31)
Theorem 2.1 is established by (2.20), (2.27), and (2.31).
Journal of Applied Mathematics 13
3. Sharp Estimate of m-Linear p-AdicHardy-Littlewood-Polya Operator
We get the following best estimate of m-linear p-adic Hardy-Littlewood-Polya operator onLebesgue spaces with power weights.
Theorem 3.1. Let m ∈ Z+, fi ∈ Lq
i (|x|αiqi/qp dx), 1 < qi < ∞, i = 1, 2, . . . , m, 1 ≤ q < ∞, 1/q =
∑mi=1 1/qi, αi < q(1 − (1/qi)) and α =
∑mi=1 αi. Then
∥∥∥T
pm
(
f1, . . . , fm)∥∥∥Lq(|x|αpdx)
= CT
∥∥f1∥∥Lq1 (|x|α1q1/qp dx) · · ·
∥∥fm∥∥Lqm (|x|αmqm/q
p dx), (3.1)
where
CT =
(
1 − p−1)m(1 − q−m
)
(
1 − p−(1/q)−(α/q))∏m
i=1(
1 − p(1/qi)+(αi/q)−1) , (3.2)
is the best constant.
In particular, when α = 0, we obtain the norm of them-linear p-adic Hardy-Littlewood-Polya operator on Lebesgue spaces.
Corollary 3.2. Letm ∈ Z+, 1 < qi < ∞, i = 1, 2, . . . , m, 1 ≤ q < ∞ and 1/q =∑m
i=1 1/qi. Then
∥∥∥T
pm
∥∥∥Lq1 (Qp)×···×Lqm (Qp)→Lq(Qp)
=
(
1 − p−1)m(1 − q−m
)
(
1 − p−1/q)∏m
i=1(
1 − p(1/qi)−1) . (3.3)
Proof of Theorem 3.1. Just as the proof of Theorem 2.1, we will only give the proof of case whenm ≥ 2.
(I) Casem = 2
By definition and the change of variables yi = xzi, i = 1, 2, we have
∥∥∥T
p
2
(
f1, f2)∥∥∥Lq(|x|αpdx)
=
⎛
⎜⎝
∫
Qp
∣∣∣∣∣∣∣
∫∫
Qp
f1(
y1)
f2(
y2)
[
max(
|x|p,∣∣y1∣∣p,∣∣y2∣∣p
)]2dy1dy2
∣∣∣∣∣∣∣
q
|x|αpdx
⎞
⎟⎠
1/q
≤
⎛
⎜⎝
∫
Qp
⎛
⎜⎝
∫∫
Qp
∣∣f1(
y1)
f2(
y2)∣∣
[
max(
|x|p,∣∣y1∣∣p,∣∣y2∣∣p
)]2dy1dy2
⎞
⎟⎠
q
|x|αpdx
⎞
⎟⎠
1/q
=
⎛
⎜⎝
∫
Qp
⎛
⎜⎝
∫∫
Qp
∣∣f1(xz1)f2(xz2)
∣∣
[
max(
1, |z1|p, |z2|p)]2
dz1dz2
⎞
⎟⎠
q
|x|αpdx
⎞
⎟⎠
1/q
.
(3.4)
14 Journal of Applied Mathematics
By Minkowski’s integral inequality and Holder’s inequality ((q/q1) + (q/q2) = 1), weget
∥∥∥T
p
2
(
f1, f2)∥∥∥Lq(|x|αpdx)
≤∫∫
Qp
(∫
Qp
∣∣f1(xz1)f2(xz2)
∣∣q|x|αpdx
)1/q1
[
max(
1, |z1|p, |z2|p)]2
dz1dz2
≤∫∫
Qp
2∏
i=1
(∫
Qp
∣∣fi(xzi)
∣∣qi |x|αiqi/q
p dx
)1/qi1
[
max(
1, |z1|p, |z2|p)]2
dz1dz2
≤
⎛
⎜⎝
∫∫
Qp
∏2i=1|zi|−(1/qi)−(αi/q)
p[
max(
1, |z1|p, |z2|p)]2
dz1dz2
⎞
⎟⎠
2∏
i=1
∥∥fi∥∥Lqi (|x|αiqi/qp dx).
(3.5)
By calculation, we have
∫∫
Qp
∏2i=1|zi|−(1/qi)−(αi/q)
p[
max(
1, |z1|p, |z2|p)]2
dz1dz2
=∫
|z1|p≤1
∫
|z2|p≤1
2∏
i=1
|zi|−(1/qi)−(αi/q)p dz1dz2
+∫
|z1|p>1
∫
|z2|p≤|z1|p|z1|−(1/q1)−(α1/q)−2
p |z2|−(1/q2)−(α2/q)p dz1dz2
+∫
|z2|p>1
∫
|z1|p<|z2|p|z1|−(1/q1)−(α1/q)
p |z2|−(1/q2)−(α2/q)−2p dz1dz2
:= L0 + L1 + L2.
(3.6)
By definition,
L0 =∫
|z1|p≤1
∫
|z2|p≤1
2∏
i=1
|zi|−(1/qi)−(αi/q)p 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|−(1/q1)−(α1/q)−2
p |z2|−(1/q2)−α2/qp dz1dz2
=
(
1 − p−1)
1 − p(1/q2)+(α2/q)−1
∫
|z1|p>1|z1|−(1/q)−(α/q)−1p dz1
=
(
1 − p−1)2p−(1/q)−(α/q)
(
1 − p(1/q2)+(α2/q)−1)(1 − p−(1/q)−(α/q)) .
(3.7)
Journal of Applied Mathematics 15
Similarly,
L2 =∫
|z2|p>1
∫
|z1|p<|z2|p|z1|−(1/q1)−(α1/q)
p |z2|−(1/q2)−(α2/q)−2p dz1dz2
=
(
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)) .
(3.8)
Substituting (3.7) and (3.8) into (3.6), we get
∫∫
Qp
∏2i=1|zi|−(1/qi)−(αi/q)
p[
max(
1, |z1|p, |z2|p)]2
dz1dz2 =
(
1 − p−1)2(1 − q−2
)
(
1 − p−(1/q)−(α/q))∏2
i=1(
1 − p(1/qi)+(αi/q)−1). (3.9)
Then (3.5) and (3.9) imply that
∥∥∥T
p
2
∥∥∥Lq1 (|x|α1q1/qp dx)×Lq2 (|x|α2q2/qp dx)→Lq(|x|αpdx)
≤(
1 − p−1)2(1 − p−2
)
(
1 − p−(1/q)−(α/q))∏2
i=1(
1 − p(1/qi)+(αi/q)−1). (3.10)
On the other hand, for 0 < ε < 1 and |ε|p > 1, we take
fεi (xi) =
⎧
⎨
⎩
0, |xi|p < 1,
|xi|−(1/qi)−(αi/q)−(q2ε/qi)p , |xi|p ≥ 1,
i = 1, 2. (3.11)
Then
∥∥fε
1
∥∥q1
Lq1(
|x|α1q1/qp dx) =∥∥fε
2
∥∥q2
Lq2(
|x|α2q2/qp dx) =
1 − p−1
1 − p−εq2. (3.12)
Since |ε|p > 1, we have
∥∥∥T
p
2
(
fε1 , f
ε2
)∥∥∥Lq(|x|αpdx)
=
⎛
⎜⎝
∫
Qp
∣∣∣∣∣∣∣
∫∫
Qp
fε1 (x1)fε
2 (x2)[
max(
|x|p, |x1|p, |x2|p)]2
dx1dx2
∣∣∣∣∣∣∣
q
|x|αpdx
⎞
⎟⎠
1/q
≥
⎛
⎜⎝
∫
|x|p≥1
⎛
⎜⎝
∫
|x1|p≥1
∫
|x2|p≥1
∏2i=1|xi|−(1/qi)−(αi/q)−(q2ε/qi)
p[
max(
|x|p, |x1|p, |x2|p)]2
dx1dx2
⎞
⎟⎠
q
|x|αpdx
⎞
⎟⎠
1/q
=
⎛
⎜⎝
∫
|x|p≥1
⎛
⎜⎝
∫
|y1|p≥1/|x|p
∫
|y2|p≥1/|x|p
∏2i=1|yi|−(1/qi)−(αi/q)−(q2ε/qi)
p[
max(
1,∣∣y1∣∣p,∣∣y2∣∣p
)]2dy1dy2
⎞
⎟⎠
q
|x|−1−q2εp dx
⎞
⎟⎠
1/q
16 Journal of Applied Mathematics
≥
⎛
⎜⎝
∫
|x|p≥|ε|p
⎛
⎜⎝
∫
|y1|p≥1/|ε|p
∫
|y2|p≥1/|ε|p
∏2i=1
∣∣yi
∣∣−(1/qi)−(αi/q)−(q2ε/qi)p
[
max(
1,∣∣y1∣∣p,∣∣y2∣∣p
)]2dy1dy2
⎞
⎟⎠
q
|x|−1−q2εp dx
⎞
⎟⎠
1/q
=2∏
i=1
∥∥fε
i
∥∥Lqi (|x|αiqi/qp dx)|ε|
−q2ε/qp
∫
|y1|p≥1/|ε|p
∫
|y2|p≥1/|ε|p
∏2i=1
∣∣yi
∣∣−(1/qi)−(αi/q)−(q2ε/qi)p
[
max(
1,∣∣y1∣∣p,∣∣y2∣∣p
)]2dy1dy2.
(3.13)
Therefore,
∥∥∥T
p
2
∥∥∥Lq1 (|x|α1q1/qp dx)×Lq2 (|x|α2q2/qp dx)→Lq(|x|αpdx)
≥ |ε|−q2ε/qp
∫
|y1|p≥1/|ε|p
∫
|y2|p≥1/|ε|p
∏2i=1
∣∣yi
∣∣−(1/qi)−(αi/q)−(q2ε/qi)p
[
max(
1,∣∣y1∣∣p,∣∣y2∣∣p
)]2dy1dy2.
(3.14)
As the calculation of (3.6)–(3.8), we obtain that
∫
|y1|p≥1/|ε|p
∫
|y2|p≥1/|ε|p
∏2i=1|yi|−(1/qi)−(αi/q)−(q2ε/qi)
p[
max(
1,∣∣y1∣∣p,∣∣y2∣∣p
)]2dy1dy2
=
(
1 − p−1)2∏2
i=1
[
1 − (p|ε|p)(1/qi)+(αi/q)+(q2ε/qi)−1]
∏2i=1(
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)) .
(3.15)
Now take ε = p−k, k ∈ Z+ and let k approach to∞, then by (3.9), (3.14), (3.15), and thefact that αi < qn(1 − (1/qi)), i = 1, 2, we have
∥∥∥T
p
2
∥∥∥Lq1 (|x|α1q1/qp dx)×Lq2 (|x|α2q2/qp dx)→Lq(|x|αpdx)
≥(
1 − p−1)2(1 − q−2
)
(
1 − p−(1/q)−(α/q))∏2
i=1(
1 − p(1/qi)+(αi/q)−1).
(3.16)
Then by (3.10) and (3.16), we get
∥∥∥T
p
2
∥∥∥Lq1 (|x|α1q1/qp dx)×Lq2 (|x|α2q2/qp dx)→Lq(|x|αpdx)
=
(
1 − p−1)2(1 − q−2
)
(
1 − p−(1/q)−(α/q))∏2
i=1(
1 − p(1/qi)+(αi/q)−1). (3.17)
Journal of Applied Mathematics 17
(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
∥∥∥T
pm
(
f1, . . . , fm)∥∥∥Lq(|x|αpdx)
≤ CT
m∏
i=1
∥∥fi∥∥Lqi (|x|αiqi/qp dx), (3.18)
where
CT =∫
|(z1,...,zm)|p≤1
∏mj=1
∣∣zj∣∣−(1/qj )−(αj/q)p
[
max(
1, |z1|p, . . . , |zm|p)]mdz1 · · ·dzm. (3.19)
Let
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
}
.
(3.20)
Obviously,
m⋃
k=1
Ek = Qp × · · · × Qp, Ei ∩ Ej = ∅, i /= j, 1 ≤ i, j ≤ m. (3.21)
Then
CT =m∑
k=0
∫
Ek
∏mj=1|zj |−(1/qj )−(αj/q)
p[
max(
1, |z1|p, . . . , |zm|p)]mdz1 · · ·dzm :=
m∑
k=0
Jk. (3.22)
Now let us calculate Jk, k = 0, 1, . . . , m, respectively,
J0 =∫
Zp
· · ·∫
Zp
m∏
j=1
∣∣zj∣∣−(1/qj )−(αj/q)p
dz1 · · ·dzm
=m∏
j=1
∫
Zp
∣∣zj∣∣−(1/qj )−(αj/q)p
dzj =m∏
j=1
(0∑
k=−∞
∫
Sk
|zj |−(1/qj )−(αj/q)p
dzj
)
=
(
1 − p−1)m
∏mj=1(
1 − p(1/qj )+(αj/q)−1) ,
18 Journal of Applied Mathematics
J1 =∫
|z1|p>1
∫
|z2|p≤|z1|p· · ·∫
|zm|p≤|z1|p|z1|−(1/q1)−(α1/q)−m
p
m∏
j=2
∣∣zj∣∣−(1/qj )−(αj/q)p
dz1 · · ·dzm
=∫
|z1|p>1|z1|−(1/q1)−(α1/q)−m
p
⎛
⎝
m∏
j=2
∫
|zj |p≤|z1|p|zj |−(1/qj )−(αj/q)
pdzj
⎞
⎠dz1
=∫
|z1|p>1|z1|−(1/q1)−(α1/q)−m
p
m∏
j=2
⎛
⎝
(
1 − p−1)|z1|−(1/qj )−(αj/q)+1
p
1 − p(1/qj )+(αj/q)−1
⎞
⎠dz1
=
(
1 − p−1)m−1
∏mj=2(
1 − p(1/qj )+(αj/q)−1)∫
|z1|p>1|z1|−(1/q)−(α/q)−1p dz1
=
(
1 − p−1)m−1
p−(1/q)−(α/q)∏m
j=2(
1 − p(1/qj)+(αj/q)−1)(1 − p−(1/q)−(α/q)) .
(3.23)
Similar to J1, for 1 < i < m, it is true that
Ji =∫
|zi|p>1|zi|−(1/qi)−(αi/q)−m
p
⎛
⎝
i−1∏
j=1
∫
|zj |p<|zi|p
∣∣zj∣∣−(1/qj )−(αj/q)p
dzj
⎞
⎠
×(
m∏
k=i+1
∫
|zk |p≤|zi|p|zk|−(1/qk)−(αk/q)
p dzk
)
dzi
=
(
1 − p−1)m−1∏i−1
j=1p(1/qj )+(αj/q)−1
∏
1≤k≤m,k /= i
(
1 − p(1/qk)+(αk/q)−1)∫
|zi|p>1|zi|−(1/q)−(α/q)−1p dzi
=
(
1 − p−1)m(∏i−1
j=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|−(1/qi)−(αi/q)−m
p
⎛
⎝
m−1∏
j=1
∫
|zj |p<|zm|p
∣∣zj∣∣−(1/qj )−(αj/q)p
dzj
⎞
⎠dzm
=
(
1 − p−1)m−1∏m−1
j=1 p(1/qj )+(αj/q)−1
∏m−1j=1(
1 − p(1/qk)+(αk/q)−1)
∫
|zm|p>1|zm|−(1/q)−(α/q)−1p dzm
=
(
1 − p−1)m(∏m−1
j=1 p(1/qj )+(αj/q)−1
)
p−(1/q)−(α/q)
(
1 − p−(1/q)−(α/q))∏m−1
j=1(
1 − p(1/qj )+(αj/q)−1).
(3.24)
Journal of Applied Mathematics 19
Consequently, we have
CT =m∑
k=0
Jk =
(
1 − p−1)m(1 − q−m
)
(
1 − p−(1/q)−(α/q))∏m
i=1(
1 − p(1/qi)+(αi/q)−1) . (3.25)
To obtain that CT is also the lower bound, for 0 < ε < 1 and |ε|p > 1, we define
fεi =
⎧
⎨
⎩
0, |xi|p < 1,
|xi|−(1/qi)−(αi/q)−(q2ε/qi)p , |xi|p ≥ 1,
i = 1, 2, . . . , m. (3.26)
By the similar discussion to that in Case m = 2, we can also get that
∥∥∥T
pm
∥∥∥Lq1 (|x|α1q1/qp dx)×···×Lqm (|x|αmqm/q
p dx)→Lq(|x|αpdx)≥ CT . (3.27)
Combining (3.18) with (3.27), we complete the proof of Theorem 3.1.
Acknowledgments
This paper was partially supported by NSF of China (Grant nos. 10871024 and 10901076) andNSF of Shandong Province (Grant nos. Q2008A01 and ZR2010AL006).
References
[1] S. Albeverio and W. Karwowski, “A random walk on p-adics—the generator and its spectrum,”Stochastic Processes and Their Applications, vol. 53, no. 1, pp. 1–22, 1994.
[2] V. A. Avetisov, A. H. Bikulov, S. V. Kozyrev, and V. A. Osipov, “p-adic models of ultrametric diffusionconstrained by hierarchical energy landscapes,” Journal of Physics A, vol. 35, no. 2, pp. 177–189, 2002.
[3] A. Khrennikov, p-Adic Valued Distributions in Mathematical Physics, vol. 309 of Mathematics and ItsApplications, Kluwer Academic, Dordrecht, The Netherlands, 1994.
[4] A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and BiologicalModels, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
[5] V. S. Varadarajan, “Path integrals for a class of p-adic Schrodinger equations,” Letters in MathematicalPhysics, vol. 39, no. 2, pp. 97–106, 1997.
[6] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, vol. 1 ofSeries on Soviet and East European Mathematics, World Scientific, River Edge, NJ, USA, 1994.
[7] V. S. Vladimirov and I. V. Volovich, “p-adic quantum mechanics,” Communications in MathematicalPhysics, vol. 123, no. 4, pp. 659–676, 1989.
[8] I. V. Volovich, “Harmonic analysis and p-adic strings,” Letters in Mathematical Physics, vol. 16, no. 1,pp. 61–67, 1988.
[9] M. H. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ, USA, 1975.[10] S. Haran, “Riesz potentials and explicit sums in arithmetic,” Inventiones Mathematicae, vol. 101, no. 3,
pp. 697–703, 1990.[11] S. Haran, “Analytic potential theory over the p-adics,” Annales de l’Institut Fourier, vol. 43, no. 4, pp.
905–944, 1993.[12] S. Albeverio, A. Yu. Khrennikov, and V. M. Shelkovich, “Harmonic analysis in the p-adic Lizorkin
spaces: fractional operators, pseudo-differential equations, p-adic wavelets, Tauberian theorems,” TheJournal of Fourier Analysis and Applications, vol. 12, no. 4, pp. 393–425, 2006.
20 Journal of Applied Mathematics
[13] N. M. Chuong and N. V. Co, “The Cauchy problem for a class of pseudodifferential equations overp-adic field,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 629–645, 2008.
[14] N. M. Chuong, Yu. V. Egorov, A. Khrennikov, Y. Meyer, and D. Mumford, Eds., Harmonic, Wavelet andp-Adic Analysis, World Scientific, Singapore, 2007.
[15] A. N. Kochubei, “A non-Archimedean wave equation,” Pacific Journal of Mathematics, vol. 235, no. 2,pp. 245–261, 2008.
[16] W. A. Zuniga-Galindo, “Pseudo-differential equations connected with p-adic forms and local zetafunctions,” Bulletin of the Australian Mathematical Society, vol. 70, no. 1, pp. 73–86, 2004.
[17] Y.-C. Kim, “Carleson measures and the BMO space on the p-adic vector space,” MathematischeNachrichten, vol. 282, no. 9, pp. 1278–1304, 2009.
[18] Y.-C. Kim, “Weak type estimates of square functions associatedwith quasiradial Bochner-Rieszmeanson certain Hardy spaces,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 266–280,2008.
[19] K. S. Rim and J. Lee, “Estimates of weighted Hardy-Littlewood averages on the p-adic vector space,”Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1470–1477, 2006.
[20] K. M. Rogers, “A van der Corput lemma for the p-adic numbers,” Proceedings of the AmericanMathematical Society, vol. 133, no. 12, pp. 3525–3534, 2005.
[21] K. M. Rogers, “Maximal averages along curves over the p-adic numbers,” Bulletin of the AustralianMathematical Society, vol. 70, no. 3, pp. 357–375, 2004.
[22] G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, Cambridge, UK,1952.
[23] W. G. Faris, “Weak Lebesgue spaces and quantum mechanical binding,” Duke Mathematical Journal,vol. 43, no. 2, pp. 365–373, 1976.
[24] M. Christ and L. Grafakos, “Best constants for two nonconvolution inequalities,” Proceedings of theAmerican Mathematical Society, vol. 123, no. 6, pp. 1687–1693, 1995.
[25] Z. W. Fu, L. Grafakos, S. Z. Lu, and F. Y. Zhao, “Sharp bounds for m-linear Hardy and Hilbertoperators,” Houston Journal of Mathematics. In press.
[26] A. Benyi and T. Oh, “Best constants for certain multilinear integral operators,” Journal of Inequalitiesand Applications, vol. 2006, Article ID 28582, 12 pages, 2006.
[27] R. R. Coifman and Y. Meyer, “On commutators of singular integrals and bilinear singular integrals,”Transactions of the American Mathematical Society, vol. 212, pp. 315–331, 1975.
[28] Z.W. Fu, Q. Y. Wu, and S. Z. Lu, “Sharp estimates for p-adic Hardy, Hardy-Littlewood-Poya operatorsand commutators,” http://arxiv.org/abs/1105.2888.
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