Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 472176, 20 pages
doi:10.1155/2011/472176
Research Article
Sharp Estimates of m-Linear p-Adic Hardy and
Hardy-Littlewood-Pólya Operators
Qingyan Wu and Zunwei Fu
Department of Mathematics, Linyi Uinverstiy, Linyi 276005, China
Correspondence should be addressed to Zunwei Fu, lyfzw@tom.com
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 Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The sharp estimates of the m-linear p-adic Hardy and Hardy-Littlewood-Pólya operators on
Lebesgue 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 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:
xy |x| y ,
p
p
p
x y ≤ max |x| , y .
p
p
p
1.1
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
x pγ
∞
j0
aj pj ,
γ γx ∈ Z,
1.2
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0. The series 1.2 converges in the p-adic norm
where aj are integers, 0 ≤ aj ≤ p − 1, a0 /
because |aj pj |p p−j . Denote by Q∗p Qp \ {0} and Zp {x ∈ Qp : |x|p ≤ 1}.
The space Qnp consists of points x x1 , x2 , . . . , xn , where xj ∈ Qp , j 1, 2, . . . , n. The
p-adic norm on Qnp is
|x|p : max xj p ,
1≤j≤n
x ∈ Qnp .
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 pγn ,
H
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 × Qnp × · · · × Qnp consists of points y1 , y2 , . . . , ym , where yi yi1 , yi2 ,
m
. . . , yin ∈ Qnp , i 1, 2, . . . , m. The p-adic norm of m-tuple y1 , y2 , . . . , ym is
y1 , y2 , . . . , ym : max yi .
p
p
1≤i≤m
1.8
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 < ∞,
Hf q ≤
L R q f q ,
L R q−1
1.9
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3
where the classical Hardy operator is defined by
1
Hfx :
x
x
1.10
ftdt,
0
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
HLq R → Lq R q
.
q−1
1.11
Faris 23 introduced the following n-dimensional Hardy operator, for nonnegative
function f on Rn ,
1
Hfx :
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 the
norm of H on Lq Rn is
HLq 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 the
m-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 . 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 mlinear 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 function f on Qnp , we define the p-adic Hardy operator as follows
1
H fx n
|x|p
p
B0,|x|p ftdt,
x ∈ Qnp \ {0},
where B0, |x|p is a ball in Qnp with center at 0 ∈ Qnp and radius |x|p .
1.15
4
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It is obvious that |Hp f| ≤ Mp f, where Mp is the Hardy-Littlewood maximal operator
17 defined by
1
M fx sup B
x
γ
γ∈Z
p
H
Bγ x
f y dy,
f ∈ L1loc Qnp .
1.16
The Hardy-Littlewood maximal operator plays an important role in harmonic analysis. The
boundedness of Mp on Lq Qnp 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-adic
Hardy-Littlewood-Pólya operator elsewhere.
Definition 1.2. Let m be a positive integer and f1 , . . . , fm be nonnegative locally integrable
functions on Qnp . The m-linear p-adic Hardy operator is defined by
p
Hm f1 , . . . , fm x 1
|x|mn
p
|y1 ,...,ym |p ≤|x|p
f1 y1 · · · fm ym dy1 · · · dym ,
1.17
where x ∈ Qnp \ {0}.
The Hardy-Littlewood-Pólya’s linear operator 26 is defined by
T fx ∞
0
f y
dy.
max x, y
1.18
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
T Lq R → Lq R q2
.
q−1
1.19
We define the p-adic Hardy-Littlewood-Pólya operator as see 28
T x p
Qp
f y
dy,
max |x|p , yp
x ∈ Q∗p .
1.20
Definition 1.3. Let m be a positive integer and f1 , . . . , fm be nonnegative locally integrable
functions on Qnp . The m-linear p-adic Hardy-Littlewood-Pólya operator is defined by
p
Tm f1 , . . . , fm x
Qp
···
Qp
f1 y1 · · · fm ym
m dy1 · · · dym ,
max |x|p , y1 p , . . . , ym p
x ∈ Q∗p .
1.21
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5
We obtain the sharp estimates of the m-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, 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
α q /q
Theorem 2.1. Let m ∈ Z , fi ∈ Lqi |x|pi i dx, 1 < qi < ∞, i 1, 2, . . . , m, 1 ≤ q < ∞,
m
1/q m
i1 1/qi , αi < qn1 − 1/qi and α i1 αi . Then
p
H f1 , . . . , fm Lq |x|αp dx
≤ CH f1 Lq1 |x|α1 q1 /q dx · · · fm Lqm |x|αm qm /q dx ,
p
p
2.1
where
CH m 1 − p−n
i1
m
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 on
Lebesgue spaces.
Corollary 2.2. Let m ∈ Z , 1 < qi < ∞, i 1, 2, . . . , m, 1 ≤ q < ∞ and 1/q m
m
1 − p−n
m .
n/qi −n
i1 1 − p
i1
1/qi . Then
p
Hm Lq1 Qnp ×···×Lqm Qnp → Lq Qnp 2.3
Proof of Theorem 2.1. Since the proof of the case when m 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 When m 2
Firstly, we claim that the operator Hp and its restriction to the functions g satisfying gx α
q
g|x|−1
p have the same operator norm on L |x|p dx. In fact, set
1
gi x 1 − p−n
|ξi |p 1
fi |x|−1
p ξi dξi ,
x ∈ Qnp , i 1, 2.
2.4
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Journal of Applied Mathematics
It’s clear that gi x gi |x|−1
p , i 1, 2, and
p
H2 g1 , g2 x 1
|x|2n
p
|y1 ,y2 |p ≤|x|p
1
1
1 − p−n
1 − p−n
1
1 − p−n
1
2
2
2
1
g1 y1 g2 y2 dy1 dy2
|x|2n
p
1
|y1 ,y2 |p ≤|x|p i1
|ξi |p 1
|y1 ,y2 |p ≤|x|p i1
|x|2n
p
2
|zi |p |yi |p
|z1 ,z2 |p ≤|x|p i1
|x|2n
|z1 ,z2 |p ≤|x|p
p
p
H2 f1 , f2 x.
−1
fi yi p ξi dξi dy1 dy2
2
|x|2n
p
1
2
|yi |p |zi |p
−n
fi zi yi p dzi dy1 dy2
−n
yi dyi f1 z1 f2 z2 dz1 dz2
p
f1 z1 f2 z2 dz1 dz2
2.5
By Hölder’s inequality, we get
gi q αi qi /q
L i |x|
dx
p
≤
1
−n
Qnp 1 − p
|ξi |
⎧
⎨
⎩
Qnp
Qnp
1 − p−n
1
1 − p−n
1
1
1 − p−n
p 1
fi |x|−1
p ξi
1
q i
|ξi |p 1
|ξi |p 1
qi
fi |x|−1
p ξi dξi
Qnp
|ξi |p 1
qi −1
⎫1/qi
⎬
α q /q
|x|pi i dx
dξi
⎭
1/qi
qi
αi qi /q
−1
dx
fi |x|p ξi dξi |x|p
1/qi
q i
1/qi
αi qi /q
dξi |x|p
dx
|zi |p |x|p
1/qi
Qnp
|x|p |zi |p
1 − p−n
fi Lqi |x|αi qi /q dx , i 1, 2.
1/qi
i qi /q−n
fi zi qi dzi |x|α
dx
p
α q /q−n
dx
|x|p i i
fi zi qi dzi
1/qi
p
2.6
Therefore,
p
p
H2 f1 , f2 q α
H2 g1 , g2 q α
L |x|p dx
L |x|p dx
≤ ,
f1 q α1 q1 /q f2 q α2 q2 /q
g1 q α1 q1 /q g2 q α2 q2 /q
L 2 |x|
dx
L 2 |x|
dx
L 1 |x|
dx
L 1 |x|
dx
p
p
p
p
2.7
Journal of Applied Mathematics
7
which implies the claim. In the following, without loss of generality, we may assume that
α q /q
fi ∈ Lqi |x|pi i dx, i 1, 2, which satisfy that fi x fi |x|−1
p , i 1, 2.
By changing of variables yi |x|−1
p zi , i 1, 2, we have
q
⎞1/q
1
α
⎝
f1 y1 f2 y2 dy1 dy2 |x|p dx⎠
2n
Qnp |x|p
|y1 ,y2 |p ≤|x|p
⎛
p
H2 f1 , f2 Lq |x|αp dx
q
1/q
−1
−1
α
f1 |x|p z1 f2 |x|p z2 dz1 dz2 |x|p dx
Qnp |z1 ,z2 |p ≤1
q
1/q
−1
−1
α
f1 |z1 |p x f2 |z2 |p x dz1 dz2 |x|p dx
.
Qnp |z1 ,z2 |p ≤1
2.8
Then using Minkowski’s integral inequality and Hölder’s inequality q/q1 q/q2 1,
we get
p
H2 f1 , f2 Lq |x|αp dx
≤
|z1 ,z2 |p ≤1
≤
Qnp
2
|z1 ,z2 |p ≤1 i1
Qnp
q i
αi qi /q
dx
fi |zi |−1
p x |x|p
1/q
dz1 dz2
1/qi
dz1 dz2
2
2
−n/qi −αi /q
fi q αi qi /q .
dz1 dz2
|zi |p
L i |x|
dx
q
−1
α
f1 |z1 |−1
p x f2 |z2 |p x |x|p dx
|z1 ,z2 |p ≤1 i1
p
i1
By calculation, we have
2
|z1 ,z2 |p ≤1 i1
−n/qi −αi /q
|zi |p
|z1 |p ≤1
2
−n/qi −αi /q
dz1 dz2
|zi |p
|z2 |p ≤|z1 |p i1
dz1 dz2
|z2 |p ≤1
2
−n/qi −αi /q
dz1 dz2
|zi |p
|z1 |p <|z2 |p i1
⎛
logp |z1 |p
−n/q1 −α1 /q ⎝ |z1 |p
|z1 |p ≤1
k−∞
⎛
−n/q2 −α2 /q
dz2 ⎠dz1
|z2 |p
Sk
⎞
logp |z2 |p −1 ⎞
−n/q2 −α2 /q ⎝
|z2 |p ≤1
|z2 |p
k−∞
−n/q1 −α1 /q
|z1 |p
Sk
dz1 ⎠dz2
2.9
8
Journal of Applied Mathematics
1 − p−n
−n/q−α/qn
dz1
|z1 |p
1 − pn/q2 α2 /q−n |z1 |p ≤1
1 − p−n pn/q1 α1 /q−n
−n/q−α/qn
dz2
|z2 |p
n/q
α
/q−n
1
1
1−p
|z2 |p ≤1
2 1 − p−n
i1
2
1 − pn/qi αi /q−n
.
2.10
Therefore,
p
H2 α q1 /q
Lq1 |x|p1
α q2 /q
dx×Lq2 |x|p2
dx → Lq |x|αp dx
1 − p−n
≤ 2 i1
2
1 − pn/qi αi /q−n
.
2.11
Now let us prove that our estimate is sharp. For 0 < < 1 and ||p > 1, we take
fi xi ⎧
⎨0,
|xi |p < 1,
⎩|x |−n/qi −αi /q−q2 /qi ,
i p
|xi |p ≥ 1,
i 1, 2.
2.12
Then by calculation, we have
q1
f 1
q
α q /q L 1 |x|p1 1 dx
q
f2 2q
L
2
α q /q
|x|p2 2 dx
1 − p−n
.
1 − p−q2
2.13
p
It is clear that when |x|p < 1, H2 f1 , f2 x 0. But when |x|p ≥ 1,
p
H2 f1 , f2 x
−n/q−α/q−q2 /q
|x|p
2 yi −n/qi −αi /q−q2 /qi dy1 dy2 .
|y1 ,y2 |p ≤1,|y1 |p ≥1/|x|p ,|y2 |p ≥1/|x|p i1
p
2.14
Journal of Applied Mathematics
9
Since ||p > 1, we get
p H2 f1 , f2 Lq |x|αp dx
−n/q−α/q−q2 /q
|x|p ≥1
|x|p
×
|y1 ,y2 |p ≤1,|y1 |p ≥1/|x|p ,|y2 |p ≥1/|x|p i1
p
|x|p
×
2 yi −n/qi −αi /q−q2 /qi dy1 dy2
|y1 ,y2 |p ≤1,|y1 |p ≥1/||p ,|y2 |p ≥1/||p i1
|y1 ,y2 |p ≤1,|y1 |p ≥1/||p ,|y2 |p ≥1/||p i1
q
p
2 yi −n/qi −αi /q−q2 /qi dy1 dy2
p
|x|p ≥||p
1/q
|x|αp dx
−n−q2
dx
|x|p
2 2 2 /q
yi −n/qi −αi /q−q2 /qi dy1 dy2 ||−q
f p
1/q
|x|αp dx
−n/q−α/q−q2 /q
|x|p ≥||p
q
≥
2 yi −n/qi −αi /q−q2 /qi dy1 dy2
i
p
|y1 ,y2 |p ≤1,|y1 |p ≥1/||p ,|y2 |p ≥1/||p i1
i1
1/q
α qi /q
Lqi |x|pi
dx
.
2.15
By the same calculation as that in 2.10, we obtain that
2 yi −n/qi −αi /q−q2 /qi dy1 dy2
|y1 ,y2 |p ≤1,|y1 |p ≥1/||p ,|y2 |p ≥1/||p i1
−n 2
#
p
n/qα/qq2 /q−2n $
1 − p||p
1−p
2 n/qi αi /qq2 /qi −n
i1 1 − p
2.16
.
Therefore,
#
n/qα/qq2 /q−2n $
2
1 − p−n 1 − p||p
2 n/qi αi /qq2 /qi −n
i1 1 − p
p
≤ H2 q α1 q1 /q
.
α2 q2 /q
α
q
q
−q2 /q |ε|p
L 1 |x|p
dx×L 2 |x|p
dx → L |x|p dx
2.17
10
Journal of Applied Mathematics
Now take p−k , k 1, 2, 3, . . .. Then ||p pk > 1. Letting k approach to ∞, then −q2 /q
approaches to 0 and ||p
i1
1−p
2
p
≤ H2 n/q α /q−n
1 − p−n
2 approaches to 1. Since αi < qn1 − 1/qi , i 1, 2, we have
i
α q1 /q
Lq1 |x|p1
i
α q2 /q
dx×Lq2 |x|p2
.
2.18
.
2.19
dx → Lq |x|αp dx
Then 2.11 and 2.18 imply that
p
H2 α q1 /q
Lq1 |x|p1
α q2 /q
dx×Lq2 |x|p2
dx → Lq |x|αp dx
2 1 − p−n
i1
2
1 − pn/qi αi /q−n
II When m ≥ 3
The proof of the upper bound in this case is similar to that of the previous case, and we can
obtain that
p
Hm f1 , . . . , fm Lq |x|αp dx
≤ CH f1 Lq1 |x|α1 q1 /q dx · · · fm Lqm |x|αm qm /q dx ,
p
p
2.20
where
CH m
|z1 ,...,zm |p ≤1 k1
−n/qk −αk /q
|zk |p
dz1 · · · dzm .
2.21
Let
D1 z1 , . . . , zm ∈ Qnp × · · · × Qnp | |z1 |p ≤ 1, |zk |p ≤ |z1 |p , 1 < k ≤ m ,
Di z1 , . . . , zm ∈ Qnp × · · · × Qnp | |zi |p ≤ 1, zj p < |zi |p , |zk |p ≤ |zi |p , 1 ≤ j < i < k ≤ m ,
Dm z1 , . . . , zm ∈ Qnp × · · · × Qnp | |zm |p ≤ 1, zj p < |zm |p , 1 ≤ j < m .
2.22
It is clear that
m
%
Dj z1 , . . . , zm ∈ Qnp × · · · × Qnp | |z1 , . . . , zm |p ≤ 1 ,
2.23
j1
and Di ∩ Dj ∅. Then
CH m j1
m
m
−n/qk −αk /q
dz1 · · · dzm :
Ij .
|zk |p
Dj k1
j1
2.24
Journal of Applied Mathematics
11
Now let us calculate Ij , j 1, 2, . . . , m, respectively,
I1 m
−n/qk −αk /q
dz1 · · · dzm
|zk |p
D1 k1
−n/q1 −α1 /q
|z1 |p
|z1 |p ≤1
|zk |p ≤|z1 |p
k2
−n/qk −αk /q
dzk
|zk |p
dz1
⎛ ⎛
⎞⎞
logp |z1 |p m
−n/q1 −α1 /q ⎝
−n/qk −αk /q
⎝
dzk ⎠⎠dz1
|z1 |p
|zk |p
m |z1 |p ≤1
m 1 − p−n
m−1
2.25
Sj
j−∞
k2
−n/q−α/qm−1n
|z1 |p
n/qk αk /q−n
|z1 |p ≤1
k2 1 − p
m
1 − p−n
m .
n/qk αk /q−n
1 pn/qα/q−mn
k2 1 − p
dz1
Similarly, for i 2, . . . , m − 1, we have
Ii m
−n/qk −αk /q
dz1 · · · dzm
|zk |p
Di k1
|zi |p ≤1
×
⎛
i−1 −n/qi −αi /q ⎝
|zi |p
j1
1 − p−n
|zk |p ≤|zi |p
−n/qk −αk /q
dzk
|zk |p
m−1 i−1
j1 p
1 − p−n
1 pn/qα/q−mn
Im n/qj αj /q−n
1 − pn/qk αk /q−n
1≤k≤m,k /
i
p
m ki1
|zj |p <|zi |p
⎞
−n/qj −αj /q
zj dzj ⎠
m i−1
j1 p
−n/q−α/qm−1n
|zi |p ≤1
1 − pn/qk αk /q−n
m−1 −n/qm −αm /q ⎝
|zm |p
|zm |p ≤1
j1 |zj |p <|zm |p
|zi |p
dzi
n/qj αj /q−n
1≤k≤m,k /
i
⎛
dzi
,
⎞
−n/qj −αj /q
zj dzj ⎠dzm
p
m m−1
n/qj αj /q−n
j1 p
.
m−1 n/qj αj /q−n
1 pn/qα/q−mn
j1 1 − p
1 − p−n
2.26
12
Journal of Applied Mathematics
Therefore,
m
1 − p−n
m
n/qk αk /q−n
1 pn/qα/q−mn
k2 1 − p
m m−1 n/q α /q−n
j
j
m−1
1 − p−n
j1 p
m−1
n/qα/q−mn
n/qk αk /q−n
i2 1 p
k1 1 − p
m m−1 n/q α /q−n
j
j
1 − p−n
j1 p
m−1
n/qj αj /q−n
1 pn/qα/q−mn
j1 1 − p
m
1 − p−n
m .
n/qi nαi /q−n
i1 1 − p
CH 2.27
To show that CH is the best constant, we should prove that it is also the lower bound
p
α q /q
α q /q
of the norm of Hm from Lq1 |x|p1 1 dx × · · · × Lqm |x|pm m dx to Lq |x|αp dx. For 0 < < 1
and ||p > 1, we take
⎧
⎨0,
fi xi |xi |p < 1,
⎩|x |−n/qi −αi /q−qm /qi ,
i p
|xi |p ≥ 1,
i 1, 2, . . . , m.
2.28
1 − p−n
.
1 − p−qm
2.29
By simple calculation, we have
q1
f α q /q Lq1 |x|p1 1 dx
1
qm
· · · fm
qm
L
α q /q
|x|pm m dx
p
x 0. But when |x|p ≥ 1,
And when |x|p < 1, Hm f1 , . . . , fm
p
Hm f1 , . . . , fm
x
−n/q−α/q−qm /q
|x|p
m yi −n/qi −αi /q−q2 /qi dy1 · · · dym .
|y1 ,...,ym |p ≤1,|y1 |p ≥1/|x|p ,...,|ym |p ≥1/|x|p i1
p
2.30
Then by the similar discussion to that in previous case, we can obtain that
p
Hm α q1 /q
Lq1 |x|p1
α qm /q
dx×···×Lqm |x|pm
dx → Lq |x|αp dx
Theorem 2.1 is established by 2.20, 2.27, and 2.31.
≥ CH .
2.31
Journal of Applied Mathematics
13
3. Sharp Estimate of m-Linear p-Adic
Hardy-Littlewood-Pólya Operator
We get the following best estimate of m-linear p-adic Hardy-Littlewood-Pólya operator on
Lebesgue spaces with power weights.
q
α q /q
Theorem 3.1. Let m ∈ Z , fi ∈ Li |x|pi i dx, 1 < qi < ∞, i 1, 2, . . . , m, 1 ≤ q < ∞, 1/q m
m
i1 1/qi , αi < q1 − 1/qi and α i1 αi . Then
p
Tm f1 , . . . , fm Lq |x|αp dx
CT f1 Lq1 |x|α1 q1 /q dx · · · fm Lqm |x|αm qm /q dx ,
p
p
3.1
where
m 1 − q−m
1 − p−1
CT m ,
1/qi αi /q−1
1 − p−1/q−α/q
i1 1 − p
3.2
is the best constant.
In particular, when α 0, we obtain the norm of the m-linear p-adic Hardy-LittlewoodPólya operator on Lebesgue spaces.
Corollary 3.2. Let m ∈ Z , 1 < qi < ∞, i 1, 2, . . . , m, 1 ≤ q < ∞ and 1/q m
i1
1/qi . Then
m 1 − q−m
1 − p−1
m .
1/qi −1
1 − p−1/q
i1 1 − p
p
Tm Lq1 Qp ×···×Lqm Qp → Lq Qp 3.3
Proof of Theorem 3.1. Just as the proof of Theorem 2.1, we will only give the proof of case when
m ≥ 2.
I Case m 2
By definition and the change of variables yi xzi , i 1, 2, we have
p
T2 f1 , f2 q
⎞1/q
f
y
f
y
1
1
2
2
⎜
⎟
α
⎝
2 dy1 dy2 |x|p dx⎠
Qp Qp max |x| , y , y 1 p
2 p
p
⎛
Lq |x|αp dx
⎛
⎜
≤⎝
⎛
⎜
⎝
⎛
Qp
⎜
⎝
⎛
Qp
⎜
⎝
⎞q
⎞1/q
f1 y1 f2 y2 ⎟
⎟
α
2 dy1 dy2 ⎠ |x|p dx⎠
Qp max |x| , y , y 1 p
2 p
p
⎞q
⎞1/q
f1 xz1 f2 xz2 ⎟
⎟
α
2 dz1 dz2 ⎠ |x|p dx⎠ .
Qp max 1, |z | , |z |
1 p
2 p
3.4
14
Journal of Applied Mathematics
By Minkowski’s integral inequality and Hölder’s inequality q/q1 q/q2 1, we
get
p
T2 f1 , f2 Lq |x|αp dx
≤
Qp
Qp i1
⎛
⎜
≤⎝
1/q
p
2
≤
Qp
f1 xz1 f2 xz2 q |x|α dx
Qp
fi xzi qi |x|pαi qi /q dx
max 1, |z1 |p , |z2 |p
1/qi
⎞
2
−1/qi −αi /q
i1 |zi |p
1
1
max 1, |z1 |p , |z2 |p
2 dz1 dz2
2 dz1 dz2
2
⎟ 2 dz1 dz2 ⎠ fi Lqi |x|αpi qi /q dx .
Qp max 1, |z | , |z |
i1
1 p
2 p
3.5
By calculation, we have
2
−1/qi −αi /q
i1 |zi |p
Qp
2 dz1 dz2
max 1, |z1 |p , |z2 |p
|z1 |p ≤1
2
|z2 |p ≤1 i1
−1/qi −αi /q
|zi |p
|z1 |p >1
|z2 |p >1
dz1 dz2
−1/q1 −α1 /q−2
|z1 |p
|z2 |p ≤|z1 |p
−1/q1 −α1 /q
|z1 |p
|z1 |p <|z2 |p
−1/q2 −α2 /q
|z2 |p
−1/q2 −α2 /q−2
|z2 |p
3.6
dz1 dz2
dz1 dz2
: L0 L1 L2 .
By definition,
L0 |z1 |p ≤1
2
|z2 |p ≤1 i1
−1/qi −αi /q
|zi |p
dz1 dz2
2
1 − p−1
,
1 − p1/q1 α1 /q−1 1 − p1/q2 α2 /q−1
−1/q −α /q−2
−1/q −α /q
L1 |z1 |p 1 1
|z2 |p 2 2 dz1 dz2
|z1 |p >1
|z2 |p ≤|z1 |p
1 − p−1
1 − p1/q2 α2 /q−1
−1/q−α/q−1
|z1 |p >1
|z1 |p
2
1 − p−1 p−1/q−α/q
.
1 − p1/q2 α2 /q−1 1 − p−1/q−α/q
dz1
3.7
Journal of Applied Mathematics
15
Similarly,
L2 |z2 |p >1
−1/q1 −α1 /q
|z1 |p <|z2 |p
|z1 |p
−1/q2 −α2 /q−2
|z2 |p
dz1 dz2
3.8
2
1 − p−1 p1/q1 α1 /q−1 p−1/q−α/q
.
1 − p1/q1 α1 /q−1 1 − p−1/q−α/q
Substituting 3.7 and 3.8 into 3.6, we get
2
2 1 − p−1 1 − q−2
2 .
2 dz1 dz2 1/qi αi /q−1
1 − p−1/q−α/q
Qp max 1, |z | , |z |
i1 1 − p
1 p
2 p
−1/qi −αi /q
i1 |zi |p
3.9
Then 3.5 and 3.9 imply that
2 1 − p−1 1 − p−2
≤
2 .
1/qi αi /q−1
1 − p−1/q−α/q
i1 1 − p
p
T2 α q1 /q
Lq1 |x|p1
α q2 /q
dx×Lq2 |x|p2
dx → Lq |x|αp dx
3.10
On the other hand, for 0 < < 1 and ||p > 1, we take
fi xi ⎧
⎨0,
|xi |p < 1,
⎩|x |−1/qi −αi /q−q2 /qi , |x | ≥ 1,
i p
i p
i 1, 2.
3.11
Then
q1
f 1
q
α q /q L 1 |x|p1 1 dx
q
f2 2q
L
2
α q /q
|x|p2 2 dx
1 − p−1
.
1 − p−q2
3.12
Since ||p > 1, we have
p T2 f1 , f2 ⎛
⎜
⎝
Lq |x|αp dx
q
⎞1/q
f
x
f
x
1
2
⎟
2
α
1
2 dx1 dx2 |x|p dx⎠
Qp Q
p max |x| , |x | , |x |
1 p
2 p
p
⎞q
⎞1/q
−1/qi −αi /q−q2 /qi |x
|
i1 i p
⎜
⎟
⎟
α
⎝
2 dx1 dx2 ⎠ |x|p dx⎠
|x|p ≥1
|x1 |p ≥1 |x2 |p ≥1
max |x|p , |x1 |p , |x2 |p
⎛
⎞q
⎛
⎞1/q
2
−1/qi −αi /q−q2 /qi |y
|
−1−q2 i1 i p
⎜
⎟
⎜
⎟
⎝
dx⎠
⎝
2 dy1 dy2 ⎠ |x|p
|x|p ≥1
|y1 |p ≥1/|x|p |y2 |p ≥1/|x|p
max 1, y1 , y2 ⎛
⎜
≥⎝
⎛
2
p
p
16
Journal of Applied Mathematics
⎛
⎞q
⎞1/q
2 −1/qi −αi /q−q2 /qi y
i1 i p
−1−q2 ⎜
⎟
⎟
dx⎠
⎝
2 dy1 dy2 ⎠ |x|p
|x|p ≥||p
|y1 |p ≥1/||p |y2 |p ≥1/||p
max 1, y1 p , y2 p
2 −1/qi −αi /q−q2 /qi 2 i1 yi p
−q
/q
2
f q αi qi /q ||p
2 dy1 dy2 .
i L i |x|p
dx
|y1 |p ≥1/||p |y2 |p ≥1/||p
i1
max 1, y1 p , y2 p
⎛
⎜
≥⎝
3.13
Therefore,
p
T2 α q1 /q
Lq1 |x|p1
−q /q
||p 2
≥
α q2 /q
dx×Lq2 |x|p2
dx → Lq |x|αp dx
|y1 |p ≥1/||p
|y2 |p ≥1/||p
2 −1/qi −αi /q−q2 /qi i1 yi p
2 dy1 dy2 .
max 1, y1 , y2 p
3.14
p
As the calculation of 3.6–3.8, we obtain that
2
|y1 |p ≥1/||p
−1/qi −αi /q−q2 /qi i1 |yi |p
|y2 |p ≥1/||p
2 dy1 dy2
max 1, y1 p , y2 p
2 2 1/qi αi /qq2 /qi −1
1 − p−1
i1 1 − p||p 2 1/qi αi /qq2 /qi −1
i1 1 − p
−1 2
3.15
p−1/q−α/q−q2 /q
1 − p1/q2 α2 /q−1 1 − p−1/q−α/q−q2 /q
1−p
2
1 − p−1 p1/q1 α1 /qq2 /q1 −1 p−1/q−α/q−q2 /q
.
1 − p1/q1 α1 /qq2 /q1 −1 1 − p−1/q−α/q−q2 /q
Now take p−k , k ∈ Z and let k approach to ∞, then by 3.9, 3.14, 3.15, and the
fact that αi < qn1 − 1/qi , i 1, 2, we have
p
T2 α q1 /q
Lq1 |x|p1
α q2 /q
dx×Lq2 |x|p2
dx → Lq |x|αp dx
2 1 − p−1 1 − q−2
≥
2 .
1/qi αi /q−1
1 − p−1/q−α/q
i1 1 − p
3.16
Then by 3.10 and 3.16, we get
p
T2 2 1 − p−1 1 − q−2
2 .
1/qi αi /q−1
1 − p−1/q−α/q
i1 1 − p
α q1 /q
Lq1 |x|p1
α q2 /q
dx×Lq2 |x|p2
dx → Lq |x|αp dx
3.17
Journal of Applied Mathematics
17
II Case m ≥ 3
The upper bound estimate for the norm can be obtained by the same way as that when m 2,
and we can obtain that
p
Tm f1 , . . . , fm Lq |x|αp dx
≤ CT
m fi q αi qi /q ,
L i |x|
dx
3.18
p
i1
where
CT |z1 ,...,zm |p ≤1
m −1/qj −αj /q
j1 zj p
m dz1 · · · dzm .
max 1, |z1 |p , . . . , |zm |p
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
%
Ek Qp × · · · × Qp ,
Ei ∩ Ej ∅,
i/
j, 1 ≤ i, j ≤ m.
3.21
k1
Then
CT m
m k0
−1/qj −αj /q
j1 |zj |p
m dz1 · · · dzm :
max 1, |z1 |p , . . . , |zm |p
Ek
m
Jk .
k0
Now let us calculate Jk , k 0, 1, . . . , m, respectively,
J0 Zp
···
Zp j1
m j1
m zj −1/qj −αj /q dz1 · · · dzm
Zp
m
0 −1/qj −αj /q
−1/qj −αj /q
zj dzj |zj |p
dzj
p
j1
1 − p−1
m j1
p
m
1 − p1/qj αj /q−1
,
k−∞
Sk
3.22
18
Journal of Applied Mathematics
J1 |z1 |p >1
···
|z2 |p ≤|z1 |p
m
−1/q1 −α1 /q−m |zm |p ≤|z1 |p
j2
⎛
m −1/q1 −α1 /q−m ⎝
|z1 |p
|z1 |p >1
j2 |zj |p ≤|z1 |p
−1/q −α /q
zj p j j dz1 · · · dzm
|z1 |p
⎞
|zj |p−1/qj −αj /q dzj ⎠dz1
⎛
−1/qj −αj /q1 ⎞
−1
m
1
−
p
|z1 |p
−1/q −α /q−m
⎝
⎠dz1
|z1 |p 1 1
1 − p1/qj αj /q−1
|z1 |p >1
j2
1 − p−1
m m−1
1 − p1/qj αj /q−1
j2
1 − p−1
−1/q−α/q−1
|z1 |p >1
|z1 |p
dz1
m−1
p−1/q−α/q
.
1 − p1/qj αj /q−1 1 − p−1/q−α/q
m j2
3.23
Similar to J1 , for 1 < i < m, it is true that
⎛
Ji −1/qi −αi /q−m ⎝
|zi |p >1
×
|zi |p
1 − p−1
|zk |p ≤|zi |p
m−1 i−1
1 − p−1
1−
1/qj αj /q−1
p1/qk αk /q−1
m i−1
j1 p
−1/q−α/q
Jm −1/q −α /q
|zk |p k k dzk
j1 p
1≤k≤m,k /i
|zm |p >1
⎞
−1/qj −αj /q
zj dzj ⎠
p
m 1−p
|zj |p <|zi |p
j1
ki1
i−1 1≤k≤m,k /
i
|zi |p
dzi
p−1/q−α/q
1 − p1/qk αk /q−1
⎛
m−1 −1/q −α /q−m ⎝
|zi |p i i
j1
−1/q−α/q−1
|zi |p >1
1/qj αj /q−1
dzi
|zj |p <|zm |p
,
⎞
−1/qj −αj /q
zj dzj ⎠dzm
p
m−1 m−1 1/q α /q−1 j
j
1 − p−1
j1 p
−1/q−α/q−1
dzm
|zm |p
m−1 1/q
α
/q−1
k
k
|zm |p >1
j1 1 − p
m m−1 1/q α /q−1 −1/q−α/q
j
j
p
1 − p−1
j1 p
.
m−1
1/qj αj /q−1
1 − p−1/q−α/q
j1 1 − p
3.24
Journal of Applied Mathematics
19
Consequently, we have
m 1 − q−m
1 − p−1
CT Jk m .
1/qi αi /q−1
1 − p−1/q−α/q
i1 1 − p
k0
m
3.25
To obtain that CT is also the lower bound, for 0 < < 1 and ||p > 1, we define
fi ⎧
⎨0,
|xi |p < 1,
⎩|x |−1/qi −αi /q−q2 /qi ,
|xi |p ≥ 1,
i p
i 1, 2, . . . , m.
3.26
By the similar discussion to that in Case m 2, we can also get that
p
Tm α q1 /q
Lq1 |x|p1
α qm /q
dx×···×Lqm |x|pm
dx → Lq |x|αp dx
≥ 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 and
NSF of Shandong Province Grant nos. Q2008A01 and ZR2010AL006.
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Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014