Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 812636, 10 pages
doi:10.1155/2010/812636
Research Article
On a New Hilbert-Hardy-Type Integral Operator
and Applications
Xingdong Liu
1
and Bicheng Yang
2
1
Department of Mathematics, Zhaoqing University, Guangdong, Zhaoqing 526061, China
2
Department of Mathematics, Guangdong Institute of Education, Guangdong, Guangzhou 510303, China
Correspondence should be addressed to Bicheng Yang, [email protected]
Received 7 September 2010; Accepted 26 October 2010
Academic Editor: Sin E. Takahasi
Copyright q 2010 X. Liu and B. Yang. 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.
By applying the way of weight functions and a Hardy’s integral inequality, a Hilbert-Hardy-type
integral operator is defined, and the norm of operator is obtained. As applications, a new Hilbert-
Hardy-type inequality similar to Hilbert-type integral inequality is given, and two equivalent
inequalities with the best constant factors as well as some particular examples are considered.
1. Introduction
In 1934, Hardy published the following theorem cf. 1, Theorem 319.
Theorem A. If kx, y≥ 0 is a homogeneous function of degree −1 in 0, ∞ × 0, ∞, p>1,
1/p 1/q 1, and k
p
g
y
dx dy < k
p
f
p
g
q
,
1.1
where the constant factor k
p
is the best possible.
Hardy 2 also published the following Hardy’s integral inequality.
2 Journal of Inequalities and Applications
Theorem B. If p>1, ρ
/
1, fx ≥ 0, and Fx :
x
ρ − 1
p
∞
0
x
p−ρ
f
p
x
dx,
1.2
where the constant factor p/|ρ − 1|
p
is the best possible (cf. [1, Theorem 330]).
In 2009, Yang 3 published the following theorem.
Theorem C. If p>1, 1/p 1/q 1, λ>0, k
λ
x, y≥ 0 is a homogeneous function of degree −λ
in 0, ∞×0, ∞, and for any r>11/r 1/s 1, 0 <k
λ
r :
∞
0
f
x
g
y
dx dy < k
λ
r
f
p,ϕ
g
q,ψ
,
1.3
where the constant factor k
λ
x y
γ
F
x
x
G
y
y
dx dy < pqB
R, S
f
p
g
q
,
λ
1,uu
α−1
du < ∞, then
∞
0
k
λ
u, 1u
λ−α−1
du kα and
0 <
∞
0
k
λ
1,u
u
α−1
|
ln u
|
du
∞
0
1,v
v
α−1
dv k
α
.
2.2
There exists β>0, satisfying α ± β ∈ λ − 1, 1 and 0 <kα ± β < ∞. Since we find
lim
u → 0
ln u
u
β
u
−β
lim
u →∞
ln u
u
β
u
−β
0,
2.3
there exists M>0, such that | ln u|≤Mu
β
|
du
≤ M
∞
0
k
λ
1,u
u
α−1
u
β
u
−β
du
M
k
α β
k
α − β
−1
,
F
λ
x
:
∞
x
1
t
λ
f
t
dt,
G
λ
y
:
∞
y
x
G
λ
y
dx dy <
k
λ
1
1 − λ
1
f
p, ϕ
G
λ
k
λ
1
1 − λ
1
f
p, ϕ
.
2.7
4 Journal of Inequalities and Applications
Proof. Setting the weight functions ωλ
1
,y and λ
2
,x as follows:
ω
λ
1
,y
:
∞
1
dy
y
1−λ
2
,
2.8
then by Lemma 2.1 ,wefind
ω
λ
1
,y
ux/y
∞
0
k
u, 1
u
λ
1
−1
du k
λ
¨
older’s inequality cf. 11 and 2.8, 2.9,weobtain
∞
0
k
λ
x, y
F
λ
x
dx
∞
0
k
λ
x, y
x
1−λ
1
/q
1−λ
1
p−1
y
1−λ
2
F
p
λ
xdx
1/p
×
y
q1−λ
2
−1
∞
0
k
λ
x, y
y
λ
2
dx
x
λ
xdx
1/p
.
2.10
Then by Fubini theorem cf. 12, it follows:
J
p
≤ k
p−1
λ
1
∞
0
k
λ
x, y
x
1−λ
1
p−1
y
1−λ
2
y
1−λ
2
dy
F
p
λ
x
dx
k
p
λ
1
∞
0
x
−pλ
1
−11
F
p
λ
p
∞
0
x
p−pλ
1
−11
fx
x
λ
p
dx
1
1 − λ
1
p
∞
0
x
p2−λ−λ
1
−1
f
λ
x
dx
ψ
1/q
y
G
λ
y
dy ≤ J
G
λ
q,ψ
2.14
then by 2.6,wefind
G
λ
q
q,ψ
J
p
I<
k
λ
1
1 − λ
1
f
p, ϕ
f
p, ϕ
.
2.15
Hence, we have 2.7, which is equivalent to 2.6.
3. A Hilbert-Hardy-Type Integral Operator and Applications
Setting a real function space as follows:
L
p
ϕ
0, ∞
:
f;
f
p, ϕ
∞
0
ft/t
λ
dt, define an integral operator T : L
p
ϕ
0, ∞ →
L
p
ψ
1−p
0, ∞ as follows:
Tf
y
:
∞
0
k
λ
x, y
F
λ
x
f
p, ϕ
≤
k
λ
1
1 − λ
1
.
3.3
6 Journal of Inequalities and Applications
Theorem 3.1. Let the assumptions of Theorem 2.2 be fulfilled, and additionally setting ψy :
y
q2−λ−λ
2
−1
. Then one has
∞
0
k
λ
x, y
g
q,ψ
, 3.4
where the constant factor kλ
1
/1 − λ
1
1 − λ
2
is the best possible. Moreover the constant factor in
2.6 and 2.7 is the best possible and then
T
k
λ
1
1 − λ
1
.
3.5
Proof. Since λ
2
1/q
<
q
1 −
q
λ
2
− 1
1
∞
0
y
q−qλ
2
−11
gy
y
λ
q
dy
Then, by 2.6, we have 3.4.
For T>2, setting
fx, gy as follows:
f
x
⎧
⎪
⎪
⎨
⎪
⎪
⎩
x
λλ
1
−2
, 1 ≤ x ≤ T,
0, 0 <x<1; x>T,
g
y
⎧
⎪
λ
dt
T
x
t
λ
1
−2
dt
1
1 − λ
1
x
λ
1
−1
− T
λ
1
−1
,
G
λ
y
k
λ
x, y
F
λ
x
G
λ
y
dx dy
1
1 − λ
1
1 − λ
2
×
T
1
dx
≥
1
1 − λ
1
1 − λ
2
I
1
− I
2
− I
3
,
3.8
where I
1
, I
2
,andI
3
are indicated as follows;
I
1
:
T
1
T
1
k
λ
x, y
y
λ
2
−1
dy
dx,
I
3
: T
λ
2
−1
T
1
T
f
p, ϕ
g
q,ψ
k
1 − λ
1
1 − λ
2
T
1
x
p2−λ−λ/r−1
ln T
I
1
−
1
ln T
I
2
I
3
<k.
3.11
8 Journal of Inequalities and Applications
Since by Fubini t heorem, we obtain
I
1
T
1
1
x
T/x
1/x
k
λ
T
1
T/u
1
1
x
dx
k
λ
1,u
u
λ
2
−1
du
ln T
1
0
k
λ
1,u
u
λ
2
−1
du −
1
ln T
T
1
k
λ
1,u
ln u
u
λ
2
−1
du
,
0 ≤ I
2
T
λ
1
T
1/u
1
x
λ
1
dx
k
λ
1,u
u
λ
2
−1
du
T
1
T/u
1
1
x
λ
k
λ
1,u
u
λ
2
−1
du
T
1
1 −
u
T
1−λ
1
k
λ
1,u
u
1,u
u
λ
2
−1
du
< ∞,
0 ≤ I
3
≤
1
1 − λ
2
2
1
0
k
λ
u, 1
u
λ
1
−1
du
1
/1 − λ
1
1 − λ
2
is the best value of 3.4.
We conclude that the constant factor in 2.6 is the best possible, otherwise we can
get a contradiction by 1.2 that the constant factor in 3.4 is not the best possible. By the
same way, if the constant factor in 2.7 is not the best possible, then by 2.13, we can get
a contradiction that the constant factor in 2.6 is not the best possible. T herefore in view of
3.3, we have 3.5.
Journal of Inequalities and Applications 9
Corollary 3.2. For λ 1, λ
1
1/q, λ
2
1/p,
F
1
x :
∞
x
1/tftdt,
G
1
y :
p
G
1
q
,
3.13
∞
0
∞
0
k
1
x, y
F
1
xdx
1
y
dx dy < pqk
p
f
p
g
q
,
3.15
where k
p
k1/q
∞
0
k
λ
u, 1u
−1/p
y
x y
λ
dx dy <
rsB
λ/r, λ/s
r − λ
s − λ
f
p, ϕ
g
q,ψ
,
s − λ
f
p, ϕ
g
q,ψ
,
∞
0
ln
x/y
F
λ
x
G
g
q,ψ
;
3.16
b if 0 <λ<1,k
λ
x, y1/|x − y|
λ
, then we have
∞
0
F
λ
x
G
λ
y
x − y
q,ψ
;
3.17
c if λ<0,k
λ
x, ymin{x, y}
−λ
, then we find
∞
0
F
λ
x
G
λ
y
min
x, y
λ
dx dy <
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