Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 4, Article 139, 2006
A MULTIPLE HARDY-HILBERT INTEGRAL INEQUALITY WITH THE BEST
CONSTANT FACTOR
HONG YONG
D EPARTMENT OF M ATHEMATICS
G UANGDONG U NIVERSITY OF B USINESS S TUDY
G UANGZHOU 510320
P EOPLE ’ S R EPUBLIC OF C HINA .
hongyong59@sohu.com
Received 14 September, 2005; accepted 14 December, 2005
Communicated by B. Yang
A BSTRACT. In this paper, by introducing the norm kxkα (x ∈ Rn ), we give a multiple HardyHilbert’s integral inequality with a best constant factor and two parameters α, λ.
Key words and phrases: Multiple Hardy-Hilbert integral inequality, the Γ-function, Best constant factor.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
R∞
R∞
If p > 1, p1 + 1q = 1, f ≥ 0, g ≥ 0, 0 < 0 f p (x)dx < +∞, 0 < 0 g q (x)dx < +∞, then
the well known Hardy-Hilbert integral inequality is given by (see [3]):
Z ∞
p1 Z ∞
1q
Z ∞Z ∞
f (x)g(x)
π
p
q
f (x)dx
g (x)dx ,
(1.1)
dxdy <
x+y
0
0
0
0
sin πp
where the constant factor
Z
∞
π
sin( π
p)
∞
Z
(1.2)
0
0
where the constant factor
is the best possible. Its equivalent form is:
f (x)
dx
x+y
π
p
p

dy < 
sin
π 
π
p
Z
∞
f p (x)dx,
0
p
sin( π
p)
is also the best possible.
Hardy-Hilbert’s inequality is valuable in harmonic analysis, real analysis and operator theory. In recent years, many valuable results (see [4] – [10]) have been obtained in the form of
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
273-05
2
H ONG YONG
generalizations and improvements of Hardy-Hilbert’s inequality. In 1999, Kuang [5] gave a
generalization with a parameter λ of (1.1) as follows:
Z ∞
p1 Z ∞
1q
f (x)g(y)
1−λ p
1−λ q
(1.3)
dxdy < hλ (p)
x f (x)dx
x g (x)dx ,
xλ + y λ
0
0
0
0
n
o
h
1 i−1
1
π
π
where max p1 , 1q < λ ≤ 1, hλ (p) = π λ sin p pλ
sin q qλ
. Because of the constant
factor hλ (p) being not the best possible, Yang [8] gave a new generalization of (1.1) in 2002 as
follows:
Z ∞Z ∞
f (x)g(y)
(1.4)
dxdy
xλ + y λ
0
0
Z ∞
p1 Z ∞
1q
π
(1−λ)(p−1) p
(1−λ)(q−1) q
<
x
f (x)dx
x
g (x)dx ,
0
0
λ sin πp
∞
Z
∞
Z
its equivalent form is:
Z
∞
y
(1.5)
λ−1
∞
Z
0
0
where the constant factors
f (x)
dx
xλ + y λ
π
λ sin( π
p)
p
p

π

dy < 
λ sin πp
in (1.4) and
π
λ sin( π
p)
Z
∞
x(1−λ)(p−1) f p (x)dx,
0
p
in (1.5) are all the best possible.
At present, because of the requirement of higher-dimensional harmonic analysis and higherdimensional operator theory, multiple
integral inequalities haveQbeen studied.
Pn Hardy-Hilbert
n
1
1
Hong [11] obtained: If a > 0,
j=1 pj , λ >
i=1 pi = 1, pi > 1, fi ≥ 0, ri = pi
1
n − 1 − r1i , i = 1, 2, . . . , n, then
a
Z
∞
(1.6)
α
∞
n
Y
Γn−2 α1
···
fi (x)dx1 · · · dxn ≤ n−1
P
λ
α Γ(λ)
[ ni=1 (xi − αi )a ] i=1
α
Z ∞
p1
n Y
i
1
1
1
1
n−1−αλ pi
×
Γ
1−
Γ λ−
n−1−
(t − α)
fi dt
.
a
r
a
r
i
i
α
i=1
Z
1
Afterwards, Bicheng Yang and Kuang Jichang etc. obtained some multiple Hardy-Hilbert
integral inequalities (see [9, 6]).
In this paper, by introducing the Γ-function, we generalize (1.3) and (1.4) into multiple
Hardy-Hilbert integral inequalities with the best constant factors.
2. S OME L EMMAS
First of all, we introduce the signs as:
Rn+ = {x = (x1 , . . . , xn ) : x1 , . . . , xn > 0},
1
kxkα = (xα1 + · · · + xαn ) α ,
J. Inequal. Pure and Appl. Math., 7(4) Art. 139, 2006
α > 0.
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M ULTIPLE H ARDY-H ILBERT I NTEGRAL I NEQUALITY
3
Lemma 2.1 (see [1]). If pi > 0, ai > 0, αi > 0, i = 1, 2, . . . , n, Ψ(u) is a measurable function,
then
α n α1
Z
Z
xn
x1
+ ··· +
(2.1)
...
Ψ
α1
α
n
x1
a1
an
n
x1 ,...,xn >0; a
+···+( x
≤1
an )
1
× xp11 −1 · · · xpnn −1 dx1 · · · dxn
ap11 · · · apnn Γ αp11 · · · Γ αpnn Z 1
p1
pn
+···+ α
−1
n
=
Ψ(u)u α1
du.
p1
pn
0
α1 · · · αn Γ α1 + · · · + αn
where Γ(·) is the Γ-function.
Lemma 2.2. If p > 1, p1 + 1q = 1, n ∈ Z+ , α > 0, λ > 0, setting the weight function ωα,λ (x, p, q)
as:

(n−λ)p
1
λq
Z
q
1
kxk
kxk
α
α


ωα,λ (x, p, q) =
dy,
1
λ + kykλ
kxk
kykα
p
Rn
α
α
+
kykα
then
kxk(n−λ)(p−1)
α
ωα,λ (x, p, q) =
(2.2)
πΓn α1
sin πp λαn−1 Γ
n
α
.
Proof. By Lemma 2.1, we have
ωα,λ (x, p, q) =
(n−λ)(p−1)+ λ
q
kxkα
(n−λ)(p−1)+ λ
q
= kxkα
r
α <r α
y1 ,...,yn >0;y1α +···+yn
y1 α
r
+ ··· +
h
y1 α
r
kxkλα + r
(n−λ)(p−1)+ λ
q
= kxkα
=
−(n−λ+ λ
1
q)
kyk
dy
α
λ
λ
kxkα + kykα
Rn
+
Z
Z
lim
···
r→+∞
h
×
Z
(n−λ)(p−1)+ λ
q
kxkα
1
yn α α
r
r Γ
r→+∞ αn Γ
lim
1
α
αn−1 Γ
1−1
1
yn α α
r
+ ··· +
n n
Γn
i−(n−λ+ λq )
1
α
n
α
Z
1
r→+∞
1
ru α
0
· · · yn1−1 dy1 · · · dyn
−(n−λ+ λq )
n
λ u α
1
−1
du
kxkλα + ru α
Z
lim
n
iλ y 1
0
r
λ
1
uλ− q −1 du
λ
+u
kxkλα
α Z ∞
n 1
λ
Γ α
(n−λ)(p−1)+ q
1
−1
λ− λ
q
du
u
= kxkα
n
λ
λ
αn−1 Γ α 0 kxkα + u
Z ∞
Γn α1
1
1
−1
(n−λ)(p−1)
p
u
= kxkα
du
n
n−1
1
+
u
λα Γ α 0
Γn α1
1
1
(n−λ)(p−1)
Γ
= kxkα
Γ 1−
p
p
λαn−1 Γ αn
J. Inequal. Pure and Appl. Math., 7(4) Art. 139, 2006
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4
H ONG YONG
=
kxkα(n−λ)(p−1)
πΓn α1
sin πp λαn−1 Γ
n
α
,
hence (2.2) is valid.
Lemma 2.3. If p > 1, p1 + 1q = 1, n ∈ Z+ , α > 0, λ > 0, 0 < ε < λ(q − 1), setting ω̃α,λ (x, q, ε)
as:
Z
(n−λ)(q−1)+n+ε
−
1
q
dy,
ω̃α,λ (x, q, ε) =
kyk
α
λ
λ
kxkα + kykα
Rn
+
then we have
ω̃α,λ (x, q, ε) =
(2.3)
−λ−ε
kxkα q q
Γn α1
λαn−1 Γ
n
α
Γ
1
ε
−
p λq
1
ε
Γ
+
.
q λq
Proof. Lemma 2.3 can be proved in the same manner as Lemma 2.2.
3. M AIN R ESULTS
Theorem 3.1. If p > 1,
Z
0<
(3.1)
Rn
+
1
p
+
1
q
= 1, n ∈ Z+ , α > 0, λ > 0, f ≥ 0, g ≥ 0, and
kxk(n−λ)(p−1)
f p (x)dx
α
Z
< ∞,
0<
Rn
+
kxk(n−λ)(q−1)
g q (x)dx < ∞,
α
then
Z
Z
(3.2)
Rn
+
Rn
+
πΓn α1
f (x)g(y)
dxdy <
kxkλα + kykλα
sin πp λαn−1 Γ
Z
×
Rn
+
Z
(3.3)
Rn
+
kykλ−n
α
where the constant factors
Z
Rn
+
n
α
! p1
kxk(n−λ)(p−1)
f p (x)dx
α
!p
f (x)
dx dy
kxkλα + kykλα

πΓn α1
< sin πp λαn−1 Γ
1
πΓn ( α
)
n
sin( π
λαn−1 Γ( α
)
p)
J. Inequal. Pure and Appl. Math., 7(4) Art. 139, 2006
and
! 1q
Z
Rn
+
kxk(n−λ)(q−1)
g q (x)dx
α
;
p
Z
n
α

Rn
+
1
πΓn ( α
)
n
sin( π
λαn−1 Γ( α
)
p)
kxk(n−λ)(p−1)
f p (x)dx,
α
p
are all the best possible.
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M ULTIPLE H ARDY-H ILBERT I NTEGRAL I NEQUALITY
5
Proof. By Hölder’s inequality, we have
Z Z
f (x)g(y)
A :=
dxdy
λ
λ
n kxk + kyk
Rn
R
α
α
+
+
n−λ

1
pqλ
Z Z
q
kxk
f (x)
kxk
α
α


=
1
1
λ + kykλ ) p
n
kykα
p
Rn
R
(kxk
+
+
kykα
α
α
n−λ

1
λ
p
kykα pq
g(y)
kykα 

dxdy
×
1
1
kxkα
(kxkλα + kykλα ) q kxkαq

 p1

(n−λ)p
1
λ
Z Z
q
f p (x)
kxkα q


 kxkα1 
≤
dxdy 
λ
λ
kxkα + kykα
kykα
Rn
Rn
+
+
kykαp

 1q

(n−λ)q
1
λ
Z Z
g q (x)
kykαp 
kykα p



×
dxdy 
1
λ
λ
kxkα + kykα
kxkα
Rn
Rn
+
+
kxkαq


  p1

(n−λ)p
1
λ
Z
Z
q
1
kxkα q  


 kxkα1 
f p (x) 
dy  dx
=
kxkλα + kykλα
kykα
p
Rn
Rn
+
+
kykα


  1q

(n−λ)q
1
λp
Z
Z
p
1
kykα


 
 kykα1 
×
g q (y) 
dx dy 
λ
λ
kxkα + kykα
kxkα
Rn
Rn
+
+
kxkαq
! p1 Z
! 1q
Z
f p (x)ωα,λ (x, p, q)dx
=
g q (y)ωα,λ (y, q, p)dy
Rn
+
,
Rn
+
according to the condition of taking equality in Hölder’s inequality, if this inequality takes the
form of an equality, then there exist constants C1 and C2 , such that they are not all zero, and
p

1
q
(n−λ)p
C1 f (x)  kxkα 
1
kxkλα + kykλα
kykαp
kxkα
kykα
q
λq

1
p
(n−λ)q
C2 g (y)  kykα 
=
1
kxkλα + kykλα
kxkαq
kykα
kxkα
λp
,
a.e. in Rn+ × Rn+ ,
C1 kxk(n−λ)(p−1)+n
f p (x) = C2 kyk(n−λ)(q−1)+n
g q (y) = C (constant),
α
α
a.e. in Rn+ × Rn+ ,
it follows that
which contradicts (3.1), hence we have
Z
A<
! p1
f p (x)ωα,λ (x, p, q)dx
Rn
+
J. Inequal. Pure and Appl. Math., 7(4) Art. 139, 2006
Z
! 1q
g q (y)ωα,λ (y, q, p)dy
.
Rn
+
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6
H ONG YONG
By Lemma 2.2 and sin
π
p
= sin( πq ), we have
 p1
! p1
Z
πΓ
kxk(n−λ)(p−1)
f p (x)dx
A< 
α
n
π
n
n−1
R+
sin p λα Γ α
# 1q Z
! 1q
"
πΓn α1
kyk(n−λ)(q−1)
g q (y)dy
×
α
π
n
n−1
sin( q )λα Γ α
Rn
+
! p1 Z
! 1q
Z
πΓn α1
=
kxk(n−λ)(p−1)
f p (x)dx
kxk(n−λ)(q−1)
g q (x)dx
.
α
α
Rn
Rn
sin πp λαn−1 Γ αn
+
+

1
α
n
Hence (3.2) is valid.
For 0 < a < b < ∞, setting
ga,b (y) =

R

λ−n
 kykα
Rn
+


f (x)
λ dx
kxkλ
α +kykα
p−1
, a < kykα < b,
0 < kykα ≤ a
0,
Z
g̃(y) = kykλ−n
α
Rn
+
f (x)
dx
λ
kxkα + kykλα
or kykα ≥ b,
!p−1
,
y ∈ Rn+ ,
by (3.1), for sufficiently small a > 0 and sufficiently large b > 0, we have
Z
q
kyk(n−λ)(q−1)
ga,b
(y)dy < ∞.
α
0<
a<kykα <b
Hence by (3.2), we have
Z
kyk(n−λ)(q−1)
g̃ q (y)dy
α
a<kykα <b
Z
kykλ−n
α
=
a<kykα <b
Z
kykλ−n
α
=
a<kykα <b
Z
Z
=
Rn
+
Rn
+
Z
Rn
+
Z
Rn
+
f (x)
dx
kxkλα + kykλα
f (x)
dx
λ
kxkα + kykλα
!p
dy
!p−1
Z
Rn
+
!
f (x)
dx dy
kxkλα + kykλα
f (x)ga,b (y)
dxdy
kxkλα + kykλα
J. Inequal. Pure and Appl. Math., 7(4) Art. 139, 2006
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M ULTIPLE H ARDY-H ILBERT I NTEGRAL I NEQUALITY
πΓn α1
<
sin πp λαn−1 Γ
Z
×
Rn
+
! p1
Z
n
α
7
Rn
+
kxk(n−λ)(p−1)
f p (x)dx
α
! 1q
q
kyk(n−λ)(q−1)
ga,b
(y)dy
α
πΓn α1
=
sin πp λαn−1 Γ
Z
×
! p1
Z
n
α
Rn
+
kxk(n−λ)(p−1)
f p (x)dx
α
kyk(n−λ)(q−1)
g̃ q (y)dy
α
1q
,
a<kykα <b
it follows that

Z
g̃ q (y)dy
kyk(n−λ)(q−1)
α
a<kykα <b
n
1
α
p
πΓ
< sin πp λαn−1 Γ
Z
n
α

Rn
+
kxk(n−λ)(p−1)
f p (x)dx.
α
For a → 0+ , b → +∞, by (3.1), we obtain
Z
0<
Rn
+

≤
kyk(n−λ)(q−1)
g̃ q (y)dy
α
n
1
α
p
πΓ
sin πp λαn−1 Γ
Z
n
α

Rn
+
kxk(n−λ)(p−1)
f p (x)dx < ∞,
α
hence by (3.2), we have
Z
Rn
+
Z
kykλ−n
α
Z
Rn
+
f (x)
dx
λ
kxkα + kykλα
!p
dy
Z
f (x)g̃(y)
dxdy
λ
λ
n kxk + kyk
Rn
R
α
α
+
+
! p1 Z
! 1q
Z
πΓn α1
<
kxk(n−λ)(p−1)
f p (x)dx
kyk(n−λ)(q−1)
g̃ q (y)dy
α
α
n
π
n
n
n−1
R+
R+
sin p λα Γ α
! p1
Z
πΓn α1
kxk(n−λ)(p−1)
f p (x)dx
=
α
n
π
Rn
sin p λαn−1 Γ α
+
"Z
!p # 1q
Z
f
(x)
kykλ−n
×
dx dy ,
α
n
kxkλα + kykλα
R+
Rn
+
=
J. Inequal. Pure and Appl. Math., 7(4) Art. 139, 2006
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8
H ONG YONG
it follows that
Z
Z
λ−n
kykα
Rn
+
Rn
+
f (x)
dx
λ
kxkα + kykλα

<
!p
dy
1
α
n
p
πΓ
sin πp λαn−1 Γ
Z
n
α

Rn
+
kxk(n−λ)(p−1)
f p (x)dx.
α
Hence (3.3) is valid.
Remark 3.2. By (3.3), we can also obtain (3.2), hence (3.3) and (3.2) are equivalent.
1
πΓn ( α
)
If the constant factor Cn,α (λ, p) := sin π λαn−1
in (3.2) is not the best possible, then
n
Γ( α
(p)
)
there exists a positive constant K < Cn,α (λ, p), such that
Z Z
f (x)g(y)
(3.4)
dxdy
kxkλα + kykλα
Rn
Rn
+
+
! p1 Z
! 1q
Z
<K
Rn
+
kxk(n−λ)(p−1)
f p (x)dx
α
Rn
+
kyk(n−λ)(q−1)
g q (y)dy
α
.
In particular, setting
f (x) =
−
kxkα
(n−λ)(p−1)+n+ε
p
,
g(y) =
−
kykα
(n−λ)(q−1)+n+ε
q
,
0 < ε < λ(q − 1),
(3.4) is still true. By the properties of limit, there exists a sufficiently small a > 0, such that
Z
Z
(n−λ)(p−1)+n+ε
(n−λ)(q−1)+n+ε
−
−
1
p
q
kxk
kyk
dxdy
α
α
λ
λ
kxkα + kykα
kxkα >a Rn
+
Z
p1
(n−λ)(p−1)
−(n−λ)(p−1)−n−ε
<K
kxkα
kxkα
dx
kxkα >a
Z
kyk(n−λ)(q−1)
kyk−(n−λ)(q−1)−n−ε
dy
α
α
×
1q
kykα >a
Z
kxk−n−ε
dx.
α
=K
kxkα >a
On the other hand, by Lemma 2.3, we have
Z
Z
(n−λ)(p−1)+n+ε
(n−λ)(q−1)+n+ε
−
−
1
p
q
kxk
kyk
dxdy
α
α
λ + kykλ
n
kxk
kxkα >a R+
α
α
Z
Z
(n−λ)(q−1)+n+ε
−n+ λ
− pε
−
1
q
q
=
kxkα
kyk
dydx
α
λ
λ
kxkα + kykα
kxkα >a
Rn
+
Z
−n+ λ − ε
=
kxkα q p ω̃α,λ (x, q, ε)dx
kxkα >a
Z
Γn α1
1
ε
1
ε
Γ
=
−
Γ
+
kxk−n−ε
dx,
α
n
n−1
p λq
q λq
λα Γ α
kxkα >a
hence we obtain
Γn α1
λαn−1 Γ
n
α
Γ
J. Inequal. Pure and Appl. Math., 7(4) Art. 139, 2006
1
ε
−
p λq
1
ε
Γ
+
< K.
q λq
http://jipam.vu.edu.au/
M ULTIPLE H ARDY-H ILBERT I NTEGRAL I NEQUALITY
9
For ε → 0+ , we have
πΓn α1
Cn,α (λ, p) =
sin πp λαn−1 Γ
n
α
≤ K,
this contradicts the fact that K < Cn,α (λ, p).Hence the constant factor in (3.2) is the best
possible.
Since (3.3) and (3.2) are equivalent, the constant factor in (3.3) is also the best possible.
Corollary 3.3. If p > 1,
1
p
+
Z
0<
(3.5)
1
q
= 1, n ∈ Z+ , α > 0, f ≥ 0, g ≥ 0, and
Z
p
f (x)dx < ∞,
0<
g q (x)dx < ∞,
Rn
+
Rn
+
then
Z
Z
(3.6)
Rn
+
Rn
+
f (x)g(y)
dxdy
kxknα + kyknα
πΓn α1
<
sin πp nαn−1 Γ
Z
Z
(3.7)
Rn
+
Rn
+
f (x)
dx
n
kxkα + kyknα
!p
! p1
Z
n
f p (x)dx
Rn
+
α
! 1q
Z
g q (x)dx
;
Rn
+
p

πΓn α1
dy <  sin πp nαn−1 Γ
Z
n
α

f p (x)dx,
Rn
+
where the constant factors in (3.6) and (3.7) are all the best possible.
Proof. By taking λ = n in Theorem 3.1, (3.6) and (3.7) can be obtained.
Corollary 3.4. If p > 1,
1
p
+
Z
0<
(3.8)
1
q
= 1, n ∈ Z+ , f ≥ 0, g ≥ 0, and
Z
p
f (x)dx < ∞,
0<
g q (x)dx < ∞,
Rn
+
Rn
+
then
Z
Z
(3.9)
Rn
+
Rn
+
f (x)g(y)
Pn
P
n
n dxdy
( i=1 xi ) + ( ni=1 yi )
<
n! sin
Z
Z
(3.10)
Rn
+
Rn
+
! p1
Z
π
π
p
f (x)
Pn
P
n
n dx
( i=1 xi ) + ( ni=1 yi )
f p (x)dx
Rn
+
!p
! 1q
Z
g q (x)dx
;
Rn
+
p

dy < 
Z
π
n! sin

π
p
f p (x)dx,
Rn
+
where the constant factors in (3.9) and (3.10) are all the best possible.
Proof. By taking α = 1 in Corollary 3.3, (3.9) and (3.10) can be obtained.
J. Inequal. Pure and Appl. Math., 7(4) Art. 139, 2006
http://jipam.vu.edu.au/
10
H ONG YONG
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[8] BICHENG YANG, On a generalization of Hardy-Hilbert’s inequality, Chin. Ann. of Math., 23
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J. Inequal. Pure and Appl. Math., 7(4) Art. 139, 2006
http://jipam.vu.edu.au/