Volume 2009, Article ID 130958,12pages doi:10.1155/2009/130958
Research Article
Multidimensional Hilbert-Type Inequalities with
a Homogeneous Kernel
Predrag Vukovi´c
Faculty of Teacher Education, University of Zagreb, Savska cesta 77, 10000 Zagreb, Croatia
Correspondence should be addressed to Predrag Vukovi´c,[email protected]
Received 11 July 2009; Revised 10 November 2009; Accepted 18 November 2009
Recommended by Radu Precup
We consider the Hilbert-type inequalities with nonconjugate parameters. The obtaining of the best possible constants in the case of nonconjugate parameters remains still open. Our generalization will include a general homogeneous kernel. Also, we obtain the best possible constants in the case of conjugate parameters when the parameters satisfy appropriate conditions. We also compare our results with some known results.
Copyrightq2009 Predrag Vukovi´c. 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.
1. Introduction
Let 1/p1/q1 p >1, f, g≥0,
0<
∞
0
fpxdx <∞, 0<
∞
0
gqxdx <∞. 1.1
The well-known Hardy-Hilbert’s integral inequalitysee1is given by
∞
0
fxgy
xy dx dy <
π
sinπ/p
∞
0
fpxdx
1/p∞
0
gqxdx
1/q
, 1.2
and an equivalent form is given by
∞
0
∞
0
fx
xydx
p
dy <
π
sinπ/p
p∞
0
fpxdx, 1.3
During the previous decades, the Hilbert-type inequalities were discussed by many authors, who either reproved them using various techniques or applied and generalized them in many different ways. For example, we refer to a paper of Yangsee2. Ifn∈N\{1}, pi>
1, ni11/pi 1, s >0, fi≥0,satisfy
0<
∞
0
xpi−s−1fipixdx <∞ i1,2, . . . , n, 1.4
then
0,∞n
n
i1fixi n j1xj
sdx1· · ·dxn<
1 Γs
n
i1
Γ
s pi
∞
0
xpi−s−1fpi i xdx
1/pi
, 1.5
where the constant factor1/Γsni1Γs/piis the best possible.
Our generalization will include a general homogeneous kernelKx1, . . . , xk:Rnk
→ R, where k ≥ 2, with k being nonconjugate parameters. The techniques that will be used in the proofs are mainly based on classical real analysis, especially on the well-known H ¨older’s inequality and on Fubini’s theorem. The obtaining of the best possible constants in the case of nonconjugate parameters seems to be a very difficult problem and it remains still open.
Let us recall the definition of nonconjugate exponentssee3. Letp andqbe real parameters, such that
p >1, q >1, 1
p
1
q≥1, 1.6
and let p and q, respectively, be their conjugate exponents, that is, 1/p1/p 1 and 1/q1/q1. Further, define
λ 1
p
1
q 1.7
and note that 0< λ≤1 for allpandqvalues as in1.6. In particular,λ1 holds if and only ifqp, that is, only whenpandqare mutually conjugate. Otherwise, 0< λ <1, and in such casespandqwill be referred to as nonconjugate exponents.
Consideringp,q, andλas in1.6and1.7, Hardy et al.1, proved that there exists a constantCp,q, dependent only on the parameterspandq, such that the following Hilbert-type
inequality holds for all nonnegative functionsf∈LpR
andg∈LqR:
∞
0
fxgy
xyλ dx dy≤Cp,q
f
Conventions
Throughout this paper we suppose that all the functions are nonnegative and measurable, so that all integrals converge. We also introduce the following notations:
Rn{x x1, x2, . . . , xn; x1, x2, . . . , xn>0},
|x|α x1αx2α· · ·xnα1/α, α >0,
1.9
and let|Sn−1|
α2nΓn1/α/αn−1Γn/αbe an area of unit sphere inRnin view ofα−norm.
2. Main Results
Before presenting our idea and results, we repeat the notion of general nonconjugate exponents from3. Letpi, i1,2, . . . , k,be the real parameters which satisfy
k
i1
1
pi ≥
1, pi>1, i1,2, . . . , k. 2.1
Further, the parameterspi,i1,2, . . . , kare defined by the equations
1
pi
1
pi 1, i1,2, . . . , k. 2.2
Sincepi >1, i1,2, . . . , k, it is obvious thatpi>1, i1,2, . . . , k. We define
λ: 1
k−1
k
i1
1
pi. 2.3
It is easy to deduce that 0 < λ ≤ 1. Also, we introduce the parameters qi, i 1,2, . . . , k,
defined by the relations
1
qi λ−
1
pi, i1,2, . . . , k. 2.4
In order to obtain our results we need to require
qi>0, i1,2, . . . , k. 2.5
It is easy to see that the above conditions do not automatically apply2.5. Further, it follows
λ
k
i1
1
qi
, 1
qi
1−λ 1 pi
Of course, ifλ 1, then ki11/pi 1; so the conditions2.1–2.4reduce to the case of
conjugate parameters.
Results in this section will be based on the following general form of Hardy-Hilbert’s inequality proven in4. All the measures are assumed to beσ-finite on some Ω measure space.
Theorem 2.1. Letk, n∈N, k ≥2,andλ, pi, pi, qi, i1,2, . . . , k, be real numbers satisfying2.1–
2.5. LetK : Ωk → Rand φ
ij : Ω → R,i, j 1, . . . , k, be nonnegative measurable functions
such thatki,j1φijxj 1. Then, for any nonnegative measurable functionsfi,i 1,2, . . . , k, the
following inequalities hold and are equivalent:
Ωk
Kλx
1, . . . , xk k
i1
fixidμ1x1· · ·dμkxk≤ k
i1
Ω
φiiFifi
pix
idμixi
1/pi
, 2.7
⎛
⎝
Ω
1 φkkFkxk
Ωk−1
Kλx
1, . . . , xk k−1
i1
fixidμ1x1· · ·dμk−1xk−1
pk
dμkxk
⎞ ⎠
1/pk
≤k−1
i1
Ω
φiiFifi
pi
xidμixi
1/pi
,
2.8
where
Fixi
⎛
⎝
Ωk−1
Kx1, . . . , xk· n
j1,j /i
φqiijxjdμ1x1· · ·dμi−1xi−1dμi1xi1· · ·dμkxk
⎞ ⎠
1/qi
,
i1, . . . , k.
2.9
In the same paper the authors discussed the case of equality in inequalities2.7and 2.8. They proved that the equality holds in2.7 and analogously in2.8if and only if
fixi Ciφiixiqi/1−λqiFixi1−λqi, Ci≥0, i1, . . . , k. 2.10
In the following theorem we give the most important case where Ω Rn, the
measures μi, i 1, . . . , k, are Lebesgue measures, Kα : 0,∞k → R is a nonnegative
homogeneous function of degree−s, s >0, and the functionsφijrepresent the formφijxj
|xj| Aij
α whereAij ∈ R, i, j 1, . . . , n. In order to obtain the generalizations of some known
results we define
kα
β1, . . . , βk−1
:
0,∞k−1Kα1, t1, . . . , tk−1t β1 1 · · ·t
βk−1
k−1dt1· · ·dtk−1, 2.11
Due to technical reasons, we introduce real parametersAij, i, j 1,2, . . . , ksatisfying
k
i1
Aij0, j 1,2, . . . , k. 2.12
We also define
αi
k
j1
Aij, i1,2, . . . , k. 2.13
Theorem 2.2. Let k, n ∈ N, k ≥ 2,and λ, pi, pi, qi, i 1,2, . . . , k, be real numbers satisfying
2.1–2.5. LetKα :0,∞k → Rbe nonnegative measurable homogeneous function of degree−s,
s > 0, and let Aij, i, j 1, . . . , k,and αi, i 1, . . . , k be real parameters satisfying 2.12and
2.13. Iffi:Rn → R,fi/0,i1, . . . , kare nonnegative measurable functions, then the following
inequalities hold and are equivalent:
Rnk
Kλα|x1|α, . . . ,|xk|α k
i1
fixidx1· · ·dxk< L k
i1
Rn
|xi|αpi/qik−1n−spiαifipixidxi
1/pi
,
Rn
|xk|−p
k/qkk−1n−s−pkαk α
Rnk−1
Kλα|x1|α, . . . ,|xk|α· k−1
i1
fixidx1· · ·dxk−1
pk
dxk
< Lpk k−1
i1
Rn
|xi|αpi/qik−1n−spiαifipixidxi
pk/pi
,
2.14
where
L S
n−1k−1λ α
2k−1nλ kα
n−1q1A12, . . . , n−1q1A1k
1/q1
·kα
s−k−1n−1−q2α2−A22, n−1q2A23, . . . , n−1q2A2k
1/q2
· · ·kα
n−1qkAk2, . . . , n−1qkAk,k−1, s−k−1n−1−qkαk−Akk
1/qk
,
2.15
Proof. Set Kx1, . . . , xk Kα|x1|α, . . . ,|xk|α and φijxj |xj|Aij in Theorem 2.1, where k
i1Aij 0 for everyj 1, . . . , k. It is enough to calculate the functionsFixi, i 1, . . . , k.
By using then-dimensional spherical coordinates we find
Fq1 1 x1
Rnk−1
Kα|x1|α, . . . ,|xk|α k
j2
xjq1A1jdx2· · ·dxk
Sn−1
k−1 α
2k−1n
0,∞k−1Kα|x1|α, t2, . . . , tk k
j2
tnj−1q1A1jdt2· · ·dtk.
2.16
Using homogeneity of the function Kα and the substitutionsui ti/|x1|α, i 2, . . . , k,we
have
Fq1 1 x1
Sn−1k−1 α
2k−1n
0,∞k−1|x1|
−s
α Kα1, u2, . . . , uk· k
j2
|x1|αuj
n−1q1A1j|x
1|kα−1du2· · ·duk
Sn−1
k−1 α
2k−1n |x1|
k−1n−sq1α1−A11 α kα
n−1q1A12, . . . , n−1q1A1k
.
2.17
Similarly, by applying the n-dimensional spherical coordinates and homogeneity of the functionKαwe have
Fq2 2 x2
Rnk−1
Kα|x1|α, . . . ,|xk|α k
j1,j /2
xjq2A2jdx
1dx3· · ·dxk
Sn−1
k−1 α
2k−1n
0,∞k−1t
−s 1 Kα
1,|x2|α t1
,t3 t1
, . . . ,tk t1
· k
j1,j /2
tn−1q2A2j
j dt1dt3· · ·dtk.
2.18
Using the change of variables
t1|x2|αu−21, ti|x2|αu−21ui, i3, . . . , k, so
∂t1, t3, . . . , tk
∂u2, u3, . . . , uk |
where∂t1, t3, . . . , tk/∂u2, u3, . . . , ukdenotes the Jacobian of the transformation, we have
Fq2 2 x2
Sn−1k−1 α
2k−1n |x2|
k−1n−sq2α2−A22 α
·
0,∞k−1Kα1, u2, . . . , uku
s−k−1n−q2α2−A22 2
k
j3
un−1q2A2j
j du2· · ·duk
Sn−1
k−1 α
2k−1n |x2|
k−1n−s−q2α2−A22 α
·kα
s−k−1n−1−q2α2−A22, n−1q2A23, . . . , n−1q2A2k
.
2.20
In a similar manner we obtain
Fqii xi
Sn−1k−1 α
2k−1n |xi|
k−1n−sqiαi−Aii α
·kα
n−1qiAi2, . . . , n−1qiAi,i−1, s−k−1n−1−qiαi−Aii,
n−1qiAi,i1, . . . , n−1qiAik
2.21
for i 3, . . . , k. This gives inequalities 2.14 with inequality sign ≤. Condition 2.10
immediately gives that nontrivial case of equality in2.14leads to the divergent integrals. This completes the proof.
Remark 2.3. Note that the kernel Kα|x1|α, . . . ,|xk|α ki1|xi|βα
−s
is a homogeneous function of degree−βs.In this case we have
kα
β1, . . . , βk−1
0,∞k−1
k−1
i1tβii
1 ki1−1tβii s
dt1· · ·dtk−1
1
βk−1ΓsΓ
s−
k−1
i1
βi1
β k−1
i1
Γ
β
i1
β
,
2.22
where we used the well-known formula for gamma functionsee, e.g.,5, Lemma 5.1. Now, by usingTheorem 2.2and2.22we obtain the result of Krni´c et al.see6.
3. The Best Possible Constants in the Conjugate Case
In this section we consider the inequalities inTheorem 2.2. In such a way we shall obtain the best possible constants for some general cases.
It follows easily thatTheorem 2.2in the conjugate caseλ 1, pi qibecomes as
Theorem 3.1. Letk, n∈N, k≥2and letp1, . . . , pkbe conjugate parameters such thatpi>1, i
1, . . . , k. LetKα : 0,∞k → Rbe nonnegative measurable homogeneous function of degree −s,
s >0, and letAij, i, j1, . . . , k,andαi, i1, . . . , kbe real parameters satisfying2.12and2.13.
If fi : Rn → R,fi/0, i 1, . . . , k are nonnegative measurable functions, then the following
inequalities hold and are equivalent:
Rnk
Kα|x1|α, . . . ,|xk|α k
i1
fixidx1· · ·dxk< M k
i1
Rn
|xi|αk−1n−spiαifipixidxi
1/pi
,
Rn
|xk|1−
pkk−1n−s−pkαk α
Rnk−1
Kα|x1|α, . . . ,|xk|α· k−1
i1
fixidx1· · ·dxk−1
pk
dxk
< Mpk k−1
i1
Rn
|xi|αk−1n−spiαifipixidxi
pk/pi
,
3.1
where
M S
n−1k−1 α
2k−1n kα
n−1p1A12, . . . , n−1p1A1k
1/p1
·kα
s−k−1n−1−p2α2−A22, n−1p2A23, . . . , n−1p2A2k
1/p2
· · ·kα
n−1pkAk2, . . . , n−1pkAk,k−1, s−k−1n−1−pkαk−Akk
1/pk
,
3.2
piAij>−n, i /jandpiAii−αi>k−1n−s.
To obtain a case of the best inequality it is natural to impose the following conditions on the parametersAij:
npjAjis−k−1n−piαi−Aii, j /i, i, j∈ {1,2, . . . , k}. 3.3
In that case the constantMfromTheorem 3.1is simplified to the following form:
M∗ S
n−1k−1 α
2k−1n kα
n−1A2, . . . , n−1Ak
, 3.4
where
Further, by using3.4and3.5, the inequalities3.1with the parametersAij,satisfying the
relation3.3, become
Rnk
Kα|x1|α, . . . ,|xk|α k
i1
fixidx1· · ·dxk< M∗ k
i1
Rn
|xi|−n−pi Ai α f
pi i xidxi
1/pi
, 3.6
⎡
⎣
Rn
|xk|1−
pk−n−pkAk α
Rnk−1
Kα|x1|α, . . . ,|xk|α· k−1
i1
fixidx1· · ·dxk−1
pk
dxk
⎤ ⎦
1/pk
< M∗
k−1
i1
Rn
|xi|−n−pi Ai
α fipixidxi
1/pi
.
3.7
Theorem 3.2. Suppose that the real parametersAij, i, j1, . . . , ksatisfy conditions inTheorem 3.1
and conditions given in3.3. If the kernelKαt1, . . . , tkis as inTheorem 3.1and for everyi2, . . . , k
Kα1, t2, . . . , ti, . . . , tk≤CKα1, t2, . . . ,0, . . . , tk, 0≤ti≤1, tj≥0, j /i 3.8
for someC >0, then the constantM∗is the best possible in inequalities3.6and3.7.
Proof. Let us suppose that the constant factorM∗given by3.4is not the best possible in the inequality3.6. Then, there exists a positive constantM1 < M∗, such that3.6is still valid
when we replaceM∗byM1.
We define the real functionsfi,ε:Rn →Rby the formulas
fi,εxi
⎧ ⎨ ⎩
0, |xi|α<1,
|xi|α
Ai−ε/pi, |x i|α≥1,
i1, . . . , k, 3.9
where 0 < ε < min1≤i≤k{pi piAi}. Now, we shall put these functions in inequality 3.6.
By using then-dimensional spherical coordinates, the right-hand side of the inequality3.6
becomes
M1
k
i1
|xi|α≥1
|xi|−αn−εdxi
1/pi
M1Sn−1α
2n
∞
1
t−1−εdt M1S
n−1 α
Further, letJdenotes the left-hand side of the inequality3.6, for the above choice of the functionsfi,ε.By applying then-dimensional spherical coordinates and the substitutions
uiti/t1, i /2,we find
J
|x1|α≥1
· · ·
|xk|α≥1
Kα|x1|α, . . . ,|xk|α k
i1
|xi| Ai−ε/pi
α dx1· · ·dxk
Sn−1
k α 2kn ∞ 1 · · · ∞ 1
Kαt1, . . . , tk k
i1
tni−1Ai−ε/pidt1· · ·dtk
Sn−1
k α
2kn
∞
1
t−11−ε/β
∞
1/t1
· · ·
∞
1/t1
Kα1, u2, . . . , uk k
i2
uni−1Ai−ε/pidu2· · ·duk
dt1.
3.11
Now, it is easy to see that the following inequality holds:
J ≥ S
n−1k α
2kn
∞
1
t−11−ε
∞
0
· · ·
∞
0
Kα1, u2, . . . , uk k
i2
uni−1Ai−ε/pidu2· · ·duk
dt1
−Sn−1
k α
2kn
∞
1
t−1−ε 1
k
j2
Ijt1dt1,
3.12
where forj2, . . . , k, Ijt1is defined by
Ijt1
Dj
Kα1, u2, . . . , uk k
i2
uni−1Ai−ε/pidu2· · ·duk, 3.13
satisfyingDj {u2, . . . , uk; 0< uj<1/t1, 0< ul<∞, l /j}.Without losing generality, we
only estimate the integralI2t1.Fork2 we have
I2t1
1/t1
0
Kα1, u2un−1 A2−ε/p2
2 du2≤C
1/t1
0
un−1A2−ε/p2
2 du2
C
nA2−
ε p2
−1
tε/p2−n−A2
1 ,
and fork >2 we find
I2t1≤C
0,∞k−2Kα1,0, u3, . . . , uk k
i3
uin−1Ai−ε/pidu3· · ·duk
1/t1
0
u2n−1A2−ε/p2du2
C
n− ε
p2
A2
−1
t1ε/p2−A2−nkα
n−1A3−
ε p3
, . . . , n−1Ak−
ε pk
,
3.15
wherekαn−1A3−ε/p3, . . . , n−1Ak−ε/pkis well defined since obviouslyA3· · ·
Ak < s−k−2n. Hence, we haveIjt1 ≤ t
ε/pj−n−Aj
1 Oj1,forε → 0, j ∈ {2, . . . , k},and
consequently
∞
1
t−11−ε
k
j2
Ijt1dt1≤O1. 3.16
We conclude, by using 3.10, 3.12, and 3.16, that M∗ ≤ M1 which is an obvious
contradiction. It follows that the constantM∗in3.6is the best possible.
Finally, the equivalence of the inequalities3.6and3.7means that the constantM∗
is also the best possible in the inequality3.7. That completes the proof.
Remark 3.3. If we putk 2, Kαx, y ln|x|α/|y|α/|x|sα− |y|sα, A1 s/q−nandA2
s/p−nin the inequalities3.6and3.7applyingTheorem 3.2, we obtain the result of Baoju Sunsee7. Further, by puttingn1 in Theorems3.1and3.2we obtain appropriate results from8. More precisely, the inequality3.6becomes
0,∞kKαx1, . . . , xk k
i1
fixidx1· · ·dxk< M∗ k
i1
∞
0
xi−1−piAifipixidxi
1/pi
. 3.17
If the kernelKαx1, . . . , xkand the parametersAij satisfy the conditions fromTheorem 3.2,
then the constant M∗ kαA2, . . . ,Ak is the best possible. For example, setting
Kαx1, . . . , xk x1 · · ·xk−s, s > 0, Ai s−pi/pi, i 2, . . . , k,in the inequality
3.17, we obtain Yang’s result1.5from introduction.
Acknowledgment
References
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