Multidimensional Hilbert-Type Inequalities with a Homogeneous Kernel

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

f^pxdx <∞, 0<

_∞

0

g^qxdx <∞. 1.1

The well-known Hardy-Hilbert’s integral inequalitysee1is given by _∞

0

fxg

y

xy dx dy < π sin

π/p _∞

0

f^pxdx

1/p_∞

0

g^qxdx 1/q

, 1.2

and an equivalent form is given by _∞

0

_∞

0

fx xydx

p

dy <

π sin

π/p _p∞

0

f^pxdx, 1.3

where the constant factorsπ/sinπ/pandπ/sinπ/p ^pare the best possible.

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}, p_i>

1, ⁿ_i11/pi 1, s >0, fi≥0,satisfy

0<

_∞

0

x^pi−s−1f_i^pixdx <∞ i1,2, . . . , n, 1.4

then

0,∞ⁿ

_n

i1fixi

nj1x_jsdx₁· · ·dx_n< 1 Γs

n i1

Γ s

pi

_∞

0

x^pi−s−1f_i^pixdx 1/pi

, 1.5

where the constant factor1/Γs_n

i1Γs/piis the best possible.

Our generalization will include a general homogeneous kernelKx1, . . . , x_k:Rn^k

→ 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∈L^pRandg∈L^qR:

_∞

0

fxg

y

xy_λ dx dy≤Cp,qf

L^pRg

L^qR. 1.8

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, . . . , x_n; x₁, x₂, . . . , x_n>0},

|x|_α x1αx2α· · ·xnα^1/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. Letp_i, i1,2, . . . , k,be the real parameters which satisfy

k i1

1

p_i ≥1, pi>1, i1,2, . . . , k. 2.1

Further, the parametersp_i,i1,2, . . . , kare defined by the equations 1

p_i 1

p_i 1, i1,2, . . . , k. 2.2

Sincep_i >1, i1,2, . . . , k, it is obvious thatp_i>1, i1,2, . . . , k. We define

λ: 1

k−1 k

i1

1

p_i. 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

p_i, i1,2, . . . , k. 2.4

In order to obtain our results we need to require

q_i>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 q_i, 1

q_i 1−λ 1

p_i, i1,2, . . . , k. 2.6

Of course, ifλ 1, then ^k_i11/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, p_i, p_i, q_i, i1,2, . . . , k, be real numbers satisfying2.1–

2.5. LetK : Ω^k → Rand φij : Ω → R,i, j 1, . . . , k, be nonnegative measurable functions such that_k

i,j1φijxj 1. Then, for any nonnegative measurable functionsfi,i 1,2, . . . , k, the following inequalities hold and are equivalent:

Ω^kK^λx1, . . . , x_k^k

i1

f_ixidμ1x1· · ·dμ_kxk≤^k

i1

Ω

φ_iiF_if_ipi

xidμixi _1/pi

, 2.7

⎛

⎝

Ω

1 φkkFkxk

Ω^k−1K^λx1, . . . , x_k^k−1

i1

f_ixidμ1x1· · ·dμ_k−1x_k−1 p_k

dμ_kxk

⎞

⎠

1/p_k

≤^k−1

i1

Ω

φiiFifi

_pi

xidμixi 1/pi

,

2.8 where

F_ixi

⎛

⎝

Ω^k−1Kx1, . . . , x_k· ⁿ

j1,j /i

φ^q_ijⁱxjdμ1x1· · ·dμ_i−1xi−1dμi1xi1· · ·dμ_kxk

⎞

⎠

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

f_ixi C_iφ_iixi^qi/1−λqiF_ixi^1−λqi, C_i≥0, i1, . . . , k. 2.10 In the following theorem we give the most important case where Ω Rn, the measures μ_i, i 1, . . . , k, are Lebesgue measures, K_α : 0,∞^k → R is a nonnegative homogeneous function of degree−s, s >0, and the functionsφijrepresent the formφijxj

|xj|^Aα^ij whereA_ij ∈ R, i, j 1, . . . , n. In order to obtain the generalizations of some known results we define

k_α

β₁, . . . , β_k−1 :

0,∞^k−1K_α1, t1, . . . , t_k−1t^β₁¹· · ·t^β_k−1^k−1dt₁· · ·dt_k−1, 2.11 where we suppose thatk_αβ1, . . . , β_k−1<∞forβ₁, . . . , β_k−1>−1 andβ₁· · ·β_k−1k < s1.

Due to technical reasons, we introduce real parametersAij, i, j 1,2, . . . , ksatisfying

k i1

A_ij0, j 1,2, . . . , k. 2.12

We also define

αi^k

j1

Aij, i1,2, . . . , k. 2.13

Theorem 2.2. Letk, n ∈ N, k ≥ 2,and λ, pi, p_i, 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 letA_ij, i, j 1, . . . , k,and α_i, i 1, . . . , k be real parameters satisfying 2.12and 2.13. Iff_i:Rn → R,f_i/0,i1, . . . , kare nonnegative measurable functions, then the following inequalities hold and are equivalent:

Rn^kK^λ_α|x1|_α, . . . ,|xk|_α^k

i1

fixidx1· · ·dxk< L k

i1

Rn|xi|^pα^{i/qik−1n−spiαi}f_i^pixidxi

_1/pi ,

Rn|xk|^−pα ^{k/qkk−1n−s−pkαk}

Rn^k−1K^λ_α|x1|_α, . . . ,|xk|_α·^k−1

i1

fixidx1· · ·dx_k−1 p_k

dxk

< L^pk k−1

i1

Rn|xi|^pα^{i/qik−1n−spiαi}f_i^pixidxi

p_k/pi

,

2.14

where

L S^n−1k−1λ

α

2^k−1nλ kα

n−1q1A12, . . . , n−1q1A1k

_1/q1

·k_α

s−k−1n−1−q₂α2−A₂₂, n−1q₂A₂₃, . . . , n−1q₂A_2k1/q2

· · ·kα

n−1qkAk2, . . . , n−1qkA_k,k−1, s−k−1n−1−qkαk−Akk_1/qk ,

2.15

q_iA_ij>−n, i /jandq_iAii−α_i>k−1n−s.

Proof. Set Kx1, . . . , xk Kα|x1|α, . . . ,|xk|α and φijxj |xj|^Aij in Theorem 2.1, where

ki1Aij 0 for everyj 1, . . . , k. It is enough to calculate the functionsFixi, i 1, . . . , k.

By using then-dimensional spherical coordinates we find

F^q₁¹x1

Rn^k−1K_α|x1|_α, . . . ,|xk|_α^k

j2

x_j^q1A1jdx₂· · ·dx_k

S^n−1k−1

α

2^k−1n

0,∞^k−1Kα|x1|_α, t2, . . . , tk^k

j2

t^n−1q_j ^1A1jdt2· · ·dtk.

2.16

Using homogeneity of the function K_α and the substitutionsu_i t_i/|x1|α, i 2, . . . , k,we have

F₁^q1x1

S^n−1k−1

α

2^k−1n

0,∞^k−1|x1|^−s_α K_α1, u2, . . . , u_k·^k

j2

|x1|_αu_jn−1q1A1j|x1|^k−1_α du₂· · ·du_k

S^n−1k−1

α

2^k−1n |x1|^k−1n−sqα ^1α1−A11kα

n−1q1A12, . . . , n−1q1A1k

.

2.17

Similarly, by applying the n-dimensional spherical coordinates and homogeneity of the functionKαwe have

F₂^q2x2

Rn^k−1Kα|x1|_α, . . . ,|xk|_α ^k

j1,j /2

xj^q2A2jdx1dx3· · ·dxk

S^n−1k−1

α

2^k−1n

0,∞^k−1t^−s₁ K_α

1,|x2|_α t₁ ,t₃

t₁, . . . ,t_k t₁

· ^k

j1,j /2

t^n−1q_j ^2A2jdt₁dt₃· · ·dt_k.

2.18

Using the change of variables

t1|x2|_αu⁻¹₂ , ti|x2|_αu⁻¹₂ ui, i3, . . . , k, so ∂t1, t3, . . . , tk

∂u2, u₃, . . . , u_k |x2|^k−1_α u^−k₂ , 2.19

where∂t1, t3, . . . , tk/∂u2, u3, . . . , ukdenotes the Jacobian of the transformation, we have

F₂^q2x2

S^n−1k−1

α

2^k−1n |x2|^k−1n−sqα ^2α2−A22

·

0,∞^k−1Kα1, u2, . . . , uku^{s−k−1n−q}₂ ^2α2−A22k

j3

u^n−1q_j ^2A2jdu2· · ·duk

S^n−1k−1

α

2^k−1n |x2|^{k−1n−s−q}α ^2α2−A22

·kα

s−k−1n−1−q2α2−A22, n−1q2A23, . . . , n−1q2A2k

.

2.20

In a similar manner we obtain

F^q_iⁱxi

S^n−1k−1

α

2^k−1n |xi|^k−1n−sqα ^iαi−Aii

·k_α

n−1q_iA_i2, . . . , n−1q_iA_i,i−1, s−k−1n−1−q_iαi−A_ii, n−1q_iA_i,i1, . . . , n−1q_iA_ik

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 K_α|x1|α, . . . ,|xk|α ^k_i1|xi|^βα^−s is a homogeneous function of degree−βs.In this case we have

kα

β1, . . . , β_k−1

0,∞^k−1

_k−1

i1t^β_iⁱ

1 ^k−1_i1 t^β_iⁱsdt1· · ·dt_k−1

1 β^k−1ΓsΓ

s−^k−1

i1

βi1 β

_k−1

i1

Γ βi1

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 follows.

Theorem 3.1. Letk, n∈N, k≥2 and letp1, . . . , pkbe conjugate parameters such thatpi>1, i 1, . . . , k. LetK_α : 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 f_i : Rⁿ → R,f_i/0, i 1, . . . , k are nonnegative measurable functions, then the following inequalities hold and are equivalent:

Rn^kK_α|x1|_α, . . . ,|xk|_α^k

i1

f_ixidx1· · ·dx_k< M k

i1

Rn|xi|^k−1n−spα ^iαif_i^pixidxi

1/pi

,

Rn|xk|^1−pα ^{kk−1n−s−pkαk}

Rn^k−1K_α|x1|_α, . . . ,|xk|_α·^k−1

i1

f_ixidx1· · ·dx_k−1 p_k

dx_k

< M^pk k−1

i1

Rn|xi|^k−1n−spα ^iαif_i^pixidxi

_p

k/pi

,

3.1 where

M S^n−1k−1

α

2^k−1n kα

n−1p1A12, . . . , n−1p1A1k

_1/p1

·k_α

s−k−1n−1−p₂α2−A₂₂, n−1p₂A₂₃, . . . , n−1p₂A_2k1/p2

· · ·k_α

n−1p_kA_k2, . . . , n−1p_kA_k,k−1, s−k−1n−1−p_kαk−A_kk1/pk ,

3.2

p_iA_ij>−n, i /jandp_iAii−α_i>k−1n−s.

To obtain a case of the best inequality it is natural to impose the following conditions on the parametersAij:

np_jA_jis−k−1n−p_iαi−A_ii, 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

2^k−1n kα

n−1A2, . . . , n−1Ak

, 3.4

where

Aip1A1i fori /1, A1pkAk1. 3.5

Further, by using3.4and3.5, the inequalities3.1with the parametersAij,satisfying the relation3.3, become

Rn^kKα|x1|_α, . . . ,|xk|_α^k

i1

fixidx1· · ·dxk< M^∗ k

i1

Rn|xi|^−n−pα ^iAif_i^pixidxi

_1/pi

, 3.6

⎡

⎣

Rn|xk|^1−pα ^k−n−pkAk

Rn^k−1Kα|x1|_α, . . . ,|xk|_α·^k−1

i1

fixidx1· · ·dx_k−1 p_k

dxk

⎤

⎦

1/p_k

< M^∗ k−1

i1

Rn|xi|^−n−pα ^iAif_i^pixidxi

_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 kernelK_αt1, . . . , t_kis 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 functionsf_i,ε:Rⁿ →Rby the formulas

fi,εxi

⎧⎨

⎩

0, |xi|_α<1,

|xi|_α^Ai−ε/pi, |xi|_α≥1,

i1, . . . , k, 3.9

where 0 < ε < min_1≤i≤k{pi p_iA_i}. 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|^−n−ε_α dxi

_1/pi

M1Sⁿ⁻¹

α

2ⁿ _∞

1

t^−1−εdt M1Sⁿ⁻¹

α

2ⁿε . 3.10

Further, letJdenotes the left-hand side of the inequality3.6, for the above choice of the functionsf_i,ε.By applying then-dimensional spherical coordinates and the substitutions u_it_i/t₁, i /2,we find

J

|x1|α≥1· · ·

|xk|α≥1K_α|x1|_α, . . . ,|xk|_α^k

i1

|xi|^Aα^i−ε/pidx₁· · ·dx_k

S^n−1k

α

2^kn _∞

1

· · · _∞

1

Kαt1, . . . , tk^k

i1

tⁿ⁻¹_i ^Ai−ε/pidt1· · ·dtk

S^n−1k

α

2^kn _∞

1

t^−1−ε/β_{1 ∞}

1/t1

· · · _∞

1/t1

K_α1, u2, . . . , u_k^k

i2

uⁿ⁻¹_i ^Ai−ε/pidu₂· · ·du_k

dt₁.

3.11

Now, it is easy to see that the following inequality holds:

J ≥ S^n−1k

α

2^kn _∞

1

t^−1−ε_{1 ∞}

0

· · · _∞

0

K_α1, u2, . . . , u_k^k

i2

uⁿ⁻¹_i ^Ai−ε/pidu₂· · ·du_k

dt₁

−S^n−1k

α

2^kn _∞

1

t^−1−ε₁ k

j2

I_jt1dt1,

3.12

where forj2, . . . , k, I_jt1is defined by

I_jt1

Dj

K_α1, u2, . . . , u_k^k

i2

uⁿ⁻¹_i ^Ai−ε/pidu₂· · ·du_k, 3.13

satisfyingD_j {u2, . . . , u_k; 0< u_j<1/t₁, 0< u_l<∞, l /j}.Without losing generality, we only estimate the integralI2t1.Fork2 we have

I₂t1

_1/t1

0

K_α1, u2uⁿ⁻¹₂ ^A2−ε/p2du₂≤C _1/t1

0

uⁿ⁻¹₂ ^A2−ε/p2du₂

C

nA₂− ε p₂

₋₁

t^ε/p₁ ^2−n−A2,

3.14

and fork >2 we find

I₂t1≤C

0,∞^k−2K_α1,0, u₃, . . . , u_k^k

i3

u_i^{n−1Ai−ε/pi}du₃· · ·du_{k 1/t1}

0

u₂^{n−1A2−ε/p2}du₂

C

n− ε p₂ A₂

₋₁

t₁^{ε/p2−A2−n}k_α

n−1A₃− ε

p₃, . . . , n−1A_k− ε p_k

,

3.15

wherekαn−1A3−ε/p3, . . . , n−1Ak−ε/pkis well defined since obviouslyA3· · · A_k < s−k−2n. Hence, we haveI_jt1 ≤ t^ε/p₁ ^j−n−AjO_j1,forε → 0, j ∈ {2, . . . , k},and consequently

_∞

1

t^−1−ε₁ 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|^s_α− |y|^s_α, 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,∞^kK_αx1, . . . , x_k^k

i1

f_ixidx1· · ·dx_k< M^∗ k

i1

_∞

0

x_i^−1−piAif_i^pixidxi

1/pi

. 3.17

If the kernelK_αx1, . . . , x_kand the parametersA_ij 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

This research is supported by the Croatian Ministry of Science, Education and Sports, Grant no. 058-1170889-1050.

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