HARDY-HILBERT TYPE INEQUALITIES WITH FRACTIONAL KERNEL IN Rn

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Hardy-Hilbert Type Inequalities M. Krnić, J. Peˇcarić, I. Perić and P. Vuković vol. 10, iss. 4, art. 115, 2009

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HARDY-HILBERT TYPE INEQUALITIES WITH FRACTIONAL KERNEL IN R

ⁿ

MARIO KRNI ´C JOSIP PE ˇCARI ´C

Faculty of Electrical Engineering and Computing Faculty of Textile Technology, University of Zagreb University of Zagreb, Unska 3, Zagreb, Croatia Pierottijeva 6, 10000 Zagreb, Croatia

EMail:mario.krnic@fer.hr EMail:pecaric@hazu.hr

IVAN PERI ´C PREDRAG VUKOVI ´C

Faculty of Textile Technology, University of Zagreb Teacher Training College ˇCakovec

Pierottijeva 6, 10000 Zagreb, Croatia Ante Starˇcevi´ca 55, 40000 ˇCakovec, Croatia

EMail:iperic@pbf.hr EMail:predrag.vukovic@vus-ck.hr

Received: 01 July, 2009

Accepted: 18 November, 2009

Communicated by: G. Sinnamon 2000 AMS Sub. Class.: 26D15.

Key words: Inequalities, multiple Hilbert’s inequality, multiple Hardy-Hilbert’s inequality, equivalent inequalities, non-conjugate parameters, gamma function, Selberg’s integral, the best possible constant, symmetric-decreasing function, general re- arrangement inequality, hypergeometric function.

Abstract: The main objective of this paper is some new special Hilbert-type and Hardy- Hilbert-type inequalities in(Rⁿ)^kwithk≥2non-conjugate parameters which are obtained by using the well known Selberg’s integral formula for fractional integrals in an appropriate form. In such a way we obtain extensions over the whole set of real numbers, of some earlier results, previously known from the lit- erature, where the integrals were taken only over the set of positive real numbers.

Also, we obtain the best possible constants in the conjugate case.

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1. Introduction

In order to obtain our general results, we need to present the definitions of non- conjugate parameters. Letp_i, i = 1,2, . . . , k,be the real parameters which satisfy

(1.1)

k

X

i=1

1 pi

≥1 and p_i >1, i= 1,2, . . . , k.

Further, the parametersp_i⁰,i= 1,2, . . . , k are defined by the equations

(1.2) 1

p_i + 1

p⁰_i = 1, i= 1,2, . . . , k.

Sincep_i >1,i= 1,2, . . . , k, it is obvious thatp⁰_i >1,i= 1,2, . . . , k. We define

(1.3) λ:= 1

k−1

k

X

i=1

1 p⁰_i.

It is easy to deduce that 0 < λ ≤ 1. Also, we introduce parameters q_i, i = 1,2, . . . , k, defined by the relations

(1.4) 1

q_i =λ− 1

p⁰_i, i= 1,2, . . . , k.

In order to obtain our results we require

(1.5) qi >0 i= 1,2, . . . , k.

It is easy to see that the above conditions do not automatically imply (1.5). The above conditions were also given by Bonsall (see [2]). It is easy to see that

λ =

k

X

i=1

1

q_i and 1

q_i + 1−λ= 1

p_i, i= 1,2, . . . , k.

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Of course, ifλ = 1, then Pk i=1

1

pi = 1, so the conditions (1.1) – (1.4) reduce to the case of conjugate parameters.

Considering the two-dimensional case of non-conjugate parameters (k = 2), Hardy, Littlewood and Pólya, (see [7]), proved that there exists a constant K, de- pendent only on the parameters p₁ andp₂ such that the following Hilbert-type inequality holds for all non-negative measurable functions f ∈ L^p¹(h0,∞i) and g ∈L^p²(h0,∞i) :

(1.6)

Z ∞ 0

f(x)g(y)

(x+y)^sdxdy≤K Z ∞

0

f^p¹(x)dx _p¹

1 Z ∞

0

g^p²(y)dy _p¹

2

. Hardy, Littlewood and Pólya did not give a specific value for the constant K in the previous inequality. An alternative proof by Levin (see [9]) established that K = B^s

1 sp10,_sp¹

20

, whereB is the beta function, but the paper did not determine whether this was the best possible constant. This question still remains open. The inequality (1.6) was also generalized by F.F. Bonsall (see [2]).

Hilbert and Hardy-Hilbert type inequalities (see [2]) are very significant weight inequalities which play an important role in many fields of mathematics. Similar inequalities, in operator form, appear in harmonic analysis where one investigates the boundedness properties of such operators. This is the reason why Hilbert’s inequality is so popular and is of great interest to numerous mathematicians.

In the last century Hilbert-type inequalities have been generalized in many differ- ent directions and numerous mathematicians have reproved them using various techniques. Some possibilities of generalizing such inequalities are, for example, various choices of non-negative measures, kernels, sets of integration, extension to the multi- dimensional case, etc. Several generalizations involve very important notions such as Hilbert’s transform, Laplace transform, singular integrals, Weyl operators.

In this paper we refer to a recent paper of Brneti´c et al, [4], where a general

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Hilbert-type and Hardy-Hilbert-type inequalities were obtained for non-conjugate parameters, wherek ≥2, with positiveσ−finite measures onΩ. However, we shall keep our attention on a result with Lebesgue measures and a special homogeneous function of degree−s. This is contained in:

Theorem 1.1. Let k ≥ 2be an integer, pi, p⁰_i, qi, i = 1,2, . . . , k, be real numbers satisfying (1.1) – (1.5) and Pk

i=1A_ij = 0, j = 1,2, . . . , k. Then the following inequalities hold and are equivalent:

(1.7) Z ∞

0

· · · Z ∞

0

Qk

i=1f_i(x_i) Pk

j=1x_jλsdx₁. . . dx_k

< K

k

Y

i=1

Z ∞ 0

x_i^pi^qi^{(k−1−s)+p}ⁱ^αⁱf_i^pⁱ(x_i)dx_i _pi¹

and

(1.8)





 Z ∞

0

x^(1−λp

0

k)(k−1−s)−p⁰_kαk

k





 Z ∞

0

· · · Z ∞

0

Qk−1 i=1 fi(xi) Pk

j=1x_jλsdx₁...dxk−1







p⁰_k

dx_k







1 p0 k

< K

k−1

Y

i=1

Z ∞ 0

x_i^pi^qi^{(k−1−s)+p}ⁱ^αⁱf_i^pⁱ(x_i)dx_i ¹

pi , where

K = 1

Γ(s)^λ

k

Y

i=1

Γ(s−k+ 1−q_iα_i+q_iA_ii)^qi¹

k

Y

i,j=1,i6=j

Γ(q_iA_ij+ 1)^qi¹,

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α_i =Pk

j=1A_ij,A_ij >−_q¹

i,i6=jandA_ii−α_i > ^k−s−1_q

i .

Our main objective is to obtain inequalities similar to the inequalities in Theorem 1.1, which will include the integrals taken over the whole set of real numbers.

The techniques that will be used in the proofs are mainly based on classical real analysis, especially on the well known Hölder inequality and on Fubini’s theorem.

Conventions. Throughout this paper we suppose that all the functions are non- negative and measurable, so that all integrals converge. Further, the Euclidean norm of the vectorx∈Rⁿwill be denoted by|x|.

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2. Preliminaries

The main results in this paper will be based on the well-known Selberg formula for the fractional integral

(2.1) Z

(Rⁿ)^k

|x_k|^α^k⁻ⁿ|x_k−xk−1|^α^k−1⁻ⁿ|xk−1−xk−2|^α^k−2⁻ⁿ· · · |x₂−x₁|^α¹⁻ⁿ

· |x₁−y|^α⁰⁻ⁿdx₁dx₂. . . dx_k= Qk

i=0Γ_n(α_i) Γ_n

Pk

i=0α_i|y|^P^kⁱ⁼⁰^αⁱ⁻ⁿ, for arbitraryk, n ∈ Nand0 < α_i < nsuch that0 < Pk

i=0α_i < n. The constant Γ_n(α)introduces then−dimensional gamma function and is defined by the formula

(2.2) Γ_n(α) = πⁿ²2^αΓ ^α₂

Γ ⁿ₂ − ^α₂ , 0< α < n,

where Γ is the well known gamma function. Further, from the definition of the n−dimensional gamma function it easily follows that

(2.3) Γn(n−α) = (2π)ⁿ

Γ_n(α), 0< α < n.

In the book [13], Stein derived the formula (2.1) with two parameters using the Riesz potential.

Multiple integrals similar to the one in (2.1) are known as Selberg’s integrals and their exact values are useful in representation theory and in mathematical physics.

These integrals have only been computed for special cases. For a treatment of Sel- berg’s integral, the reader can consult Section 17.11 of [11].

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Now, by using the integral equality (2.1), we can easily compute the integral Z

(Rⁿ)^k−1

Qk−1

i=1 |x_i|^−βⁱ

Pk i=1x_i

s dx₁dx₂. . . dxk−1, where0< βi < n,0< s < nand

(k−1)n−

k−1

X

i=1

β_i < s < kn−

k−1

X

i=1

β_i.

Such an integral will be more suitable for our computations. Namely, by using the substitutionx₁ =t₁−x_kandx_i =t_i−ti−1,i= 2,3, . . . , k−1(see also [5]), one obtains the formula

(2.4) Z

(Rⁿ)^k−1

Qk−1 i=1 |x_i|^−βⁱ

Pk i=1x_i

s dx₁dx₂. . . dx_k−1

= Γ_n(n−s)Qk−1

i=1 Γ_n(n−β_i) Γ_n

kn−s−Pk−1

i=1 β_i |x_k|^{(k−1)n−s−}^P^k−1ⁱ⁼¹ ^βⁱ, x_k6=0 where0 < β_i < n, 0 < s < nand (k −1)n −Pk−1

i=1 β_i < s < kn−Pk−1 i=1 β_i. Obviously, if0 < β_i < nand0 < s < nthen the condition s < kn−Pk−1

i=1 β_i is trivially satisfied.

We shall use the relation (2.4) in the next section, to obtain generalizations of the multiple Hilbert inequality, over the set of real numbers.

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3. Basic Result

As we have already mentioned, we shall obtain some extensions of the multiple Hilbert inequality on the whole set of real numbers. We also obtain the equivalent inequality, usually called the Hardy-Hilbert inequality. For more details about equivalent inequalities the reader can consult [7]. To obtain our results we introduce the real parametersAij, i, j = 1,2, . . . , ksatisfying

(3.1)

k

X

i=1

Aij = 0, j = 1,2, . . . , k.

We also define

(3.2) α_i =

k

X

j=1

A_ij, i= 1,2, . . . , k.

The main result of this paper is as follows:

Theorem 3.1. Letk ≥2be an integer andp_i, p⁰_i, q_i,i= 1,2, . . . , k, be real numbers satisfying (1.1) – (1.5). Further, letA_ij, i, j = 1,2, . . . , kbe real parameters defined by (3.1) and (3.2). Then the following inequalities hold and are equivalent:

(3.3) Z

(Rⁿ)^k

Qk

i=1fi(xi)

Pk i=1x_i

λsdx₁dx₂. . . dx_k

≤K

k

Y

i=1

Z

Rⁿ

|x_i|^pi

(k−1)n−pis qi +piαi

f_i^pⁱ(x_i)dx_i _pi¹

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and

(3.4) Z

Rⁿ

|xk|⁻^pk

0

qk[(k−1)n−s]−p_k⁰αk

·





 Z

(Rⁿ)^k−1

Qk−1 i=1 f_i(x_i)

Pk i=1x_i

λsdx₁dx₂. . . dxk−1







pk0

dx_k











1 pk0

≤K

k−1

Y

i=1

Z

Rⁿ

|x_i|^pi

, for any0 < s < n,A_ij ∈

−_qⁿ

i,0

, α_i −A_ii < ^s−(k−1)n_q

i ,where the constantK is given by the formula

K = 1

Γ^λ_n(s)

k

Y

i,j=1,i6=j

Γ_n(n+q_iA_ij)^qi¹

k

Y

i=1

Γ_n(s−(k−1)n−q_iα_i+q_iA_ii)^qi¹ . Proof. We start with the inequality (3.3). The left-hand side of the inequality (3.3) can easily be transformed in the following way

Z

(Rⁿ)^k

Qk

i=1fi(xi)

Pk i=1x_i

= Z

(Rⁿ)^k k

Y

i=1





|x_i|^pⁱ^AⁱⁱQk

j=1,j6=i|x_j|^qⁱ^A^ij

Pk j=1x_j

s F_i^pⁱ^−qⁱ(x_i)f_i^pⁱ(x_i)





1 qi

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·

" _k Y

i=1

|xi|^pⁱ^Aⁱⁱ(Fifi)^pⁱ(xi)

#^1−λ

dx1dx2. . . dxk, where

F_i(x_i) =



 Z

(Rⁿ)^k−1

Qk

j=1,j6=i|x_j|^qⁱ^A^ij

Pk j=1x_j

s dx₁dx₂. . . dxi−1dx_i+1. . . dx_n





1 qi

. Now by using Selberg’s integral formula (2.4) it follows easily that

(3.5) F_i(x_i) =

" Qk

j=1,j6=iΓn(n+qiAij)Γn(n−s) Γ_n(kn+q_iα_i−q_iA_ii−s)

#_qi¹

|x_i|

(k−1)n−s

qi +αi−A_ii

. Further, sincePk

i=1 1

qi + 1−λ = 1, q_i > 0and0 < λ≤ 1, we can apply Hölder’s inequality with conjugate parameters q₁, q₂, . . . , q_k and _1−λ¹ , on the above transfor- mation. In such a way, we obtain the inequality

Z

(Rⁿ)^k

Qk

i=1f_i(x_i)

Pk i=1x_i

≤

k

Y

i=1

Z

Rⁿ

|xi|^pⁱ^Aⁱⁱ(Fifi)^pⁱ(xi)dxi

_qi¹ ^k Y

i=1

Z

Rⁿ

|xi|^pⁱ^Aⁱⁱ(Fifi)^pⁱ(xi)dxi

1−λ

=

k

Y

i=1

Z

Rⁿ

|x_i|^pⁱ^Aⁱⁱ(F_if_i)^pⁱ(x_i)dx_i _pi¹

,

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since _q¹

i + 1−λ = _p¹

i. Finally, by using definition (3.5) of the functions F_i, i = 1,2, . . . , k, one obtains the inequality (3.3).

Let us show that the inequalities (3.3) and (3.4) are equivalent. Suppose that the inequality (3.3) is valid. If we put the functionf_n :Rⁿ7→R, defined by

f_k(x_k) =|x_k|⁻^pk

0

qk[(k−1)n−s]−p_k⁰α_k





 Z

(Rⁿ)^k−1

Qk−1 i=1 f_i(x_i)

Pk i=1x_i

λsdx₁dx₂. . . dxk−1







pk0 pk

in the inequality (3.3), we obtain I(x_k)^p⁰^k ≤K

k−1

Y

i=1

Z

Rⁿ

|x_i|^pi

I(x_k)

p0 k pk,

where I(x_k) denotes the left-hand side of the inequality (3.4). This gives the inequality (3.4).

It remains to prove that the inequality (3.3) is a consequence of the inequality (3.4). For this purpose, let us suppose that the inequality (3.4) is valid. Then the left-hand side of the inequality (3.3) can be transformed in the following way:

Z

(Rⁿ)^k

Qk

i=1f_i(x_i)

Pk i=1x_i

λsdx1dx2. . . dxk = Z

Rⁿ

|xk|

(k−1)n−s qk +αk

fk(xk)

·





|xk|⁻

(k−1)n−s qk −α_kZ

(Rⁿ)^k−1

Qk−1 i=1 f_i(x_i)

Pk i=1x_i

λsdx1dx2. . . dxk−1





dxk.

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Applying Hölder’s inequality with conjugate parametersp_kandp⁰_kto the above trans- formation, we have

Z

(Rⁿ)^k

Qk

i=1f_i(x_i)

Pk i=1x_i

≤ Z

Rⁿ

|x_k|^pk

(k−1)n−pks qk +pkαk

f_k^p^k(x_k)dx_k ¹

pk ·I(x_k), and the result follows from (3.4). Hence, we have shown that the inequalities (3.3) and (3.4) are equivalent. Since the first inequality is valid, the second one is also valid. This completes the proof.

Clearly, by puttingn= 1in Theorem3.1, we obtain inequalities which are similar to the inequalities in Theorem1.1. The integrals are taken over the whole set of real numbers, the weight functions are the same and the constant is of the same form as in Theorem1.1, where the ordinary gamma function is replaced withΓ₁(α).

Remark 1. Observe that equality in the inequality (3.3) holds if and only if it holds in Hölder’s inequality. By using the notation from Theorem 3.1, it means that the functions

|x_i|^pⁱ^Aⁱⁱ

k

Y

j=1,j6=i

|x_j|^qⁱ^A^ij

k

X

j=1

x_j

−s

F_i^pⁱ^−qⁱ(x_i)f_i^pⁱ(x_i), i= 1,2, . . . , k and

k

Y

i=1

|x_i|^pⁱ^Aⁱⁱ(F_if_i)^pⁱ(x_i)

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are effectively proportional. So, if we suppose that the functionsf_i, i = 1,2, . . . , k are not equal to zero, straightforward computation (see also [4, Remark 1]) leads to the condition

k

X

i=1

x_i

−s

=C

k

Y

i=1

|x_i|^{(k−1)n−s+q}ⁱ^(αⁱ^−Aⁱⁱ⁾,

whereCis an appropriate constant, and that is a contradiction. So equality in The- orem3.1 holds if and only if at least one of the functionsf_i is identically equal to zero. Otherwise, for non-negative and non-zero functions, the inequalities (3.3) and (3.4) are strict.

Remark 2. If the parametersp_i,i= 1,2, . . . , k are chosen in such a way that (3.6) q_j >0, for somej ∈ {1,2, . . . n}, q_i <0, i6=j and λ <1 or

(3.7) q_i <0, i= 1,2, . . . , n

then the exponents from the proof of Theorem3.1 fulfill the conditions for the reverse Hölder inequality (for details see e.g. [12, Chapter V]), which gives the reverse of the inequalities (3.3) and (3.4).

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4. The Best Possible Constants in the Conjugate Case

In this section we shall focus on the case of the conjugate exponent, to obtain the best possible constants in Theorem 3.1, for some general cases. It seems to be a difficult problem to obtain the best possible constant in the case of non-conjugate parameters.

It follows easily that the constantK from the previous theorem, in the conjugate case (λ= 1,p_i =q_i), takes the form

K = 1

Γ_n(s)

k

Y

i,j=1,i6=j

Γ_n(n+p_iA_ij)^pi¹

k

Y

i=1

Γ_n(s−(k−1)n−p_iα_i+p_iA_ii)^pi¹ . However, we shall deal with an appropriate form of the inequalities obtained in the previous section in the conjugate case. The main idea is to simplify the above constantK, i.e. to obtain the constant without exponents. For this sake, it is natural to consider real parametersAij satisfying the following constraint

(4.1) s−(k−1)n+piAii−piαi =n+pjAji, j 6=i, i, j ∈ {1,2, . . . , k}.

In this case, the above constantK takes the form

(4.2) K^∗ = 1

Γ_n(s)

k

Y

i=1

Γ_n(n+fA_i), where

(4.3) fA_i =p_jA_ji, j 6=i and −n <fA_i <0.

It is easy to see that the parametersfA_i satisfy the relation (4.4)

k

X

i=1

fAi =s−kn.

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Further, the inequalities (3.3) and (3.4) with the parametersA_ij, satisfying (4.1), become

(4.5) Z

(Rⁿ)^k

Qk

i=1fi(xi)

Pk i=1x_i

s dx₁dx₂. . . dx_k≤K^∗

k

Y

i=1

Z

Rⁿ

|x_i|^−n−pⁱ^A^fⁱf_i^pⁱ(x_i)dx_i _pi¹

and

(4.6)





 Z

Rⁿ

|x_k|^(1−p⁰^k^)(−n−p^k^A^f^k⁾·



 Z

(Rⁿ)^k−1

Qk−1 i=1 f_i(x_i)

Pk i=1x_i

s dx₁dx₂...dxk−1





pk0

dx_k







1 pk0

≤K^∗

k−1

Y

i=1

Z

Rⁿ

|x_i|^−n−pⁱ^A^fⁱf_i^pⁱ(x_i)dx_i _pi¹

. We shall see that the constant K^∗ in (4.5) and (4.6) is the best possible in the sense that we cannot replace the constantK^∗ in inequalities (4.5) and (4.6) with the smaller constant, so that inequalities are fulfilled for all non-negative measurable functions. Before we prove the facts we have to establish the following two lemmas:

Lemma 4.1. Letk ≥2be an integer,x_k ∈Rⁿ, andx_k6=0. We define I₁^ε(x_k) =

Z

Kⁿ(ε)

|x₁|^A^f¹



 Z

(Rⁿ)^k−2

Qk−1 i=2 |x_i|^f^Aⁱ

Pk i=1x_i

s dx₂. . . dxk−1



dx₁,

where ε > 0, Kⁿ(ε)is the closedn−dimensional ball of radiusε and parameters fA_i,i= 1,2, . . . , kare defined by (4.3). Then there exists a positive constantC_ksuch that

(4.7) I₁^ε(x_k)≤C_kε^n+fÂ¹|x_k|⁻²ⁿ⁻Â^f¹⁻Â^f^k, when ε →0.

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Proof. We treat two cases. Ifk = 2we have I₁^ε(x₂) =

Z

Kⁿ(ε)

|x₁|^A^f¹

|x₁+x₂|^sdx₁.

By lettingε→0, we easily conclude that there exists a positive constantc₂such that I₁^ε(x2)≤c2|x2|^−s

Z

Kⁿ(ε)

|x1|^A^f¹dx1.

The previous integral can be calculated by using n−dimensional spherical coordinates. More precisely, we have

Z

Kⁿ(ε)

|x₁|^A^f¹dx₁ (4.8)

= Z π

0

· · · Z π

0

Z 2π 0

Z ε 0

r^n+f^A¹⁻¹sinⁿ⁻²θn−1sinⁿ⁻³θn−2· · ·sinθ2drdθ1. . . dθn−1

= Z ε

0

rⁿ⁺^A^f¹⁻¹dr Z

Sn

dS = |S_n|ε^n+f^A¹ n+Af₁ ,

where|S_n|= 2πⁿ²Γ⁻¹(ⁿ₂)is the Lebesgue measure of the unit sphere inRⁿ.Conse- quently,

I₁^ε(x₂)≤ c₂|S_n|ε^n+f^A¹

n+Af₁ |x₂|^−s,

so the inequality holds whenε → 0, since−2n−Af₁−Af₂ =−sholds fork = 2.

Further, if k > 2, then by letting ε → 0, since |x₁| → 0, we easily conclude that

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there exists a positive constantc_ksuch that (4.9) I₁^ε(x_k)≤c_k

Z

Kⁿ(ε)

|x₁|^A^f¹dx₁



 Z

(Rⁿ)^k−2

Qk−1 i=2 |x_i|^A^fⁱ

Pk i=2x_i

s dx₂. . . dxk−1



. We have already calculated the first integral in the inequality (4.9), and the second one is the Selberg integral. Namely, by using the formulas (2.3), (2.4) and (4.4) we have

(4.10) Z

(Rⁿ)^k−2

Pk i=2x_i

s dx₂. . . dxk−1

= Γ_n(2n+Af₁+Af_k)Qk−1

i=2 Γ_n(n+Af_i)

Γ_n(s) |x_k|⁻²ⁿ⁻^A^f¹⁻^A^f^k. Finally, by using (4.8), (4.9) and (4.10), we obtain the inequality (4.7) and the proof is completed.

Similarly, we have

Lemma 4.2. Letk ≥2be an integer andx_k∈Rⁿ. We define I₁^ε⁻¹(xk) =

Z

Rⁿ\Kⁿ(ε⁻¹)

|x1|^A^f¹



 Z

(Rⁿ)^k−2

Pk i=1xi

s dx2. . . dxk−1



dx1, where ε > 0 and parametersAf_i, i = 1,2, . . . , k are defined by (4.3). Then there exists a positive constantD_ksuch that

(4.11) I₁^ε⁻¹(x_k)≤D_kεⁿ⁺^A^f^k, when ε →0.

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Proof. We treat again two cases. Ifk= 2we have I₁^ε⁻¹(x₂) =

Z

Rⁿ\Kⁿ(ε⁻¹)

|x₁|^A^f¹

|x₁+x₂|^sdx₁.

Ifε →0, then|x₁| → ∞, so we easily conclude that there exists a positive constant d₂ such that

I₁^ε⁻¹(x₂)≤d₂ Z

Rⁿ\Kⁿ(ε⁻¹)

|x₁|^A^f¹^−sdx₁,

and by using spherical coordinates for calculating the integral on the right-hand side of the previous inequality, we obtain

I₁^ε⁻¹(x₂)≤ d₂|S_n|

n+Af₂εⁿ⁺^A^f². Further, ifk > 2, then by using (2.3), (2.4) and (4.4), we have (4.12)

Z

(Rⁿ)^k−2

Pk i=1x_i

s dx₂. . . dxk−1

= Γ_n(2n+Af₁ +Af_k)Qk−1

i=2 Γ_n(n+fA_i)

Γ_n(s) |x₁ +x_k|⁻²ⁿ⁻^A^f¹⁻^A^f^k. So, we get

(4.13) I₁^ε⁻¹(x_k) = Γ_n(2n+Af₁+Af_k)Qk−1

i=2 Γ_n(n+fA_i) Γn(s)

· Z

Rⁿ\Kⁿ(ε⁻¹)

|x₁|Â^f¹|x₁+x_k|⁻²ⁿ⁻Â^f¹⁻Â^f^kdx₁.

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By lettingε→0, then|x₁| → ∞, so there exists a positive constantd_k such that I₁^ε⁻¹(x_k)≤d_k

Z

Rⁿ\Kⁿ(ε⁻¹)

|x₁|⁻²ⁿ⁻^A^f^kdx₁. Since,

Z

Rⁿ\Kⁿ(ε⁻¹)

|x₁|⁻²ⁿ⁻^A^f^kdx₁ = |S_n|εⁿ⁺^A^f^k n+Af_k , the inequality (4.11) holds.

Now, we are able to obtain the main result, i.e. the best possible constants in the inequalities (4.5) and (4.6). Clearly, inequalities (4.5) and (4.6) do not contain parametersA_ij, i, j = 1,2, . . . , k, so we can regard these inequalities withAe_i, i = 1,2, . . . , k, as primitive parameters. More precisely, we have

Theorem 4.3. SupposeAe_i, i= 1,2, . . . , k, are real parameters fulfilling constraint (4.4) and−n <fAi < 0, i = 1,2, . . . , k. Then, the constantK^∗ is the best possible in both inequalities (4.5) and (4.6).

Proof. Let us denote by Kⁿ(ε)the closed n−dimensional ball of radius ε with the center in0.

Let 0 < ε < 1. We define the functionsfe_i : Rⁿ 7→ R, i = 1,2, . . . , k in the following way

fe_i(x_i) =

( |x_i|^A^fⁱ, x_i ∈Kⁿ(ε⁻¹)\Kⁿ(ε), 0, otherwise.

If we put defined functions in the inequality (4.5), then the right-hand side of the

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inequality (4.5) becomes K^∗

k

Y

i=1

Z

Kⁿ(ε⁻¹)\Kⁿ(ε)

|x_i|⁻ⁿdx_i _pi¹

=K^∗ Z

|x_i|⁻ⁿdx_i. By usingn−dimensional spherical coordinates we obtain for the above integral

Z

|x_i|⁻ⁿdx_i = Z ε⁻¹

ε

r⁻¹dr Z

Sn

dS =|S_n|ln 1 ε²,

where|S_n|= 2πⁿ²Γ⁻¹(ⁿ₂)is the Lebesgue measure of the unit sphere inRⁿ.So for the above choice of functionsf_ithe right-hand side of the inequality (4.5) becomes

(4.14) K^∗|S_n|ln 1

ε².

Now letJ denote the left-hand side of the inequality (4.5). By using Fubini’s theorem, for the above choice of functionsf_i, we have

J = Z

(Kⁿ(ε⁻¹)\Kⁿ(ε))^k

Qk

i=1|xi|^A^fⁱ

Pk i=1x_i

sdx₁dx₂. . . dx_k

= Z

|x_k|^A^f^k

·



 Z

(Kⁿ(ε⁻¹)\Kⁿ(ε))^k−1

Qk−1 i=1 |xi|^A^fⁱ

Pk i=1x_i

s dx₁dx₂. . . dxk−1



dx_k. Note that the integralJ can be transformed in the following way:J =J₁−J₂−J₃,

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where J1 =

Z

|xk|^A^f^k



 Z

(Rⁿ)^k−1

Pk i=1x_i

s dx1dx2. . . dxk−1



dxk,

J₂ = Z

|x_k|^A^f^k

k−1

X

j=1

I_j^ε(x_k)dx_k,

J₃ = Z

|x_k|^A^f^k

k−1

X

j=1

I_j^ε⁻¹(x_k)dx_k.

Here, forj = 1,2, . . . , k−1, the integralsI_j^ε(x_k)andI_j^ε⁻¹(x_k)are defined by I_j^ε(xk) =

Z

Pj

Pk i=1x_i

s dx1dx2. . . dxk−1,

satisfyingP_j ={(U₁, U₂, . . . , Uk−1) ;U_j =Kⁿ(ε), U_l=Rⁿ, l 6=j},and I_j^ε⁻¹(x_k) =

Z

Qj

Pk i=1x_i

s dx₁dx₂. . . dxk−1,

satisfyingQ_j ={(U₁, U₂, . . . , Uk−1) ;U_j =Rⁿ\Kⁿ(ε⁻¹), U_l =Rⁿ, l 6=j}.

Now, the main idea is to find the lower bound forJ. The first partJ₁can easily be computed. Namely by using Selberg’s integral formula (2.4) and since the relation (4.4) holds for parametersfAi, it easily follows that

Z

(Rⁿ)^k−1

Pk i=1x_i

s dx₁dx₂. . . dx_k−1 =K^∗|x_k|⁻^A^f^k⁻ⁿ,