Multidimensional Hilbert-Type Inequalities with a Homogeneous Kernel

Abstract

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.

1. Introduction

Let

(1.1)

The well-known Hardy-Hilbert's integral inequality (see [1]) is given by

(1.2)

and an equivalent form is given by

(1.3)

where the constant factors and are 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 Yang (see [2]). If satisfy

(1.4)

then

(1.5)

where the constant factor is the best possible.

Our generalization will include a general homogeneous kernel , where , with being nonconjugate parameters. The techniques that will be used in the proofs are mainly based on classical real analysis, especially on the well-known Hölder'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 exponents (see [3]). Let and be real parameters, such that

(1.6)

and let and , respectively, be their conjugate exponents, that is, and . Further, define

(1.7)

and note that for all and values as in (1.6). In particular, holds if and only if , that is, only when and are mutually conjugate. Otherwise, , and in such cases and will be referred to as nonconjugate exponents.

Considering , , and as in (1.6) and (1.7), Hardy et al. [1], proved that there exists a constant , dependent only on the parameters and , such that the following Hilbert-type inequality holds for all nonnegative functions and :

(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:

(1.9)

and let be an area of unit sphere in in view of norm.

2. Main Results

Before presenting our idea and results, we repeat the notion of general nonconjugate exponents from [3]. Let be the real parameters which satisfy

(2.1)

Further, the parameters , are defined by the equations

(2.2)

Since , , it is obvious that , . We define

(2.3)

It is easy to deduce that . Also, we introduce the parameters , , defined by the relations

(2.4)

In order to obtain our results we need to require

(2.5)

It is easy to see that the above conditions do not automatically apply (2.5). Further, it follows

(2.6)

Of course, if , then ; so the conditions (2.1)–(2.4) reduce to the case of conjugate parameters.

Results in this section will be based on the following general form of Hardy-Hilbert's inequality proven in [4]. All the measures are assumed to be -finite on some measure space.

Theorem 2.1.

Let and , be real numbers satisfying (2.1)–(2.5). Let and , , be nonnegative measurable functions such that . Then, for any nonnegative measurable functions , , the following inequalities hold and are equivalent:

(2.7)

(2.8)

where

(2.9)

In the same paper the authors discussed the case of equality in inequalities (2.7) and (2.8). They proved that the equality holds in (2.7) (and analogously in (2.8)) if and only if

(2.10)

In the following theorem we give the most important case where , the measures are Lebesgue measures, is a nonnegative homogeneous function of degree , and the functions represent the form where , . In order to obtain the generalizations of some known results we define

(2.11)

where we suppose that for and .

Due to technical reasons, we introduce real parameters satisfying

(2.12)

We also define

(2.13)

Theorem 2.2.

Let and , be real numbers satisfying (2.1)–(2.5). Let be nonnegative measurable homogeneous function of degree , , and let and be real parameters satisfying (2.12) and (2.13). If , , are nonnegative measurable functions, then the following inequalities hold and are equivalent:

(2.14)

where

(2.15)

and .

Proof.

Set and in Theorem 2.1, where for every . It is enough to calculate the functions , . By using the -dimensional spherical coordinates we find

(2.16)

Using homogeneity of the function and the substitutions we have

(2.17)

Similarly, by applying the -dimensional spherical coordinates and homogeneity of the function we have

(2.18)

Using the change of variables

(2.19)

where denotes the Jacobian of the transformation, we have

(2.20)

In a similar manner we obtain

(2.21)

for . This gives inequalities (2.14) with inequality sign . Condition (2.10) immediately gives that nontrivial case of equality in (2.14) leads to the divergent integrals. This completes the proof.

Remark 2.3.

Note that the kernel is a homogeneous function of degree In this case we have

(2.22)

where we used the well-known formula for gamma function (see, e.g., [5, Lemma ]). Now, by using Theorem 2.2 and (2.22) we obtain the result of Krnić et al. (see [6]).

3. The Best Possible Constants in the Conjugate Case

In this section we consider the inequalities in Theorem 2.2. In such a way we shall obtain the best possible constants for some general cases.

It follows easily that Theorem 2.2 in the conjugate case () becomes as follows.

Theorem 3.1.

Let and let be conjugate parameters such that Let be nonnegative measurable homogeneous function of degree , , and let and be real parameters satisfying (2.12) and (2.13). If , , are nonnegative measurable functions, then the following inequalities hold and are equivalent:

(3.1)

where

(3.2)

and .

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

(3.3)

In that case the constant from Theorem 3.1 is simplified to the following form:

(3.4)

where

(3.5)

Further, by using (3.4) and (3.5), the inequalities (3.1) with the parameters satisfying the relation (3.3), become

(3.6)

(3.7)

Theorem 3.2.

Suppose that the real parameters satisfy conditions in Theorem 3.1 and conditions given in (3.3). If the kernel is as in Theorem 3.1 and for every

(3.8)

for some , then the constant is the best possible in inequalities (3.6) and (3.7).

Proof.

Let us suppose that the constant factor given by (3.4) is not the best possible in the inequality (3.6). Then, there exists a positive constant , such that (3.6) is still valid when we replace by .

We define the real functions by the formulas

(3.9)

where . Now, we shall put these functions in inequality (3.6). By using the -dimensional spherical coordinates, the right-hand side of the inequality (3.6) becomes

(3.10)

Further, let denotes the left-hand side of the inequality (3.6), for the above choice of the functions By applying the -dimensional spherical coordinates and the substitutions we find

(3.11)

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

(3.12)

where for is defined by

(3.13)

satisfying Without losing generality, we only estimate the integral For we have

(3.14)

and for we find

(3.15)

where is well defined since obviously . Hence, we have for and consequently

(3.16)

We conclude, by using (3.10), (3.12), and (3.16), that which is an obvious contradiction. It follows that the constant in (3.6) is the best possible.

Finally, the equivalence of the inequalities (3.6) and (3.7) means that the constant is also the best possible in the inequality (3.7). That completes the proof.

Remark 3.3.

If we put and in the inequalities (3.6) and (3.7) applying Theorem 3.2, we obtain the result of Baoju Sun (see [7]). Further, by putting in Theorems 3.1 and 3.2 we obtain appropriate results from [8]. More precisely, the inequality (3.6) becomes

(3.17)

If the kernel and the parameters satisfy the conditions from Theorem 3.2, then the constant is the best possible. For example, setting in the inequality (3.17), we obtain Yang's result (1.5) from introduction.