- Research Article
Multidimensional Hilbert-Type Inequalities with a Homogeneous Kernel
Journal of Inequalities and Applicationsvolume 2009, Article number: 130958 (2009)
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.
The well-known Hardy-Hilbert's integral inequality (see ) is given by
and an equivalent form is given by
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 ). If satisfy
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 ). Let and be real parameters, such that
and let and , respectively, be their conjugate exponents, that is, and . Further, define
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. , 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 :
Throughout this paper we suppose that all the functions are nonnegative and measurable, so that all integrals converge. We also introduce the following notations:
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 . Let be the real parameters which satisfy
Further, the parameters , are defined by the equations
Since , , it is obvious that , . We define
It is easy to deduce that . Also, we introduce the parameters , , defined by the relations
In order to obtain our results we need to require
It is easy to see that the above conditions do not automatically apply (2.5). Further, it follows
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 . All the measures are assumed to be -finite on some measure space.
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:
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
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
where we suppose that for and .
Due to technical reasons, we introduce real parameters satisfying
We also define
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:
Set and in Theorem 2.1, where for every . It is enough to calculate the functions , . By using the -dimensional spherical coordinates we find
Using homogeneity of the function and the substitutions we have
Similarly, by applying the -dimensional spherical coordinates and homogeneity of the function we have
Using the change of variables
where denotes the Jacobian of the transformation, we have
In a similar manner we obtain
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.
Note that the kernel is a homogeneous function of degree In this case we have
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.
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:
To obtain a case of the best inequality it is natural to impose the following conditions on the parameters :
In that case the constant from Theorem 3.1 is simplified to the following form:
Further, by using (3.4) and (3.5), the inequalities (3.1) with the parameters satisfying the relation (3.3), become
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
for some , then the constant is the best possible in inequalities (3.6) and (3.7).
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
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
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
Now, it is easy to see that the following inequality holds:
where for is defined by
satisfying Without losing generality, we only estimate the integral For we have
and for we find
where is well defined since obviously . Hence, we have for and consequently
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.
If we put and in the inequalities (3.6) and (3.7) applying Theorem 3.2, we obtain the result of Baoju Sun (see ). Further, by putting in Theorems 3.1 and 3.2 we obtain appropriate results from . More precisely, the inequality (3.6) becomes
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.
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This research is supported by the Croatian Ministry of Science, Education and Sports, Grant no. 058-1170889-1050.