Multidimensional Hilbert-Type Inequalities with a Homogeneous Kernel
© Predrag Vuković. 2009
Received: 11 July 2009
Accepted: 18 November 2009
Published: 24 November 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
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
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 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 :
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
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
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.
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
We also define
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.
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.
This research is supported by the Croatian Ministry of Science, Education and Sports, Grant no. 058-1170889-1050.
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