- Research Article
- Open Access

# On a Multiple Hilbert-Type Integral Operator and Applications

- Qiliang Huang
^{1}Email author and - Bicheng Yang
^{1}

**2009**:192197

https://doi.org/10.1155/2009/192197

© Q. Huang and B. Yang. 2010

**Received:**19 August 2009**Accepted:**9 December 2009**Published:**12 January 2010

## Abstract

By using the way of weight functions and the technic of real analysis, a multiple Hilbert-type integral operator with the homogeneous kernel of -degree ( ) and its norm are considered. As for applications, two equivalent inequalities with the best constant factors, the reverses, and some particular norms are obtained.

## Keywords

- Weight Function
- Measurable Function
- Integral Operator
- Convergence Theorem
- Equivalent Form

## 1. Introduction

where the constant factor is the best possible. Define the Hardy-Hilbert's integral operator as follows: for Then in view of (1.2), it follows that and Since the constant factor in (1.2) is the best possible, we find that (cf.[2])

where is the Beta function ( ,

If , then we have the following inequality:

where the constant factor is the best possible. For , and in (1.6), we obtain (1.3) and (1.4). Inequality (1.6) is some extensions of the results in [6, 8–11]. In 2006, reference [12] also considered a multiple Hilbert-type integral operator with the homogeneous kernel of -degree and its inequality with the norm, which is the best extension of (1.2).

In this paper, by using the way of weight functions and the technic of real analysis, a new multiple Hilbert-type integral operator with the norm is considered, which is an extension of the result in [12]. As for applications, an extended multiple Hilbert-type integral inequality and the equivalent form, the reverses, and some particular norms are obtained.

## 2. Some Lemmas

Lemma 2.1.

Proof.

and then (2.1) is valid.

Definition 2.2.

If is a measurable function in such that for any and then call the homogeneous function of -degree in

Lemma 2.3.

Proof.

Setting in the above integral, we obtain Setting in (2.4), we find that

Lemma 2.4.

then there exist and such that for any if and only if is continuous at

Proof.

By mathematical induction, we prove that for is continuous at

Lemma 2.5.

Proof.

Then by combination with (2.15), we have (2.10).

Lemma 2.6.

( ) for the reverse of (2.18) is obtained.

Proof.

Then by (2.4), we have (2.18) (note that for we do not use H lder's inequality again). ( ) For by the reverse H lder's inequality and the same way, we obtain the reverses of (2.18).

## 3. Main Results and Applications

As for the assumption of Lemma 2.6, setting then we find that If then define the following real function spaces:

and a multiple Hilbert-type integral operator as follows: for

Then by (2.18), it follows that , is bounded, , and where

Define the formal inner product of and as

Theorem 3.1.

where the constant factor is the best possible; ( ) for using the formal symbols of the case in the equivalent reverses of (3.6) and (3.7) with the best constant factor are given.

Proof.

and then (3.6) is valid, which is equivalent to (3.7).

and Hence is the best value of (3.7). We conform that the constant factor in (3.6) is the best possible; otherwise, we can get a contradiction by (3.9) that the constant factor in (3.7) is not the best possible. Therefore

( ) For by using the reverse H lder's inequality and the same way, we have the equivalent reverses of (3.6) and (3.7) with the same best constant factor.

Example 3.2.

It is obvious that (3.13) and (3.14) are equivalent in which the constant factors are all the best possible. Hence for , we can show that

Example 3.3.

Hence for , we can show that

Example 3.4.

Then by mathematical induction, (3.17) is valid for

Hence for ), we can show that

## Declarations

### Acknowledgments

This work is supported by the Emphases Natural Science Foundation of Guangdong Institution, Higher Learning, College and University (no. 05Z026), and Guangdong Natural Science Foundation (no. 7004344).

## Authors’ Affiliations

## References

- Hardy GH, Littlewood JE, Pólya G:
*Inequalities*. Cambridge University Press, Cambridge, UK; 1934.MATHGoogle Scholar - Carleman T: Sur les Equations Integrals Singulieres a Noyau Reed et Symetrique. Volume 923. Uppsala universitet, Uppsala, Sweden;Google Scholar
- Mitrinović DS, Pečarić JE, Fink AM:
*Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications*.*Volume 53*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+587.View ArticleMATHGoogle Scholar - Zhang K:
**A bilinear inequality.***Journal of Mathematical Analysis and Applications*2002,**271**(1):288–296. 10.1016/S0022-247X(02)00104-XMathSciNetView ArticleMATHGoogle Scholar - Yang B:
**On a extansion of Hilbert's integral inequality with some parameters.***The Australian Journal of Mathematical Analysis and Applications*2004,**1**(1, article 11):1–8.Google Scholar - Yang B, Brnetić I, Krnić M, Pečarić J:
**Generalization of Hilbert and Hardy-Hilbert integral inequalities.***Mathematical Inequalities & Applications*2005,**8**(2):259–272.MathSciNetMATHGoogle Scholar - Yang B:
*The Norm of Operator and Hilbert-Type Inequalities*. Science Press, Beijin, China; 2009.Google Scholar - Hong Y:
**All-sided generalization about Hardy-Hilbert integral inequalities.***Acta Mathematica Sinica*2001,**44**(4):619–625.MathSciNetMATHGoogle Scholar - He L, Yu J, Gao M:
**An extension of the Hilbert's integral inequality.***Journal of Shaoguan University ( Natural Science)*2002,**23**(3):25–30.Google Scholar - Yang B:
**A multiple Hardy-Hilbert integral inequality.***Chinese Annals of Mathematics. Series A*2003,**24**(6):25–30.MathSciNetGoogle Scholar - Yang B, Rassias ThM:
**On the way of weight coefficient and research for the Hilbert-type inequalities.***Mathematical Inequalities & Applications*2003,**6**(4):625–658.MathSciNetView ArticleMATHGoogle Scholar - Bényi Á, Choonghong O:
**Best constants for certain multilinear integral operators.***Journal of Inequalities and Applications*2006,**2006:**-12.Google Scholar - Kuang J:
*Introduction to Real Analysis*. Hunan Education Press, Chansha, China; 1996.Google Scholar - Kuang J:
*Applied Inequalities*. Shangdong Science Technic Press, Jinan, China; 2004.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.