# On a Multiple Hilbert-Type Integral Operator and Applications

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

where Open image in new window is the Beta function ( Open image in new window , Open image in new window

If Open image in new window , Open image in new window Open image in new window then we have the following inequality:

where the constant factor Open image in new window is the best possible. For Open image in new window , and Open image in new window in (1.6), we obtain (1.3) and (1.4). Inequality (1.6) is some extensions of the results in [6, 8, 9, 10, 11]. In 2006, reference [12] also considered a multiple Hilbert-type integral operator with the homogeneous kernel of Open image in new window -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 Open image in new window is a measurable function in Open image in new window such that for any Open image in new window and Open image in new window then call Open image in new window the homogeneous function of Open image in new window -degree in Open image in new window

Lemma 2.3.

Proof.

Setting Open image in new window in the above integral, we obtain Open image in new window Setting Open image in new window in (2.4), we find that Open image in new window

Lemma 2.4.

then there exist Open image in new window and Open image in new window Open image in new window such that for any Open image in new window Open image in new window if and only if Open image in new window is continuous at Open image in new window

Proof.

By mathematical induction, we prove that for Open image in new window Open image in new window is continuous at Open image in new window

Lemma 2.5.

Proof.

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

Lemma 2.6.

( Open image in new window ) for Open image in new window the reverse of (2.18) is obtained.

Proof.

Then by (2.4), we have (2.18) (note that for Open image in new window we do not use H Open image in new window lder's inequality again). ( Open image in new window ) For Open image in new window by the reverse H Open image in new window 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 Open image in new window Open image in new window then we find that Open image in new window If Open image in new window Open image in new window then define the following real function spaces:

and a multiple Hilbert-type integral operator Open image in new window as follows: for Open image in new window

Then by (2.18), it follows that Open image in new window , Open image in new window is bounded, Open image in new window , and Open image in new window where

Define the formal inner product of Open image in new window and Open image in new window as

Theorem 3.1.

where the constant factor Open image in new window is the best possible; ( Open image in new window ) for Open image in new window using the formal symbols of the case in Open image in new window 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 Open image in new window Hence Open image in new window is the best value of (3.7). We conform that the constant factor Open image in new window 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 Open image in new window

( Open image in new window ) For Open image in new window by using the reverse H Open image in new window 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 Open image in new window , we can show that Open image in new window

Example 3.3.

Hence for Open image in new window , we can show that Open image in new window

Example 3.4.

Then by mathematical induction, (3.17) is valid for Open image in new window

Hence for Open image in new window Open image in new window ), we can show that Open image in new window

## Notes

### 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).

## References

- 1.Hardy GH, Littlewood JE, Pólya G:
*Inequalities*. Cambridge University Press, Cambridge, UK; 1934.MATHGoogle Scholar - 2.Carleman T:
*Sur les Equations Integrals Singulieres a Noyau Reed et Symetrique*.*Volume 923*. Uppsala universitet, Uppsala, Sweden;Google Scholar - 3.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.CrossRefMATHGoogle Scholar - 4.Zhang K:
**A bilinear inequality.***Journal of Mathematical Analysis and Applications*2002,**271**(1):288–296. 10.1016/S0022-247X(02)00104-XMathSciNetCrossRefMATHGoogle Scholar - 5.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 - 6.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 - 7.Yang B:
*The Norm of Operator and Hilbert-Type Inequalities*. Science Press, Beijin, China; 2009.Google Scholar - 8.Hong Y:
**All-sided generalization about Hardy-Hilbert integral inequalities.***Acta Mathematica Sinica*2001,**44**(4):619–625.MathSciNetMATHGoogle Scholar - 9.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 - 10.Yang B:
**A multiple Hardy-Hilbert integral inequality.***Chinese Annals of Mathematics. Series A*2003,**24**(6):25–30.MathSciNetGoogle Scholar - 11.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.MathSciNetCrossRefMATHGoogle Scholar - 12.Bényi Á, Choonghong O:
**Best constants for certain multilinear integral operators.***Journal of Inequalities and Applications*2006,**2006:**-12.Google Scholar - 13.Kuang J:
*Introduction to Real Analysis*. Hunan Education Press, Chansha, China; 1996.Google Scholar - 14.Kuang J:
*Applied Inequalities*. Shangdong Science Technic Press, Jinan, China; 2004.Google Scholar

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