# On a Hilbert-Type Operator with a Class of Homogeneous Kernels

- Bicheng Yang
^{1}Email author

**2009**:572176

**DOI: **10.1155/2009/572176

© Bicheng Yang 2009

**Received: **15 September 2008

**Accepted: **20 February 2009

**Published: **4 March 2009

## Abstract

By using the way of weight coefficient and the theory of operators, we define a Hilbert-type operator with a class of homogeneous kernels and obtain its norm. As applications, an extended basic theorem on Hilbert-type inequalities with the decreasing homogeneous kernels of -degree is established, and some particular cases are considered.

## 1. Introduction

where the constant factor is the best possible. We named (1.2) Hardy-Hilbert's inequality. In 1934, Hardy et al. [3] gave some applications of (1.1)-(1.2) and a basic theorem with the general kernel (see [3, Theorem 318]).

Theorem 1.

where the constant factors and are the best possible.

Note.

Hardy did not prove this theorem in [3]. In particular, we find some classical Hilbert-type inequalities as,

(i)for in (1.3), it reduces (1.2),

In 1951, Bonsall [4] considered (1.7) in the case of general kernel; in 1991, Mitrinović et al. [5] summarized the above results.

where the constant factor is the best possible.

where the constant factor is the best possible; Krnić and Pečarić [10] also considered (1.11) in the general homogeneous kernel, but the best possible property of the constant factor was not proved by [10].

Note.

For in [10, inequality (37)], it reduces to the equivalent result of (3.1) in this paper.

In 2006-2007, some authors also studied the operator expressing of (1.3) and (1.4).

where the constant factor is the best possible. In particular, for being -degree homogeneous, inequalities (1.15) reduce to (1.3)-(1.4) (in the symmetric kernel). Yang [12] also considered (1.15) in the real space .

In this paper, by using the way of weight coefficient and the theory of operators, we define a new Hilbert-type operator and obtain its norm. As applications, an extended basic theorem on Hilbert-type inequalities with the decreasing homogeneous kernel of -degree is established; some particular cases are considered.

## 2. On a New Hilbert-Type Operator and the Norm

If is a measurable function, satisfying for then we call the homogeneous function of -degree.

For setting we find Hence, the following two words are equivalent: (a) is decreasing in and strictly decreasing in a subinterval of ; (b) for any , is decreasing in and strictly decreasing in a subinterval of . The following two words are also equivalent: is decreasing in and strictly decreasing in a subinterval of ; for any , is decreasing in and strictly decreasing in a subinterval of .

Lemma 2.1.

Proof.

Lemma 2.2.

then .

- (i)
Setting by the assumption, we obtain (ii) Setting and in the integrals and respectively, in view of (i), we still find that

For we set and Define the real space as and then we may also define the spaces and

Lemma 2.3.

As the assumption of Lemma 2.2, for setting , if and are decreasing in and strictly decreasing in a subinterval of , then

Proof.

Therefore, .

In view of Lemma 2.3, and then exists. If there exists such that for any then is bounded and Hence by (2.4), we find and is bounded.

Theorem 2.4.

As the assumption of Lemma 2.3, it follows

Proof.

Hence is the best value of (2.7). We conform that is the best value of (2.4). Otherwise, we can get a contradiction by (2.6) that the constant factor in (2.7) is not the best possible. It follows that

## 3. An Extended Basic Theorem on Hilbert-Type Inequalities

Still setting , and we have the following theorem.

Theorem 3.1.

where the constant factors and are the best possible.

Proof.

In view of (2.7) and (2.4), we have (3.1) and (3.2). Based on Theorem 2.4, it follows that the constant factors in (3.1) and (3.2) are the best possible.

and we have (3.2). Hence (3.1) and (3.2) are equivalent.

- (a)
For (3.1) and (3.2) reduce, respectively, to (1.6) and (1.7). Hence, Theorem 3.1 is an extension of Theorem A.

- (b)
Replacing the condition " and are decreasing in and strictly decreasing in a subinterval of " by "for and are decreasing in and strictly decreasing in a subinterval of ," the theorem is still valid. Then in particular,

and then it deduces to (1.11);

## Authors’ Affiliations

## References

- Weyl H:
*Singulare Integralgleichungen mit besonderer Beriicksichtigung des Fourierschen Integraltheorems, Inaugeral dissertation*. University of Göttingen, Göttingen, Germany; 1908.Google Scholar - Hardy GH:
**Note on a theorem of Hilbert concerning series of positive terms.***Proceedings of the London Mathematical Society*1925,**23**(2):45–46.Google Scholar - Hardy GH, Littlewood JE, Pólya G:
*Inequalities*. Cambridge University Press, Cambridge, UK; 1934.Google Scholar - Bonsall FF:
**Inequalities with non-conjugate parameters.***The Quarterly Journal of Mathematics*1951,**2**(1):135–150. 10.1093/qmath/2.1.135MathSciNetView ArticleMATHGoogle Scholar - Mitrinović DS, Pečarić JE, Fink AM:
*Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications (East European Series)*.*Volume 53*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+587.View ArticleGoogle Scholar - Yang B:
**A generalization of the Hilbert double series theorem.***Journal of Nanjing University Mathematical Biquarterly*2001,**18**(1):145–152.MATHGoogle Scholar - Yang B:
**An extension of Hardy-Hilbert's inequality.***Chinese Annals of Mathematics. Series A*2002,**23**(2):247–254.MathSciNetMATHGoogle Scholar - Yang B:
**On new extensions of Hilbert's inequality.***Acta Mathematica Hungarica*2004,**104**(4):291–299.MathSciNetView ArticleMATHGoogle Scholar - Yang B:
**On best extensions of Hardy-Hilbert's inequality with two parameters.***Journal of Inequalities in Pure and Applied Mathematics*2005,**6**(3, article 81):1–15.Google Scholar - Krnić M, Pečarić J:
**General Hilbert's and Hardy's inequalities.***Mathematical Inequalities & Applications*2005,**8**(1):28–51.MATHGoogle Scholar - Yang B:
**On the norm of a Hilbert's type linear operator and applications.***Journal of Mathematical Analysis and Applications*2007,**325**(1):529–541. 10.1016/j.jmaa.2006.02.006MathSciNetView ArticleMATHGoogle Scholar - Yang B:
**On the norm of a self-adjoint operator and applications to the Hilbert's type inequalities.***Bulletin of the Belgian Mathematical Society*2006,**13**(4):577–584.MathSciNetMATHGoogle Scholar - Kuang J:
*Applied Inequalities*. Shandong Science and Technology Press, Jinan, China; 2004.Google Scholar - Kuang J:
*Introduction to Real Analysis*. Hunan Education Press, Changsha, China; 1996.Google Scholar

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