- Research Article
- Open Access

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

- Bicheng Yang
^{1}Email author

**2009**:572176

https://doi.org/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.

## Keywords

- Constant Factor
- Beta Function
- Homogeneous Function
- Dual Form
- Real Sequence

## 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

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## Copyright

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