On a Multiple Hilbert-Type Integral Operator and Applications

Open Access
Research Article

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

If  Open image in new window , Open image in new window  =  Open image in new window then we have the following famous Hardy-Hilbert's integral inequality and its equivalent form (cf. [1]):

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

Inequalities (1.1) and (1.2) are important in analysis and its applications (cf. [3]). In 2002, reference [4] considered the property of Hardy-Hilbert's integral operator and gave an improvement of (1.1) (for Open image in new window ). In 2004-2005, introducing another pair of conjugate exponents Open image in new window and an independent parameter Open image in new window [5, 6] gave two best extensions of (1.1) as follows:

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

). In 2009, [7, Theorem Open image in new window ] gave the following multiple Hilbert-type integral inequality: suppose that Open image in new window Open image in new window is a measurable function of Open image in new window ree in Open image in new window and for any Open image in new window satisfies Open image in new window and

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.

We find that

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.

As for the assumption of Lemma 2.1, if Open image in new window is a homogeneous function of Open image in new window degree in Open image in new window

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.

As for the assumption of Lemma 2.3, setting

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.

The sufficiency property is obvious. We prove the necessary property of the condition by mathematical induction. For Open image in new window , since
and Open image in new window then by Lebesgue control convergence theorem (cf. [13]), it follows that Open image in new window Assuming that for Open image in new window is continuous at Open image in new window then for Open image in new window in view of the result for Open image in new window we have that
then by the assumption for Open image in new window it follows that

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.

As for the assumption of Lemma 2.4, if Open image in new window , then for Open image in new window

Proof.

then by (2.11), it follows that
Without loses of generality, we estimate that Open image in new window In fact, setting Open image in new window such that Open image in new window since Open image in new window there exists Open image in new window such that Open image in new window Open image in new window and then by Fubini theorem, it follows that
Hence by (2.13), we have that
By Lemma 2.4, we find that

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.

( Open image in new window ) For Open image in new window by H Open image in new window lder's inequality (cf. [14] ) and (2.4), it follows that
For Open image in new window by H Open image in new window lder's inequality again, it follows that

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.

( Open image in new window ) For Open image in new window if (2.18) takes the form of equality, then for Open image in new window in (2.21), there exist constants Open image in new window and Open image in new window such that they are not all zero and
viz. Open image in new window in Open image in new window Assuming that Open image in new window then Open image in new window which contradicts Open image in new window . (Note that for Open image in new window we only consider (2.19) for Open image in new window in the above). Hence we have (3.6). By H Open image in new window lder's inequality, it follows that
and then by (3.6), we have (3.7). Assuming that (3.7) is valid, setting
then Open image in new window By (2.18), it follows that Open image in new window If Open image in new window then (3.6) is naturally valid. Assuming that Open image in new window by (3.7), it follows that

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

For Open image in new window small enough, setting Open image in new window as follows: Open image in new window Open image in new window , if there exists Open image in new window , such that (3.7) is still valid as we replace Open image in new window by Open image in new window then in particular, by Lemma 2.5, we have that

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.

For Open image in new window we obtain Open image in new window (cf. [7, Open image in new window ]. By Theorem 3.1, it follows that Open image in new window , and then by (3.7), we find that

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.

For Open image in new window we obtain Open image in new window (cf. [7, Open image in new window ]. By Theorem 3.1, it follows that Open image in new window , and then by (3.7), we find that

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

Example 3.4.

For Open image in new window Open image in new window by mathematical induction, we can show that
Assuming that for Open image in new window (3.17) is valid, then for Open image in new window , it follows that

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

By Theorem 3.1, it follows that Open image in new window , and by (3.7), we find that

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).

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Copyright information

© Q. Huang and B. Yang. 2010

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.

Authors and Affiliations

  1. 1.Department of MathematicsGuangdong Institute of EducationGuangzhouChina

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