Open Access

On a Multiple Hilbert-Type Integral Operator and Applications

Journal of Inequalities and Applications20102009:192197

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

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.

1. Introduction

If  ,  =  then we have the following famous Hardy-Hilbert's integral inequality and its equivalent form (cf. [1]):
(1.1)
(1.2)

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])

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 ). In 2004-2005, introducing another pair of conjugate exponents and an independent parameter [5, 6] gave two best extensions of (1.1) as follows:
(1.3)
(1.4)

where is the Beta function ( ,

). In 2009, [7, Theorem ] gave the following multiple Hilbert-type integral inequality: suppose that is a measurable function of ree in and for any satisfies and
(1.5)

If , then we have the following inequality:

(1.6)

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, 811]. 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.

If then
(2.1)

Proof.

We find that
(2.2)

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.

As for the assumption of Lemma 2.1, if is a homogeneous function of degree in
(2.3)
and then each and for any , it follows that
(2.4)

Proof.

Setting in the integral we find that
(2.5)

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

Lemma 2.4.

As for the assumption of Lemma 2.3, setting
(2.6)

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

Proof.

The sufficiency property is obvious. We prove the necessary property of the condition by mathematical induction. For , since
(2.7)
and then by Lebesgue control convergence theorem (cf. [13]), it follows that Assuming that for is continuous at then for in view of the result for we have that
(2.8)
then by the assumption for it follows that
(2.9)

By mathematical induction, we prove that for is continuous at

Lemma 2.5.

As for the assumption of Lemma 2.4, if , then for
(2.10)

Proof.

Setting we find that
(2.11)
Setting and
(2.12)
then by (2.11), it follows that
(2.13)
Without loses of generality, we estimate that In fact, setting such that since there exists such that and then by Fubini theorem, it follows that
(2.14)
Hence by (2.13), we have that
(2.15)
By Lemma 2.4, we find that
(2.16)

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

Lemma 2.6.

Suppose that then is a measurable function of ree in such that
(2.17)
If are measurable functions in then ( ) for
(2.18)

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

Proof.

( ) For by H lder's inequality (cf. [14] ) and (2.4), it follows that
(2.19)
(2.20)
For by H lder's inequality again, it follows that
(2.21)

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:

(3.1)

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

(3.2)

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

(3.3)

Define the formal inner product of and as

(3.4)

Theorem 3.1.

Suppose that is a measurable function of -degree in and for any it satisfies and
(3.5)
If , then ( ) for and the following equivalent inequalities are obtained:
(3.6)
(3.7)

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.

( ) For if (2.18) takes the form of equality, then for in (2.21), there exist constants and such that they are not all zero and
(3.8)
viz. in Assuming that then which contradicts . (Note that for we only consider (2.19) for in the above). Hence we have (3.6). By H lder's inequality, it follows that
(3.9)
and then by (3.6), we have (3.7). Assuming that (3.7) is valid, setting
(3.10)
then By (2.18), it follows that If then (3.6) is naturally valid. Assuming that by (3.7), it follows that
(3.11)

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

For small enough, setting as follows: , if there exists , such that (3.7) is still valid as we replace by then in particular, by Lemma 2.5, we have that
(3.12)

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.

For we obtain (cf. [7, ]. By Theorem 3.1, it follows that , and then by (3.7), we find that
(3.13)
Setting and in (3.13), we obtain , and
(3.14)

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.

For we obtain (cf. [7, ]. By Theorem 3.1, it follows that , and then by (3.7), we find that
(3.15)
Setting and in (3.15), we obtain , and
(3.16)

Hence for , we can show that

Example 3.4.

For by mathematical induction, we can show that
(3.17)
In fact, for we obtain
(3.18)
Assuming that for (3.17) is valid, then for , it follows that
(3.19)

Then by mathematical induction, (3.17) is valid for

By Theorem 3.1, it follows that , and by (3.7), we find that
(3.20)
Setting and in (3.20), we obtain and
(3.21)

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

(1)
Department of Mathematics, Guangdong Institute of Education

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Copyright

© 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.