- Research Article
- Open access
- Published:
On a Multiple Hilbert-Type Integral Operator and Applications
Journal of Inequalities and Applications volume 2009, Article number: 192197 (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]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ2_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ3_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ4_HTML.gif)
where is the Beta function (
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_IEq19_HTML.gif)
). 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ5_HTML.gif)
If ,
then we have the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ6_HTML.gif)
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, 8–11]. 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ7_HTML.gif)
Proof.
We find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_IEq45_HTML.gif)
and then each
and for any
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ10_HTML.gif)
Proof.
Setting in the integral
we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ14_HTML.gif)
then by the assumption for it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ15_HTML.gif)
By mathematical induction, we prove that for is continuous at
Lemma 2.5.
As for the assumption of Lemma 2.4, if , then for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ16_HTML.gif)
Proof.
Setting we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ17_HTML.gif)
Setting and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ18_HTML.gif)
then by (2.11), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ19_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ20_HTML.gif)
Hence by (2.13), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ21_HTML.gif)
By Lemma 2.4, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ23_HTML.gif)
If are measurable functions in
then (
) for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ24_HTML.gif)
() for
the reverse of (2.18) is obtained.
Proof.
() For
by H
lder's inequality (cf. [14] ) and (2.4), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ25_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ26_HTML.gif)
For by H
lder's inequality again, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ27_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ28_HTML.gif)
and a multiple Hilbert-type integral operator as follows: for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ29_HTML.gif)
Then by (2.18), it follows that ,
is bounded,
, and
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ30_HTML.gif)
Define the formal inner product of and
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ31_HTML.gif)
Theorem 3.1.
Suppose that is a measurable function of
-degree in
and for any
it satisfies
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ32_HTML.gif)
If ,
then (
) for
and the following equivalent inequalities are obtained:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ33_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ36_HTML.gif)
and then by (3.6), we have (3.7). Assuming that (3.7) is valid, setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ37_HTML.gif)
then By (2.18), it follows that
If
then (3.6) is naturally valid. Assuming that
by (3.7), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ38_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ39_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ40_HTML.gif)
Setting and
in (3.13), we obtain
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ41_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ42_HTML.gif)
Setting and
in (3.15), we obtain
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ43_HTML.gif)
Hence for , we can show that
Example 3.4.
For by mathematical induction, we can show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ44_HTML.gif)
In fact, for we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ45_HTML.gif)
Assuming that for (3.17) is valid, then for
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ46_HTML.gif)
Then by mathematical induction, (3.17) is valid for
By Theorem 3.1, it follows that , and by (3.7), we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ47_HTML.gif)
Setting and
in (3.20), we obtain
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F192197/MediaObjects/13660_2009_Article_1913_Equ48_HTML.gif)
Hence for ), we can show that
References
Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge, UK; 1934.
Carleman T: Sur les Equations Integrals Singulieres a Noyau Reed et Symetrique. Volume 923. Uppsala universitet, Uppsala, Sweden;
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.
Zhang K: A bilinear inequality. Journal of Mathematical Analysis and Applications 2002,271(1):288–296. 10.1016/S0022-247X(02)00104-X
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.
Yang B, Brnetić I, Krnić M, Pečarić J: Generalization of Hilbert and Hardy-Hilbert integral inequalities. Mathematical Inequalities & Applications 2005,8(2):259–272.
Yang B: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijin, China; 2009.
Hong Y: All-sided generalization about Hardy-Hilbert integral inequalities. Acta Mathematica Sinica 2001,44(4):619–625.
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.
Yang B: A multiple Hardy-Hilbert integral inequality. Chinese Annals of Mathematics. Series A 2003,24(6):25–30.
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.
Bényi Á, Choonghong O: Best constants for certain multilinear integral operators. Journal of Inequalities and Applications 2006, 2006:-12.
Kuang J: Introduction to Real Analysis. Hunan Education Press, Chansha, China; 1996.
Kuang J: Applied Inequalities. Shangdong Science Technic Press, Jinan, China; 2004.
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Huang, Q., Yang, B. On a Multiple Hilbert-Type Integral Operator and Applications. J Inequal Appl 2009, 192197 (2010). https://doi.org/10.1155/2009/192197
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/192197