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On a Multiple Hilbert-Type Integral Operator and Applications
Journal of Inequalities and Applications volume 2009, Article number: 192197 (2010)
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.
If , = then we have the following famous Hardy-Hilbert's integral inequality and its equivalent form (cf. ):
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.)
Inequalities (1.1) and (1.2) are important in analysis and its applications (cf. ). In 2002, reference  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:
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
If , then we have the following inequality:
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  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 . 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
We find that
and then (2.1) is valid.
If is a measurable function in such that for any and then call the homogeneous function of -degree in
As for the assumption of Lemma 2.1, if is a homogeneous function of degree in
and then each and for any , it follows that
Setting in the integral we find that
Setting in the above integral, we obtain Setting in (2.4), we find that
As for the assumption of Lemma 2.3, setting
then there exist and such that for any if and only if is continuous at
The sufficiency property is obvious. We prove the necessary property of the condition by mathematical induction. For , since
and then by Lebesgue control convergence theorem (cf. ), it follows that Assuming that for is continuous at then for in view of the result for we have that
then by the assumption for it follows that
By mathematical induction, we prove that for is continuous at
As for the assumption of Lemma 2.4, if , then for
Setting we find that
then by (2.11), it follows that
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
Hence by (2.13), we have that
By Lemma 2.4, we find that
Then by combination with (2.15), we have (2.10).
Suppose that then is a measurable function of ree in such that
If are measurable functions in then () for
() for the reverse of (2.18) is obtained.
() For by Hlder's inequality (cf.  ) and (2.4), it follows that
For by Hlder's inequality again, it follows that
Then by (2.4), we have (2.18) (note that for we do not use Hlder's inequality again). () For by the reverse Hlder'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:
and a multiple Hilbert-type integral operator as follows: for
Then by (2.18), it follows that , is bounded, , and where
Define the formal inner product of and as
Suppose that is a measurable function of -degree in and for any it satisfies and
If , then () for and the following equivalent inequalities are obtained:
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.
() 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
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 Hlder's inequality, it follows that
and then by (3.6), we have (3.7). Assuming that (3.7) is valid, setting
then By (2.18), it follows that If then (3.6) is naturally valid. Assuming that by (3.7), it follows that
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
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 Hlder's inequality and the same way, we have the equivalent reverses of (3.6) and (3.7) with the same best constant factor.
For we obtain (cf. [7,]. By Theorem 3.1, it follows that , and then by (3.7), we find that
Setting and in (3.13), we obtain , and
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
For we obtain (cf. [7, ]. By Theorem 3.1, it follows that , and then by (3.7), we find that
Setting and in (3.15), we obtain , and
Hence for , we can show that
For by mathematical induction, we can show that
In fact, for we obtain
Assuming that for (3.17) is valid, then for , it follows that
Then by mathematical induction, (3.17) is valid for
By Theorem 3.1, it follows that , and by (3.7), we find that
Setting and in (3.20), we obtain and
Hence for ), we can show that
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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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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
- Weight Function
- Measurable Function
- Integral Operator
- Convergence Theorem
- Equivalent Form