Abstract
Let , in this paper, by using the method of weight functions and technique of real analysis; it is proved that the equivalent parameter condition for the validity of multiple integral Hilbert-type inequality with homogeneous kernel of order is , and the calculation formula of its optimal constant factor is obtained. The basic theory and method of constructing a Hilbert-type multiple integral inequality with the homogeneous kernel and optimal constant factor are solved.
1. Preliminary
Assuming that , , and , define
Particularly, denote and . If , , and , then there holds the well-known Hilbert’s integral inequality [1] where the constant factor is optimal.
In general, let , , , be a nonnegative measurable function, and be a constant; we call the Hilbert-type multiple integral inequality.
Fruitful results have been obtained for the Hilbert-type inequality [2–12], but the results of the multi-integral form are much less, especially for the parameters’ conditions and the best constant factor for the Hilbert-type multi-integral inequality. The research for these problems is natural and important. However, the related references are less. In this paper, we will discuss the cases for the homogeneous integral kernel.
Lemma 1. Let be an integer, , be a homogeneous nonnegative measurable function of order , , and . Denote where . Then, we have and .
Proof. Since is a homogeneous function of order , we have and then
Setting , then we find
Thus, (5) holds, because for any , we have
Setting and , then it follows that and . So we get
Hence, we obtain .
Lemma 2. Let , then
Proof. Since , we have and , so
2. The Equivalent Conditions for a Hilbert-Type Multiple Integral Inequality Holding
Theorem 3. Let be an integer, , , be a homogeneous nonnegative measurable function of order , and is convergent, such that Then (i)For all , there exists a constant , such that the Hilbert-type multiple integral inequalityholds true if and only if . (ii)If (15) holds, then the best constant factor is .
Proof. (i) Suppose that there exists a constant such that (15) holds. Denote .
If , then for , we set
where . We find
Thus, by (15), we get
Since and , is divergent to . So we get a contradiction that , namely, , cannot be held.
If , then for , we set
where . Similarly, we get
Since and , is divergent to ; also, we get a contradiction that , namely, , cannot be held.
From the above discussions, we get ; that is, .
Conversely, assume that holds. Note that
By Hlder’s inequality and Lemma 1, we find
So, for all , (15) holds.
(ii) Next, we prove that when the equality (15) holds, . Otherwise, there exists a constant , such that
For a sufficient small and , let
and when , we let
Thus, we get
We still have
Combining this with (23) and (26), we obtain
Let , and then by the Lebesgue dominated convergence theorem, we have
Taking , we get
This is a contradiction compared with , and then .
3. Applications
Taking some different integral kernels and different parameters, we can get a great deal of Hilbert-type inequalities in former literatures and other some new equalities. Moreover, the necessary and sufficient conditions for the existence of these inequalities are obtained.
Corollary 4. Let the integer , , , , and convergence. Then, there exists a constant , such that the necessary and sufficient condition for the equality hold is . And when equation (32) holds, the best constant factor is .
Proof. Let ; then, certainly is a homogeneous nonnegative measurable function of order . By Theorem 3, the corollary holds.
Corollary 5. Let , , , , and . Then, there exists a constant , such that the necessary and sufficient condition for the equality hold is . And when equation (33) holds, the best constant factor is .
Proof. Since , by Lemma 2, we have Combining this with the case of Corollary 4, the proof is completed.
Remark 6. (i) Letting in Corollary 5, we can get
where the constant factor is the best possible. The above equality is the main result in [2].
(ii) Letting in Corollary 5, we get
where the constant factor is optimal.
(iii) Letting and in Corollary 5, then we have
where the constant factor is the best possible.
(iv) Letting and in Corollary 5, we obtain
where the constant factor is optimal.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflict of interests.
Acknowledgments
The first author was supported by the National Natural Science Foundation of China (No. 11401113) and the Characteristic Innovation Project (Natural Science) of Guangdong Province (No. 2017KTSCX133).