# Equivalent Conditions of a Hilbert-Type Multiple Integral Inequality Holding.

1. Preliminary

Assuming that r > 1, f(x) [greater than or equal to] 0, and [alpha] [member of] R, define

[mathematical expression not reproducible]. (1)

Particularly, denote [L.sub.r](0, +[infinity]) = [L.sub.r,0](0, +[infinity]) and [parallel]f[[parallel].sub.r] = [parallel]f[[parallel].sub.r,0]. If (1/p) + (1/q) = 1 (P > 1), f [member of] [L.sub.p] (0, +[infinity]) and g [member of] [L.sub.q] (0, [infinity]), then there holds the well-known Hilbert's integral inequality 

[mathematical expression not reproducible], {2)

where the constant factor [pi]/(sin ([pi]/p)) is optimal.

In general, let [mathematical expression not reproducible] be a nonnegative measurable function, and M be a constant; we call

[mathematical expression not reproducible], {3)

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 n [greater than or equal to] 2 bean integer, [[alpha].sub.i] [member of] R (i = 1, 2, ..., n), K ([x.sub.1], ..., [x.sub.n]) be a homogeneous nonnegative measurable function of order [lambda], [[SIGMA].sup.n.sub.i=1] 1/[p.sub.i] = 1 ([p.sub.i] > 1)> and [[SIGMA].sup.n.sub.i=1][[alpha].sub.i]/[p.sub.i] = [lambda] + n - 1. Denote

[mathematical expression not reproducible], (4)

where j = 1, 2, ..., n. Then, we have

[mathematical expression not reproducible], (5)

and [W.sub.1] = [W.sub.2] = ... = [W.sub.n].

Proof. Since K([x.sub.1], ..., [x.sub.n]) is a homogeneous function of order [lambda], we have

K([x.sub.1], ..., [x.sub.n]) = [x.sup.[lambda].sub.j]K ([x.sub.1]/[x.sub.j], ..., [x.sub.j-1]/[x.sub.j], 1, [x.sub.j- 1]/[x.sub.j], ..., [x.sub.n]/[x.sub.j]), (6)

and then

[mathematical expression not reproducible].

Setting [x.sub.i]/[x.sub.j] = [u.sub.i] (I = 1, ..., j - 1, j + 1, ..., n), then we find

[mathematical expression not reproducible]. (8)

Thus, (5) holds, because for any j, we have

[mathematical expression not reproducible] (9)

Setting [u.sub.2]/[u.sub.1] - [v.sub.2], ..., [u.sub.j-1]/[u.sub.1] = [v.sub.j-1], 1/[u.sub.1] = [v.sub.j] [u.sub.j+1]/[u.sub.1] = [v.sub.j+1], ..., and [u.sub.n]/[u.sub.1] = [v.sub.n], then it follows that [u.sub.1] = 1/[v.sub.j], [u.sub.2] = (1/[v.sub.j])[v.sub.2], ..., [u.sub.j-1] = [u.sub.j-1] = (1/[v.sub.j])[v.sub.j+l], ..., and [u.sub.n] = (1/[v.sub.j])[v.sub.n]. So we get

[mathematical expression not reproducible]. (10)

Hence, we obtain [W.sub.1] = [W.sub.2] = ... = [W.sub.n].

Lemma 2. Let 0 < a < 2, then

[mathematical expression not reproducible]. (11)

Proof. Since 0 < a < 2, we have a - 1 < 1 and a + 1 > 1, so

[mathematical expression not reproducible]. (12)

2. The Equivalent Conditions for a Hilbert-Type Multiple Integral Inequality Holding

Theorem 3. Let n [greater than or equal to] 2 be an integer, [[SIGMA].sup.n.sub.i=1] 1/[p.sub.i] ([p.sub.i] > 1), [a.sub.i] [member of] R (i = 1, 2, ..., n), K([x.sub.1], ..., [x.sub.n]) be a homogeneous nonnegative measurable function of order [lambda], and

[mathematical expression not reproducible], (13)

is convergent, such that

[mathematical expression not reproducible]. (14)

Then

(i) For all [mathematical expression not reproducible], there exists a constant M, such that the Hilbert-type multiple integral inequality

[mathematical expression not reproducible], (15)

holds true if and only if [[SIGMA].sup.n.sub.i=1] [[alpha].sub.i]/[p.sub.i] = [lambda] + n - 1.

(ii) If (15) holds, then the best constant factor is inf M = [W.sub.n].

Proof, (i) Suppose that there exists a constant M such that

(15) holds. Denote c = [[SIGMA].sup.n.sub.i=1] [[alpha].sub.i]/[p.sub.i] - ([lambda] + n - 1).

If c > 0, then for 0 < [epsilon] < c, we set

[mathematical expression not reproducible], (16)

where i = 1, 2, ..., n. We find

[mathematical expression not reproducible]. (17)

Thus, by (15), we get

[mathematical expression not reproducible]. (18)

Since [W.sup.(1).sub.n] > 0 and [mathematical expression not reproducible] is divergent to +[infinity]. So we get a contradiction that +[infinity] [less than or equal to] M/[epsilon], namely, c > 0, cannot be held.

If c <0, then for 0 < [epsilon] < -c, we set

[mathematical expression not reproducible], (19)

where i = 1, 2, ..., n. Similarly, we get

[mathematical expression not reproducible]. (20)

Since [W.sup.(2).sub.n] > 0 and [mathematical expression not reproducible] is divergent to +[infinity]; also, we get a contradiction that +[infinity][less than or equal to]M/[epsilon], namely, c < 0, cannot be held.

From the above discussions, we get c = 0; that is, [[SIGMA].sup.n.sub.i=1] [[alpha].sub.1]/[p.sub.i] = [lambda] + n - 1.

Conversely, assume that [[SIGMA].sup.n.sub.i=1] [[alpha].sub.i]/[p.sub.i] = [lambda] + n - 1 holds. Note that

[mathematical expression not reproducible]. (21)

By Holder's inequality and Lemma 1, we find

[mathematical expression not reproducible]. (22)

So, for all M [greater than or equal to] [W.sub.n], (15) holds.

(ii) Next, we prove that when the equality (15) holds, inf M = [W.sub.n]. Otherwise, there exists a constant [M.sub.0] < [W.sub.n], such that

[mathematical expression not reproducible]. (23)

For a sufficient small [epsilon] > 0 and [delta] > 0, let

[mathematical expression not reproducible], (24)

and when i = 2, 3, ..., n, we let

[mathematical expression not reproducible]. (25)

Thus, we get

[mathematical expression not reproducible]. (26)

We still have

[mathematical expression not reproducible]. (27)

Combining this with (23) and (26), we obtain

[mathematical expression not reproducible]. (28)

Let [epsilon] [right arrow] [0.sup.+], and then by the Lebesgue dominated convergence theorem, we have

[mathematical expression not reproducible]. (29)

Taking [delta] [right arrow] [0.sup.+], we get

[mathematical expression not reproducible]. (30)

This is a contradiction compared with [M.sub.0] < [W.sub.n], and then inf M = [W.sub.n].

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 n [greater than or equal to] 2, [[SIGMA].sup.n.sub.i=1] 1/[p.sub.i] = 1 ([p.sub.i] > 1), [[alpha].sub.i] [member of] R, [mathematical expression not reproducible], and

[mathematical expression not reproducible], (31)

convergence. Then, there exists a constant M, such that the necessary and sufficient condition for the equality hold

[mathematical expression not reproducible], (32)

is [[SIGMA].sup.n.sub.i=1] [[alpha].sub.i]/[p.sub.i] = n - 1. And when equation (32) holds, the best constant factor is inf M = [W.sub.n].

Proof. Let K ([x.sub.1], ..., [x.sub.n]) = min {[x.sub.1], ..., [x.sub.n]}/max {[x.sub.1], ..., [x.sub.n]}; then, K([x.sub.1], ..., [x.sub.n]) certainly is a homogeneous nonnegative measurable function of order [lambda] (=0). By Theorem 3, the corollary holds.

Corollary 5. Let (1/p) + (1/q) = 1 (p > 1), -1 < [alpha] < 2p - 1, 1 < [beta] < 2q - 1, f (x) [member of] [L.sub.p,[alpha]] (0, +[infinity]), and g(y) [member of] [L.sub.q,[beta]] (0, +[infinity]). Then, there exists a constant M, such that the necessary and sufficient condition for the equality hold

[mathematical expression not reproducible], (33)

is ([alpha]/p) + ([beta]/q) = 1. And when equation (33) holds, the best constant factor is inf M = (p/([alpha] + 1)) + (q/([beta] + 1)).

Proof. Since 0 < ([beta] +1)/q < 2, by Lemma 2, we have

[mathematical expression not reproducible]. (34)

Combining this with the case n = 2 of Corollary 4, the proof is completed.

Remark 6. (i) Letting [[alpha].sub.i] = [p.sub.i] - 1 (i = 1,...,n) in Corollary 5, we can get

[mathematical expression not reproducible]. (35)

where the constant factor n! is the best possible. The above equality is the main result in .

(ii) Letting [alpha] = [beta] = 1 in Corollary 5, we get

[mathematical expression not reproducible]. (36)

where the constant factor pq/2 is optimal.

(iii) Letting [alpha] = p/2 and [beta] = q/2 in Corollary 5, then we have

[mathematical expression not reproducible], (37)

where the constant factor 8pq/((p + 2)(q + 2)) is the best possible.

(iv) Letting [alpha] = (3/2)p - 2 and [beta] = (3/2) q - 2 in Corollary 5, we obtain

[mathematical expression not reproducible] (38)

where the constant factor 8pq/((3p - 2)(3q - 2)) is optimal.

https://doi.org/10.1155/2020/3050952

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).

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Jianquan Liao, (1) Yong Hong [ID], (2) and Bicheng Yang (1)

(1) Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China

(2) Department of Mathematics, Guangdong Baiyun University, Guangzhou 510450, China

Correspondence should be addressed to Yong Hong; yonghonglbz@163.com

Received 28 September 2019; Accepted 18 March 2020; Published 15 April 2020

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