1. Introduction
In 1925, by introducing one pair of conjugate exponents
Hardy [
1] established a well-known extension of Hilbert’s integral inequality as follows.
If
then:
where the constant factor
is the best possible.
Inequalities (
1) as well as Hilbert’s integral inequality (for
in (
1), cf. [
2]) are important in analysis and its applications (cf. [
3,
4]).
Almost ten years later, in 1934, Hardy et al. proved an extension of (
1) with the general homogeneous kernel of degree
as
(cf. [
3], Theorem 319). The following Hilbert-type integral inequality with the general nonhomogeneous kernel was established.
If
then:
where the constant factor
is the best possible (cf. [
3], Theorem 350).
In 1998, by introducing an independent parameter
Yang proved an extension of Hilbert’s integral inequality with the kernel
(cf. [
5,
6]). In 2004, by introducing another pair of conjugate exponents
Yang [
7] was able to estabish an extension of (
1) with the kernel
In the paper [
8], a further extension of (
1) was proved along with the result of the paper [
5] with the kernel
Several papers (cf. [
9,
10,
11,
12,
13,
14]) provided some extensions of (
1) with parameters. In 2009, Yang presented the following extension of (
1) (cf. [
15,
16]).
If
is a non-negative homogeneous function of degree
, satisfying:
and:
then we have:
where the constant factor
is the best possible.
For
(
3) reduces to (
1). The following extension of (
2) was proven:
where the constant factor
is the best possible (cf. [
17]).
For
(
4) reduces to (
2).
Some equivalent inequalities of (
3) and (
4) are considered in [
16]. In 2013, Yang [
17] also studied the equivalency between (
3) and (
4) by adding a condition. In 2017, Hong [
18] proved an equivalent condition between (
3) and a few parameters. Some similar results were obtained in [
19,
20,
21,
22,
23,
24,
25,
26,
27,
28].
Remark 1 (cf. [
17]).
If for then:and (4) reduces to the following Hardy-type integral inequality with nonhomogeneous kernel:where the constant factor is the best possible.If for then:and (4) reduces to the following kind of Hardy-type integral inequality with nonhomogeneous kernel:where the constant factor is the best possible. In this paper, using weight functions as well as employing various techniques from real analysis, we establish a few equivalent conditions of two kinds of Hardy-type integral inequalities with the nonhomogeneous kernel:
To prove our results, we also deduce a few equivalent conditions of two kinds of Hardy-type integral inequalities with a homogeneous kernel in the form of applications. We additionally consider operator expressions. Analytic inequalities of this nature and especially the techniques involved have far reaching applications in various areas in which symmetry plays a prominent role, including aspects of physics and engineering.
2. Two Lemmas
For
we set
For
by the Lebesgue term-by-term integration theorem, we derive that:
Setting
in the above integral, we obtain:
where:
stands for the gamma function and:
which is a function very well known for its applications in analytic number theory.
For
setting
by (
7), we obtain that:
In the sequel, we assume that
Lemma 1. If there exists a constant , such that for any non-negative measurable functions and in the following inequality:holds true. Then, we have and Proof. If
then for
we set the following two functions:
and deduce that:
Setting
we obtain:
and then by (
9), we have:
Since
it follows that:
By (
10), in view of:
we deduce that
which is a contradiction.
If
then for
we set the following two functions:
and obtain:
Setting
we obtain:
and then by Fubini’s theorem and (
9), we have:
Since
it follows that:
By (
11), in view of the fact that
we obtain that
which is a contradiction.
Hence, we conclude that
For
we reduce (
11) as follows:
Since:
is non-negative and increasing in
by Levi’s theorem, we derive that:
This completes the proof of the lemma. □
Lemma 2. If there exists a constant , such that for any non-negative measurable functions and in the following inequality:holds true. Then, we have and Proof. If
then for
we set two functions
and
as in Lemma 1, and derive that:
Setting
we obtain:
and then by (
13), we deduce that:
Since
it follows that:
By (
14), in view of
we have
which is a contradiction.
If
then for
we set two sequences of
and
as in Lemma 1, and obtain:
Setting
we obtain:
and then, by Fubini’s theorem and (
13), we have:
Since
it follows that
By (
15), in view of the fact that:
we have
which is a contradiction.
Hence, we conclude the fact that
For
we reduce (
15) as follows:
Since:
is non-negative and increasing in
still by Levi’s theorem, we have:
This completes the proof of the Lemma. □
3. Main Results and Corollaries
Theorem 1. If then the following conditions are equivalent.
(i) There exists a constant such that for any satisfying:we have the following Hardy-type integral inequality of the first kind with nonhomogeneous kernel: (ii) There exists a constant such that for any satisfying:we have the following inequality: (iii)
If Condition (iii) holds, then and the constant factor:in (17) and (18) is the best possible. Proof. . By Hölder’s inequality (cf. [
29,
30]), we obtain:
Then by (
17), we have (
18).
. By Lemma 1, we have
. Setting
we obtain the following weight function:
By Hölder’s inequality with weight and (
20), for
we have:
If (
21) takes the form of equality for some
, then (cf. [
30]) there exist constants
A and
B, such that they are not all zero and:
We suppose that
(otherwise
). It follows that:
which contradicts the fact that:
Hence, (
21) takes the form of strict inequality.
For
by (
21) and Fubini’s theorem, we obtain:
Setting
(
17) follows.
Therefore, Condition (i), Condition (ii) and Condition (iii) are equivalent.
When Condition (iii) is satisfied, if there exists a constant factor
such that (
18) is valid, then by Lemma 1 we have
. Then, the constant factor
in (
18) is the best possible. The constant factor
in (
17) is still the best possible. Otherwise, by (
19) (for
), we can conclude that the constant factor
in (
18) is not the best possible. □
Setting , in Theorem 1, then replacing Y (resp. ) by y (resp. we derive the following Corollary.
Corollary 1. If then the following conditions are equivalent.
(i) There exists a constant such that for any satisfying:we have the following Hardy-type inequality of the first kind with homogeneous kernel: (ii) There exists a constant such that for any satisfying:we have the following inequality: (iii)
If Condition (iii) holds, then we have and the constant in (22) and (23) is the best possible. Similarly, we obtain the following weight function:
and then in view of Lemma 2 and in a similar manner, we obtain the following theorem:
Theorem 2. If then the following conditions are equivalent.
(i) There exists a constant such that for any satisfying:we have the following Hardy-type inequality of the second kind with the nonhomogeneous kernel: (ii) There exists a constant such that for any satisfying:we have the following inequality: (iii)
If Condition (iii) holds, then we have and the constant factor:in (24) and (25) is the best possible. Setting:
in Theorem 2, then replacing
Y (resp.
) by
y (resp.
we derive the following Corollary.
Corollary 2. If then the following conditions are equivalent.
(i) There exists a constant such that for any satisfying:we have the following Hardy-type inequality of the second kind with homogeneous kernel: (ii) There exists a constant such that for any satisfyingwe have the following inequality: (iii)
If Condition (iii) holds, then we have and the constant in (26) and (27) is the best possible. 4. Operator Expressions
For
we set the following functions:
and:
Define the following real normed linear spaces:
(a) In view of Theorem 1 (setting
, for
setting:
by (
17), we have:
Definition 1. Define a Hardy-type integral operator of the first kind with the nonhomogeneous kernel:as follows. For any there exists a unique representation:satisfying for any . In view of (
28), it follows that:
and then the operator
is bounded satisfying
If we define the formal inner product of
and
g as follows:
then we can rewrite Theorem 1 as follows.
Theorem 3. For the following conditions are equivalent.
(i) There exists a constant such that for any we have the following inequality: (ii) There exists a constant such that for any we have the following inequality: We also have that
(b) In view of Corollary 1 (setting
), for
considering the function:
by (
22), we have:
Definition 2. Define a Hardy-type integral operator of the first kind with the homogeneous kernel:as follows. For any there exists a unique representation:satisfying for any . In view of (
31), it follows that:
and then the operator
is bounded satisfying:
If we define the formal inner product of
and
g as follows:
then we can rewrite Corollary 1 as follows.
Corollary 3. For the following conditions are equivalent.
(i) There exists a constant such that for any we have the following inequality: (ii) There exists a constant such that for any we have the following inequality: We still have
(c) In view of Theorem 2 (setting
, for
considering the function:
by (
24), we have:
(A. Raigorodskii)
Definition 3. Define a Hardy-type integral operator of the second kind with the nonhomogeneous kernel:as follows. For any there exists a unique representation:satisfying for any . In view of (
34), it follows that:
and then the operator
is bounded satisfying:
If we define the formal inner product of
and
g as follows.
then we can rewrite Theorem 2 as follows.
Theorem 4. For the following conditions are equivalent.
(i) There exists a constant such that for any we have the following inequality: (ii) There exists a constant such that for any we have the following inequality: We still have
(d) In view of Corollary 2 (setting
), for
considering the function:
by (
26), we have:
Definition 4. Define a Hardy-type integral operator of the second kind with the homogeneous kernel:as follows. For any there exists a unique representation:satisfying for any In view of (
37), it follows that:
and then the operator
is bounded satisfying.
If we define the formal inner product of
and
g as follows:
then we can rewrite Corollary 2 as follows.
Corollary 4. For the following conditions are equivalent.
(i) There exists a constant such that for any we have the following inequality: (ii) There exists a constant such that for any we have the following inequality: We still have