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On a HilbertType Operator with a Class of Homogeneous Kernels
Journal of Inequalities and Applications volume 2009, Article number: 572176 (2009)
Abstract
By using the way of weight coefficient and the theory of operators, we define a Hilberttype operator with a class of homogeneous kernels and obtain its norm. As applications, an extended basic theorem on Hilberttype inequalities with the decreasing homogeneous kernels of degree is established, and some particular cases are considered.
1. Introduction
In 1908, Weyl published the wellknown Hilbert's inequality as the following. If are real sequences, and then [1]
where the constant factor is the best possible. In 1925, Hardy gave an extension of (1.1) by introducing one pair of conjugate exponents as [2]. If , , and then
where the constant factor is the best possible. We named (1.2) HardyHilbert's inequality. In 1934, Hardy et al. [3] gave some applications of (1.1)(1.2) and a basic theorem with the general kernel (see [3, Theorem 318]).
Theorem 1.
Suppose that is a homogeneous function of degree, and is a positive number. If both and are strictly decreasing functions for , , and then one has the following equivalent inequalities:
where the constant factors and are the best possible.
Note.
Hardy did not prove this theorem in [3]. In particular, we find some classical Hilberttype inequalities as,
(i)for in (1.3), it reduces (1.2),
(ii)for in (1.3), it reduces to (see [3, Theorem 341])
(iii)for in (1.3), it reduces to (see [3, Theorem 342])
Hardy also gave some multiple extensions of (1.3) (see [3, Theorem 322]). About introducing one pair of nonconjugate exponents in (1.1), Hardy et al. [3] gave that if then
In 1951, Bonsall [4] considered (1.7) in the case of general kernel; in 1991, Mitrinović et al. [5] summarized the above results.
In 2001, Yang [6] gave an extension of (1.1) as for
where the constant is the best possible ( is the Beta function). For (1.8) reduces to (1.1). And Yang [7] also gave an extension of (1.2) as
where the constant factor is the best possible.
In 2004, Yang [8] published the dual form of (1.2) as follows:
where is the best possible. For both (1.10) and (1.2) reduce to (1.1). It means that there are more than two different best extensions of (1.1). In 2005, Yang [9] gave an extension of (1.8)–(1.10) with two pairs of conjugate exponents , and two parameters as
where the constant factor is the best possible; Krnić and Pečarić [10] also considered (1.11) in the general homogeneous kernel, but the best possible property of the constant factor was not proved by [10].
Note.
For in [10, inequality (37)], it reduces to the equivalent result of (3.1) in this paper.
In 20062007, some authors also studied the operator expressing of (1.3) and (1.4).
Suppose that is a symmetric function with and is a positive number independent of Define an operator as follows. For there exists only satisfying
Then the formal inner product of and are defined as follows:
In 2007, Yang [11] proved that if for small enough, is strictly decreasing for the integral is also a positive number independent of and
then in this case, if then we have two equivalent inequalities as
where the constant factor is the best possible. In particular, for being degree homogeneous, inequalities (1.15) reduce to (1.3)(1.4) (in the symmetric kernel). Yang [12] also considered (1.15) in the real space .
In this paper, by using the way of weight coefficient and the theory of operators, we define a new Hilberttype operator and obtain its norm. As applications, an extended basic theorem on Hilberttype inequalities with the decreasing homogeneous kernel of degree is established; some particular cases are considered.
2. On a New HilbertType Operator and the Norm
If is a measurable function, satisfying for then we call the homogeneous function of degree.
For setting we find Hence, the following two words are equivalent: (a) is decreasing in and strictly decreasing in a subinterval of ; (b) for any , is decreasing in and strictly decreasing in a subinterval of . The following two words are also equivalent: is decreasing in and strictly decreasing in a subinterval of ; for any , is decreasing in and strictly decreasing in a subinterval of .
Lemma 2.1.
If is decreasing in and strictly decreasing in a subinterval of , and then
Proof.
By the assumption, we find and there exists such that Hence,
Lemma 2.2.
If is a homogeneous function of degree, and is a positive number, then (i)(ii) for setting the weight functions as
then .
Proof.

(i)
Setting by the assumption, we obtain (ii) Setting and in the integrals and respectively, in view of (i), we still find that
For we set and Define the real space as and then we may also define the spaces and
Lemma 2.3.
As the assumption of Lemma 2.2, for setting , if and are decreasing in and strictly decreasing in a subinterval of , then
Proof.
By Hölder's inequality [13] and Lemmas 2.12.2, we obtain
Therefore, .
For define a Hilberttype operator as satisfying
In view of Lemma 2.3, and then exists. If there exists such that for any then is bounded and Hence by (2.4), we find and is bounded.
Theorem 2.4.
As the assumption of Lemma 2.3, it follows
Proof.
For by Hölder's inequality [12], we find
Then by (2.4), we obtain
For setting , as for if there exists a constant such that (2.7) is still valid when we replace by then by Lemma 2.1,
In view of (2.8) and (2.9), setting , by Fubini's theorem [13], it follows
Setting in the above inequality, by Fatou's lemma [14], we find
Hence is the best value of (2.7). We conform that is the best value of (2.4). Otherwise, we can get a contradiction by (2.6) that the constant factor in (2.7) is not the best possible. It follows that
3. An Extended Basic Theorem on HilbertType Inequalities
Still setting , and we have the following theorem.
Theorem 3.1.
Suppose that is a homogeneous function of degree, is a positive number, both and are decreasing in and strictly decreasing in a subinterval of . If , then one has the equivalent inequalities as
where the constant factors and are the best possible.
Proof.
In view of (2.7) and (2.4), we have (3.1) and (3.2). Based on Theorem 2.4, it follows that the constant factors in (3.1) and (3.2) are the best possible.
If (3.2) is valid, then by (2.6), we have (3.1). Suppose that (3.1) is valid. By (2.4), If then (3.2) is naturally valid; if setting then By (3.1), we obtain
and we have (3.2). Hence (3.1) and (3.2) are equivalent.
Remark 3.2.

(a)
For (3.1) and (3.2) reduce, respectively, to (1.6) and (1.7). Hence, Theorem 3.1 is an extension of Theorem A.

(b)
Replacing the condition " and are decreasing in and strictly decreasing in a subinterval of " by "for and are decreasing in and strictly decreasing in a subinterval of ," the theorem is still valid. Then in particular,
(i)for () in (3.1), we find
and then it deduces to (1.11);
(ii)for in (3.1), we find
and then it deduces to the best extension of (1.5) as
(iii)for in (3.1), we find [3]
and , and then it deduces to the best extension of (1.6) as
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Yang, B. On a HilbertType Operator with a Class of Homogeneous Kernels. J Inequal Appl 2009, 572176 (2009). https://doi.org/10.1155/2009/572176
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Keywords
 Constant Factor
 Beta Function
 Homogeneous Function
 Dual Form
 Real Sequence