On a Hilbert-Type Operator with a Class of Homogeneous Kernels

Abstract

By using the way of weight coefficient and the theory of operators, we define a Hilbert-type operator with a class of homogeneous kernels and obtain its norm. As applications, an extended basic theorem on Hilbert-type inequalities with the decreasing homogeneous kernels of -degree is established, and some particular cases are considered.

1. Introduction

In 1908, Weyl published the well-known Hilbert's inequality as the following. If are real sequences, and then [1]

(1.1)

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

(1.2)

where the constant factor is the best possible. We named (1.2) Hardy-Hilbert'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:

(1.3)

(1.4)

where the constant factors and are the best possible.

Note.

Hardy did not prove this theorem in [3]. In particular, we find some classical Hilbert-type inequalities as,

(i)for in (1.3), it reduces (1.2),

(ii)for in (1.3), it reduces to (see [3, Theorem 341])

(1.5)

(iii)for in (1.3), it reduces to (see [3, Theorem 342])

(1.6)

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

(1.7)

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

(1.8)

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

(1.9)

where the constant factor is the best possible.

In 2004, Yang [8] published the dual form of (1.2) as follows:

(1.10)

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

(1.11)

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 2006-2007, 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

(1.12)

Then the formal inner product of and are defined as follows:

(1.13)

In 2007, Yang [11] proved that if for small enough, is strictly decreasing for the integral is also a positive number independent of and

(1.14)

then in this case, if then we have two equivalent inequalities as

(1.15)

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 Hilbert-type operator and obtain its norm. As applications, an extended basic theorem on Hilbert-type inequalities with the decreasing homogeneous kernel of -degree is established; some particular cases are considered.

2. On a New Hilbert-Type 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

(2.1)

Proof.

By the assumption, we find and there exists such that Hence,

(2.2)

Lemma 2.2.

If is a homogeneous function of -degree, and is a positive number, then (i)(ii) for setting the weight functions as

(2.3)

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 2.1-2.2, we obtain

(2.4)

Therefore, .

For define a Hilbert-type operator as satisfying

(2.5)

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

(2.6)

Then by (2.4), we obtain

(2.7)

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,

(2.8)

(2.9)

In view of (2.8) and (2.9), setting , by Fubini's theorem [13], it follows

(2.10)

Setting in the above inequality, by Fatou's lemma [14], we find

(2.11)

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 Hilbert-Type 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

(3.1)

(3.2)

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

(3.3)

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

(3.4)

and then it deduces to (1.11);

(ii)for in (3.1), we find

(3.5)

and then it deduces to the best extension of (1.5) as

(3.6)

(iii)for in (3.1), we find [3]

(3.7)

and , and then it deduces to the best extension of (1.6) as

(3.8)