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A New Hilbert-Type Linear Operator with a Composite Kernel and Its Applications

Open Access
Research Article

Abstract

A new Hilbert-type linear operator with a composite kernel function is built. As the applications, two new more accurate operator inequalities and their equivalent forms are deduced. The constant factors in these inequalities are proved to be the best possible.

Keywords

Linear Operator Weight Function Kernel Function Differentiable Function Constant Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

In 1908, Weyl [1] published the well-known Hilbert's inequality as follows:

where the constant factor Open image in new window is the best possible.

Under the same conditions, there are the classical inequalities [2]

where the constant factors Open image in new window and Open image in new window are the best possible also. Expression (1.2) is called a more accurate form of(1.1). Some more accurate inequalities were considered by [3, 4, 5]. In 2009, Zhong [5] gave a more accurate form of (1.3).

where Open image in new window is a linear operator, Open image in new window . Open image in new window is the norm of the sequence Open image in new window with a weight function Open image in new window . Open image in new window is a formal inner product of the sequences Open image in new window and Open image in new window .

By setting two monotonic increasing functions Open image in new window and Open image in new window , a new Hilbert-type inequality, which is with a composite kernel function Open image in new window , and its equivalent are built in this paper. As the applications, two new more accurate Hilbert-type inequalities incorporating the linear operator and the norm are deduced.

Firstly, the improved Euler-Maclaurin's summation formula [6] is introduced.

2. Lemmas

Lemma 2.1.

Then, it has the following.

(1)The functions Open image in new window , Open image in new window satisfy the conditions of (1.6). It means that
when Open image in new window . It is easy to find that
Similarly, it can be shown that Open image in new window , Open image in new window ( Open image in new window , Open image in new window ). These tell us that (2.6) holds and the functions Open image in new window , Open image in new window satisfy the conditions of (1.6).
  1. (2)
    Set Open image in new window . With the partial integration, it has
     
By (2.1), it has
In view of (2.12)~(2.14), it has
It means that Open image in new window . Similarly, it can be shown that Open image in new window . The expression (2.7) holds.
  1. (3)
    By (2.5), (2.12), (2.13), and Open image in new window , Open image in new window , it has
     

The expression (2.8) holds, and Lemma 2.1 is proved.

Lemma 2.2.

Then, it has

(1)The functions Open image in new window , Open image in new window satisfy the conditions of (1.6). It means that
Proof.
  1. (1)

    Letting Open image in new window , Open image in new window , it can be proved that Open image in new window satisfy (2.19) as in [5]. Similarly, it can be shown that Open image in new window satisfy (2.19) also.

     
  2. (2)
     
With (2.22)~(2.24), it has
So Open image in new window holds. Similarly, it can be shown that Open image in new window .
  1. (3)
    In view of (2.22), (2.23), by Open image in new window , Open image in new window , it has
     

It means that (2.21) holds. The proof for Lemma 2.2 is finished.

3. Main Results

Set Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window as two pairs of conjugate exponents. Open image in new window is a measurable kernel function. Both Open image in new window and Open image in new window are strictly monotonic increasing differentiable functions in Open image in new window such that Open image in new window , Open image in new window . Give some notations as follows:

(2) Open image in new window set

and call Open image in new window a real space of sequences, where
is called the norm of the sequence with a weight function Open image in new window . Similarly, the real spaces of sequences Open image in new window , Open image in new window and the norm Open image in new window can be defined as well,
  1. (3)

    define a Hilbert-type linear operator Open image in new window , for all Open image in new window ,

     
  1. (4)
     

define two weight coefficients Open image in new window and Open image in new window as

Then it has some results in the following theorems.

Theorem 3.1.

then for all Open image in new window and Open image in new window , it has the following:

It means that Open image in new window ,

(2) Open image in new window is a bounded linear operator and

where Open image in new window , Open image in new window are defined by (3.4), Open image in new window is defined as (3.3).

Proof.

By using Hölder's inequality [7] and (3.6), (3.7), it has Open image in new window and
And by Open image in new window , it follows that

This means that Open image in new window , Open image in new window , and Open image in new window . Open image in new window is a bounded linear operator.

But on the other side, by (3.8), it has
The series is uniformly convergent for Open image in new window , so it has
By (3.14) and (3.8), when Open image in new window , it has

In view of (3.13) and (3.18), letting Open image in new window , it has Open image in new window . This means that Open image in new window ; that is, Open image in new window . Theorem 3.1 is proved.

Theorem 3.2.

Here, Open image in new window , Open image in new window satisfy the conditions as in Theorem 3.1. Set
If (a) Open image in new window is a homogeneous measurable kernel function of " Open image in new window '' degree in Open image in new window , such that
  1. (b)
    functions Open image in new window , Open image in new window satisfy the conditions of (1.6); that is,
     
  1. (c)
    there exists Open image in new window , such that
     

then it has

(2)if Open image in new window and Open image in new window , then

where inequality (3.27) is equivalent to (3.26) and the constant factor Open image in new window is the best possible.

Proof.

By (3.24), (1.6), it has
Letting Open image in new window and Open image in new window in the integral of (3.28) and (3.29), respectively, by (3.23), it has
(where, letting Open image in new window , it has Open image in new window ). In view of (3.28), (3.30), (3.20), (3.22), and with (3.25), it has
Similarly, with (3.29), (3.31), (3.21), and (3.25), it has
also. By Theorem 3.1, it has
and (3.27) holds. In view of

(3.26) holds also.

Letting Open image in new window in (3.37), it has Open image in new window , and it means that Open image in new window and Open image in new window . Therefore, the inequality (3.36) keeps the form of the strict inequality when Open image in new window . In view of Open image in new window , inequality (3.27) holds and (3.27) is equivalent to (3.26). By Open image in new window , it is obvious that the constant factor Open image in new window is the best possible. This completes the proof of Theorem 3.2.

4. Applications

Example 4.1.

Set Open image in new window , Open image in new window be two pairs of conjugate exponents and Open image in new window , Open image in new window , Open image in new window , Open image in new window . Then it has the following.

(2)If Open image in new window , then

where the constant factors Open image in new window and Open image in new window are both the best possible. Inequality (4.2) is equivalent to (4.1).

Proof.

Setting Open image in new window , Open image in new window , it is a homogeneous measurable kernel function of " Open image in new window '' degree. Letting Open image in new window , it has
Setting Open image in new window , Open image in new window , then both Open image in new window and Open image in new window are strictly monotonic increasing differentiable functions in Open image in new window and satisfy
with (2.1)~(2.8), it has

When Open image in new window and Open image in new window ; that is, Open image in new window , Open image in new window and Open image in new window , Open image in new window , by Theorem 3.2, inequality (4.1) holds, so does (4.2). And (4.2) is equivalent to (4.1), and the constant factors Open image in new window and Open image in new window are both the best possible.

Example 4.2.

Set Open image in new window , Open image in new window be two pairs of conjugate exponents and Open image in new window , Open image in new window , Open image in new window , Open image in new window . Then it has the following.

(2)If Open image in new window , then

where inequality (4.8) is equivalent to (4.7) and the constant factors Open image in new window and Open image in new window are both the best possible.

Proof.

Setting Open image in new window Open image in new window , it is a homogeneous measurable kernel function of " Open image in new window '' degree. Letting Open image in new window , it has [2]
Setting Open image in new window , Open image in new window , then both Open image in new window and Open image in new window are strictly monotonic increasing differentiable functions in Open image in new window and satisfy
with (2.18)~(2.21), it has

When Open image in new window and Open image in new window ; that is, Open image in new window , Open image in new window and Open image in new window , Open image in new window , by Theorem 3.2, inequality (4.7) holds, so does (4.8). And (4.8) is equivalent to (4.7), and the constant factors Open image in new window and Open image in new window are both the best possible.

Remark 4.3.

It can be proved similarly that, if the conditions " Open image in new window " in Lemma 2.1 and " Open image in new window " in Lemma 2.2 are changed into " Open image in new window " and " Open image in new window ", respectively, Lemmas 2.1 and 2.2 are also valid. So the conditions " Open image in new window " in Example 4.1 and " Open image in new window " in Example 4.2 can be replaced by " Open image in new window , Open image in new window " and " Open image in new window , Open image in new window ", respectively.

Notes

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (no. 10871073). The author would like to thank the anonymous referee for his or her suggestions and corrections.

References

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Copyright information

© Wuyi Zhong. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsGuangdong Institute of EducationGuangzhouChina

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