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A new discrete Hardy-type inequality with kernels and monotone functions
- Aigerim Kalybay^{1},
- Lars-Erik Persson^{2, 3}Email author and
- Ainur Temirkhanova^{4}
https://doi.org/10.1186/s13660-015-0843-9
© Kalybay et al. 2015
- Received: 16 April 2015
- Accepted: 25 September 2015
- Published: 6 October 2015
Abstract
A new discrete Hardy-type inequality with kernels and monotone functions is proved for the case \(1< q< p<\infty\). This result is discussed in a general framework and some applications related to Hölder’s summation method are pointed out.
Keywords
- inequality
- Hardy-type inequality
- kernel
- matrix operator
- monotone sequence
- Oinarov condition
MSC
- 26D10
- 26D15
- 39B82
1 Introduction
It is also of great interest to investigate all the problems above when the inequalities are studied only on the cone of non-decreasing functions. In particular, such inequalities give precise results concerning embeddings between weighted Lorentz spaces described by different (quasi-)norms. Here, we mention the fundamental paper [8] from 1990 by Sawyer, where he proved his famous ‘Sawyer duality principle’ for reduction of an inequality for monotone functions to the corresponding inequality for non-negative functions.
There has been a similar development for Hardy-type inequalities in the discrete case. For clarity and as an introduction of the main result of this paper we briefly describe this development in Section 2, where we also compare with the continuous case and formulate our main result. This main result is proved in Section 4. In order to prepare for this proof we state some auxiliary results in Section 3. Finally, we give some applications connected to Hölder’s summation method (see [9]) in Section 5.
2 The discrete case - formulation of the main result
When \(n=1\) the operator \(S_{n-1}\) becomes the standard discrete Hardy operator \((S_{0}f)_{i}=\sum_{j=1}^{i} f_{j}\). The validity of inequality (6) with the standard discrete Hardy operator has been in detail investigated for non-negative sequences f and different relations between the parameters p and q. A thorough analysis and review of the development of this problem can be found e.g. in [4, 12] and [13]. For this case and when the sequence \(\{f_{k}\}\) is non-increasing we refer to the important paper [14] and the references given there.
However, so far no such characterization in the case \(1< q< p<\infty\) is known and it is the main aim of this paper to fill in this gap. Our main result reads as follows.
Theorem 2.1
Moreover, \(E_{12}\approx C\) when \(V_{\infty}=\infty\) and \(E_{13}\approx C\) when \(V_{\infty}<\infty\), where C is the best constant in (6).
3 Notations and auxiliary statements
Let \(\frac{1}{p}+\frac{1}{p'}=1\) and \(\frac{1}{q}+\frac{1}{q'}=1\). The symbol \(A\ll B\) means that \(A\leq CB\) with some constant C, which may depend only on the parameters p and q. Moreover, if \(A\ll B\ll A\), then we write \(A\approx B\).
For all \(i\geq j\geq1\) we suppose that \(A_{l, m}(i, j)=1\) when \(l< m\) and \(A_{l, m}(i, j)=\sum_{k_{l}=j}^{i}\omega_{l, k_{l}} \sum_{k_{l-1}=k_{l}}^{i}\omega_{l-1,k_{l-1}}\cdots \sum_{k_{m}=k_{m+1}}^{i}\omega_{m, k_{m}}\) for \(n-1\geq l\geq m\geq1\). Moreover, for all \(i< j\) we suppose that \(A_{l, m}(i, j)=0\) when \(l, m\geq1\).
In [15] the following lemma was proved.
Lemma A
We also need the following discrete analog of the ‘Sawyer duality principle’, which was proved in [10].
Theorem A
Moreover, \(\widetilde{C}\approx C\) when \(V_{\infty}=\infty\) and \(\overline{C}\approx C\) when \(V_{\infty}<\infty\), where C, C̃ and C̅ are the best constants in (12), (13) and (14), respectively.
We also need the following well-known result (see e.g. [4], p.58).
Theorem B
Moreover, \(H\approx C\), where C is the best constant in (15).
We also need two theorems from [16].
The first theorem presents conditions for the validity of inequality (6) for only non-negative sequences. Here we consider absolutely the same problem but with monotonicity restriction. Thus, it helps us to compare the results with and without monotonicity restriction. In addition, we need it to illustrate some applications given in the last section of the presented paper.
Theorem C
Moreover, \(A(n)\approx C\), where C is the best constant in (6).
Theorem D
Moreover, \(B(n)\approx C\), where C is the best constant in (17).
4 Proof of Theorem 2.1
We start our proof from the case \(V_{\infty}=\infty\).
There are n inequalities in (20). All of these n inequalities can be characterized by Theorem D. It means that the condition \(E_{1}<\infty\) is necessary and sufficient for the validity of (20). Moreover, in view of Theorem B, Theorem D and (21), we see that \(E_{12}=\max\{E_{1},E_{2}\}\approx C\), where C is the best constant in (6).
5 Applications
In the theory of series the estimates of norms of summable matrices are very important problems. One of the important methods of summation is Hölder’s method by \((H,n-1)\) defined as follows:
This method, introduced in 1882 by Hölder in [9], is a generalization of the summation method of arithmetic averages. It is obvious that \((H,1)\) is the method of arithmetic averages in the ordinary sense.
Theorem 5.1
Let \(1< q< p<\infty\) and \(n\geq1\). Then inequality (27) holds for all non-negative sequences \(f=\{f_{i}\}_{i=1}^{\infty}\) from \(l_{p, v}\) if and only if \(A(n)<\infty\) (see (16)), where \(A_{n-1,1}\) is defined in (26) and \(u_{i}=\frac{\widetilde{u}_{i}}{i}\). Moreover, \(A(n)\approx C\), where C is the best constant in (27).
Theorem 5.2
Let \(1< q< p<\infty\) and \(n\geq1\). Suppose that \(V_{k}=\sum_{i=1}^{k} v^{p}_{i}\) when \(k\geq1\) and \(V_{\infty}=\lim_{k\rightarrow\infty }V_{k}\). Then inequality (27) holds for all non-negative non-increasing sequences \(f=\{f_{i}\}_{i=1}^{\infty}\) from \(l_{p, v}\) if and only if \(E_{12}=\max\{E_{1},E_{2}\}<\infty\) when \(V_{\infty}=\infty\) (see (7) and (8)) and \(E_{13}=\max\{E_{1},E'_{2},E_{3}\} <\infty\) when \(V_{\infty}<\infty\) (see (7), (9), and (10)), where \(A_{n-1,1}\) is defined in (26) and \(u_{i}=\frac{\widetilde{u}_{i}}{i}\). Moreover, \(E_{12}\approx C\) when \(V_{\infty}=\infty\) and \(E_{13}\approx C\) when \(V_{\infty}<\infty\), where C is the best constant in (27).
Declarations
Acknowledgements
The authors would like to thank Professor Ryskul Oinarov for his generous suggestions which have improved this paper. The paper was written under financial support by the Scientific Committee of the Ministry of Education and Science of Kazakhstan, Grant No. 5495/GF4 on priority area ‘Intellectual potential of the country’.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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