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A new discrete Hardytype inequality with kernels and monotone functions
Journal of Inequalities and Applications volume 2015, Article number: 321 (2015)
Abstract
A new discrete Hardytype 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.
Introduction
Hardy’s famous inequality reads
where \(f(x)\) is a pintegrable nonnegative function on \((0,\infty)\). The constant \((\frac{p}{p1} )^{p}\) is sharp. Hardy stated his inequality in 1920 (see [1]) and finally proved it in 1925 (see [2]). Since then there has been an enormous development of this inequality to what today is called Hardytype inequalities. One important reason is that this theory is very important for various applications both in mathematics and the technical sciences. See e.g. [3–5] and the references given there. Let us just mention some important steps in this development. One central problem was to characterize the weights \(u(x)\) and \(v(x)\) so that the more general Hardytype inequality
holds for various parameters p and q, and to also have some good estimates of the best constant C (the operator norm). Here and in the sequel C does not depend on f. This problem is nowadays solved for most cases (see e.g. the books mentioned above). However, for various applications it is also important to consider the mapping properties of the more general operator \(H_{k}(f)(x):=\int_{0}^{x}k(x,t)f(t)\,dt\), where \(k(x,t)\) is a kernel (a nonnegative and measurable function on \(\{(x,t):0\leq x<\infty, 0<t\leq x\}\)). In applications \(k=k(x,t)\) is sometimes called the Greenkernel or unit impulse answer when solving technical problems modeled by linear differential equations. In the case with weighted Lebesgue spaces this problem is equivalent to study the inequality (2) with \(H_{1}\) replaced by \(H_{k}\). It is still a wellknown open problem to characterize (2) with \(H_{1}\) replaced by \(H_{k}\) without restrictions on k. However, the problem is more or less solved for some special cases, e.g. the following:

(a)
\(k(x,t)\equiv1\) (the classical case; see e.g. [3, 4] and [5]);

(b)
\(k(x,t)\) is of product type (see e.g. [5] and the references therein);

(c)
\(k(x,t)\) satisfies the socalled Oinarov condition (see e.g. [3], Chapter 2, especially p.77);

(d)
\(k(x,t)\) satisfies some generalized Oinarov condition (see [6] and [7]).
It is also of great interest to investigate all the problems above when the inequalities are studied only on the cone of nondecreasing 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 nonnegative functions.
There has been a similar development for Hardytype 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.
The discrete case  formulation of the main result
The discrete form of Hardy’s inequality reads
where \(f=\{f_{k}\}\) is any sequence of nonnegative numbers. The constant \((\frac{p}{p1} )^{p}\) is sharp also in this case. By applying (1) with stepfunctions we obtain (3). The discrete inequality corresponding to (2) reads
where \(u=\{u_{n}\}\) and \(v=\{v_{n}\}\) are sequences of nonnegative numbers. Also in this case the characterization of (4) is known for almost all reasonable parameters p and q (see e.g. [4] and the references therein). However, in this case the result does not follow automatically from the corresponding result in the continuous case but must be proved separately. Also in this case it is important to study the more general case when the operator \(H(f)(n)=\sum_{k=1}^{n}f_{k}\) is replaced by a more general discrete kernel (matrix) operator \(H_{w}(f)(n)=\sum_{k=1}^{n}w_{n,k}f_{k}\), where \(w_{n,k}\), \(1\leq n\leq\infty\), \(1\leq k\leq n\), are nonnegative numbers. The case when \(\{f_{k}\}\) is a nondecreasing sequence is of special interest in this case too and gives different characterizations. The corresponding result to ‘Sawyer duality principle’ for this discrete case was proved in [10]. This result is important for our investigations in this paper. Also in this discrete case it is an open question how to characterize (4) with H replaced by \(H_{w}\) without any further restrictions on the matrix sequence \(\{w_{n,k}\}\). However, several results corresponding to (a)(d) in the continuous case are known. So, a characterization corresponding to (d) for the case \(1< p\leq q<\infty\) was proved in [11]. A characterization for the case \(1< q< p<\infty \) is an open problem. However, we can consider the setting corresponding to some subclass of (d) that is as follows:
We consider the operator
and call it a nmultiple discrete Hardy operator with weights.
Let us notice that if in (5) we change the order of summation, then the operator \(S_{n1}\) can be rewritten in the form
where
for \(n>1\) and \(A_{n1,1}(i, j)=1\) for \(n=1\).
One main problem is to find necessary and sufficient conditions on the weight sequences \(\omega_{i}\), \(i=1, 2,\ldots, n1\), u and v for the validity of the inequality
for nonnegative nonincreasing sequences \(f=\{f_{k}\}_{k=1}^{\infty}\) in the case \(1< q< p<\infty\).
When \(n=1\) the operator \(S_{n1}\) 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 nonnegative 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 nonincreasing we refer to the important paper [14] and the references given there.
Concerning the general inequality (6) the following is known:
 (α):

for the case when \(\{f_{k}\}\) is a nonnegative sequence a characterization of (6) for the case \(1< p\leq q< \infty\) was derived in [15] while a corresponding characterization for the case \(1< q< p<\infty\) can be found in [16];
 (β):

for the case when \(\{f_{k}\}\) is a nonincreasing sequence a characterization of (6) for the case \(1< p\leq q< \infty\) was derived in [17].
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
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 (6) holds for all nonnegative nonincreasing sequences \(f=\{f_{i}\}_{i=1}^{\infty}\) from \(l_{p, v}\) if and only if \(E_{12}=\max\{E_{1},E_{2}\}<\infty\) when \(V_{\infty}=\infty\) and \(E_{13}=\max\{E_{1},E'_{2},E_{3}\}<\infty\) when \(V_{\infty}<\infty\), where
and
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).
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 \(A_{l, m}(i, j)=\sum_{k_{l}=j}^{i}\omega_{l, k_{l}} \sum_{k_{l1}=k_{l}}^{i}\omega_{l1,k_{l1}}\cdots \sum_{k_{m}=k_{m+1}}^{i}\omega_{m, k_{m}}\) for \(n1\geq l\geq m\geq1\). 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
For all i, j, τ: \(1\leq j\leq\tau\leq i\) the estimates
hold when \(n1\geq l\geq m\geq1\).
We also need the following discrete analog of the ‘Sawyer duality principle’, which was proved in [10].
Theorem A
Let \(1< p, q<\infty\). Let \((a_{i,j})\) be a triangle matrix with \(a_{i,j}\geq0\) when \(i\geq j\geq 1\) and \(a_{i,j}=0\) when \(i< j\). Let \(V_{k}=\sum_{i=1}^{k} v^{p}_{i}\) when \(k\geq1\) and \(V_{\infty}=\lim_{k\rightarrow\infty }V_{k}\). Then the inequality
for all nonnegative nonincreasing sequences \(f=\{f_{i}\}_{i=1}^{\infty}\) from \(l_{p, v}\) is equivalent to the inequality
for all nonnegative sequences \(g=\{g_{i}\}_{i=1}^{\infty}\) when \(V_{\infty}=\infty\), and it is equivalent to the inequality
for all nonnegative sequences \(g=\{g_{i}\}_{i=1}^{\infty}\) when \(V_{\infty}<\infty\).
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 wellknown result (see e.g. [4], p.58).
Theorem B
Let \(1< q< p<\infty\). Let \(\{\alpha_{i}\}_{i=1}^{\infty}\) be a nonnegative sequence. Then the inequality
holds for all nonnegative sequences \(f=\{f_{i}\}_{i=1}^{\infty}\) from \(l_{p, v}\) if and only if
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 nonnegative 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.
The second theorem studies inequality (6) for the dual operator \(S^{*}_{n1}\) to the operator \(S_{n1}\) defined by
Theorem C
Let \(1< q< p<\infty\) and \(n\geq1\). Then inequality (6) holds for all nonnegative sequences \(f=\{f_{i}\}_{i=1}^{\infty}\) from \(l_{p, v}\) if and only if
where \(\Delta^{+}_{i}E_{j,i}=E_{j,i}E_{j,i+1}\).
Moreover, \(A(n)\approx C\), where C is the best constant in (6).
Theorem D
Let \(1< q< p<\infty\) and \(n\geq1\). Then the inequality
holds for all nonnegative sequences \(f=\{f_{i}\}_{i=1}^{\infty}\) from \(l_{p, v}\) if and only if
Moreover, \(B(n)\approx C\), where C is the best constant in (17).
Proof of Theorem 2.1
We start our proof from the case \(V_{\infty}=\infty\).
Assuming \(a_{i,j}\equiv A_{n1,1}(i,j)\), by Theorem A we see that inequality (6) for all nonnegative nonincreasing sequences \(f=\{f_{i}\}_{i=1}^{\infty}\) from \(l_{p, v}\) is equivalent to the inequality
for all nonnegative sequences \(g=\{g_{i}\}_{i=1}^{\infty}\).
Denote
It is obvious that
Since \(A_{n1,1}(i,j)\) and \(g_{i}\) are nonnegative, we can change the order of summation in both terms:
Let us, respectively, denote
where \(\widetilde{A}_{i}=\sum_{j=1}^{i} A_{n1,1}(i,j)\), and
where \(\widetilde{A}_{n1,1}(i,k)=\sum_{j=1}^{k} A_{n1,1}(i,j)\).
Next, we will work with \(\widetilde{A}_{n1,1}(i,k)\). Since \(i\geq k\geq j\), then from (11) we see that
Substitute it in \(\widetilde{A}_{n1,1}(i,k)\) and we find that
where \(b_{k,r}=\sum_{j=1}^{k}A_{n1,r+1}(k,j)\).
Therefore, the validity of inequality (18) is equivalent to the simultaneous validity of the following inequalities:
and
where \(0\leq r\leq n1\). Moreover, if \(\widetilde{C}_{2}=\max_{0\leq r\leq n1}\widetilde{C}_{2,r}\), then
It is obvious that inequality (19) is a standard Hardy inequality. Hence, according to Theorem B, inequality (19) holds if and only if \(E_{2}<\infty\). Moreover, we take into account the following fact:
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).
Let us turn to the case \(V_{\infty}<\infty\). The difference from the previous situation is that by Theorem A inequality (6) holds if and only if (18) holds with some other constant \(\overline{C}_{1}\) instead of C̃, and
simultaneously holds for all nonnegative sequences \(g=\{g_{i}\} _{i=1}^{\infty}\). Moreover,
By arguing as in the previous case, we see that inequality (18) holds if and only if the conditions \(E_{1}<\infty\) and \(E'_{2}<\infty\) hold. Let us notice that the difference of the expressions \(E_{2}\) and \(E'_{2}\) is only in the fact that we need to take into account the relation
instead of the relation (22).
Next, we study inequality (23). Since \(A_{n1,1}(i,j)\) and \(g_{i}\) are nonnegative, we can change the order of summation in the first bracket of (23) and obtain
If we apply the reverse Hölder inequality to the last expression, we get
Therefore, inequality (23) holds if and only if \(E_{3}<\infty\). Moreover, by Theorem B, Theorem D and (24), we see that \(E_{13}=\max\{E_{1},E'_{2},E_{3}\}\approx C\), where C is the best constant in (6). The proof of Theorem 2.1 is complete.
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 \((H,n1)\) defined as follows:
The series \(\sum_{j=1}^{\infty}f_{j}\) is summable by Hölder’s \((H,n1)\) method with the sum S if \(\lim_{i\rightarrow\infty }(H_{n1}f)_{i}=S\), where
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.
In [17] it was shown that \(H_{n1}\) can be presented in the following form:
If \(w_{m,k_{m}}\equiv\frac{1}{k_{m}}\) for \(1\leq m\leq n2\) and \(w_{n1,k_{n1}}\equiv1\), then we see that the operator \(H_{n1}\) is a particular case of our operator \(S_{n1}\), namely:
If we change the order of summation, then we find that for this particular operator \(S_{n1}\) the summation matrix \(A_{n1,1}\) has the following form:
Therefore, we can consider the following inequality:
which is equivalent to inequality (6), where, in view of (25), \(u_{i}=\frac{\widetilde{u}_{i}}{i}\). Thus, according to Theorem C and Theorem 2.1 we have the following results.
Theorem 5.1
Let \(1< q< p<\infty\) and \(n\geq1\). Then inequality (27) holds for all nonnegative sequences \(f=\{f_{i}\}_{i=1}^{\infty}\) from \(l_{p, v}\) if and only if \(A(n)<\infty\) (see (16)), where \(A_{n1,1}\) 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 nonnegative nonincreasing 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 \(A_{n1,1}\) 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).
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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’.
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All three authors have on an equal level discussed and posed the research questions in this paper. AK has substantially helped to prove the main results and to type the manuscript. AT is the main author concerning the proofs of the main results and typing of the manuscript. LEP has put the results into a more general frame in the introduction and instructed how to write the paper in this final form. All authors read and approved the final manuscript.
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Kalybay, A., Persson, LE. & Temirkhanova, A. A new discrete Hardytype inequality with kernels and monotone functions. J Inequal Appl 2015, 321 (2015). https://doi.org/10.1186/s1366001508439
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DOI: https://doi.org/10.1186/s1366001508439
MSC
 26D10
 26D15
 39B82
Keywords
 inequality
 Hardytype inequality
 kernel
 matrix operator
 monotone sequence
 Oinarov condition