One-dimensional discrete Hardy and Rellich inequalities on integers

In this paper, we consider a weighted version of one-dimensional discrete Hardy inequalities with power weights of the form $n^\alpha$. We prove the inequality when $\alpha$ is an even natural number with the sharp constant and remainder terms. We also find explicit constants in standard and weighted Rellich inequalities and its higher order versions. As a by-product of this work we derive a combinatorial identity using purely analytic methods. This suggests a correlation between combinatorial identities and functional identities.


Introduction
The classical Hardy inequality on the positive half-line reads as for u P C 8 0 p0, 8q, the space of smooth and compactly supported functions.This inequality first appeared in Hardy's proof of Hilbert's theorem [9] and then in the book [10].It was later extended to higher order derivatives by Birman [3]: Let k P N and u P C 8 0 p0, 8q, then where u pkq denotes the k th derivative of u and p2k ´1q!! " p2k ´1qp2k ´3qp2k ´5q . . . 3 ¨1.Inequality (1.2) for k " 2 is referred to as Rellich inequality.Note that constants in (1.1) and (1.2) are sharp, that is, these inequalities fail to hold true for a strictly bigger constant.
The main goal of this paper is to study a discrete analogue of (1.1) and (1.2) as well as their weighted versions on integers.A well known discrete variant of (1.1) states: Let N 0 denotes the set of non-negative integers and u : N 0 Ñ R be a finitely supported function with up0q " 0. Let Dupnq :" upnq ´upn ´1q denote the first order difference operator on N 0 .Then (1. 3) The constant in (1.3) is sharp.This inequality was developed alongside the integral inequality (1.1) during the period 1906-1928: [19] contains many stories and contributions of other mathematicians such as E. Landau, G. Polya, I. Schur and M. Riesz in the development of Hardy inequality.We would also like to mention some recent proofs of Hardy inequality (1.3) [7,13,17,18,20] as well as [2, 4, 5, 8, 11, 14-16, 22, 23], where various variants of (1.3) have been studied and applied: extensions of (1.3) to higher dimensional integer lattice, on combinatorial trees, general weighted graphs, etc.
In this paper, we are concerned with an extension of inequality (1.3) in two directions.First, we consider a weighted version of (1.3) with power weights n α : for some positive constant c.We prove inequality (1.4) with the sharp constant and furthermore improve it by adding lower order remainder terms in the RHS.This is done when α is a nonnegative even integer.This problem has been studied previously: in [21], (1.4) was proved when α P p0, 1q and recently it was extended to α ą 5 in [8].In this paper, we provide a new method to prove these inequalities, which extends and improves previously known results.
Secondly, we consider the higher order versions of inequality (1.3), in which the "discrete derivative" on the LHS of (1.3) is replaced by higher order operators.In other words, we prove a discrete analogue of inequalities (1.2).In particular, we find a constant cpkq in the following Rellich inequality for finitely supported functions on non-negative integers: where the second order difference operator on N 0 , called Laplacian is given by ∆upnq :" This is a well known discrete analogue of second order derivative.Inequality (1.5) has been considered in the past in a general setting of graphs [12,15].In these papers, authors developed a general theory to tackle problems of the kind (1.5), however; one cannot deduce the Rellich inequality (1.5) from their general theory.To the author's best knowledge, this is the first time an explicit constant has been computed in the discrete Rellich inequality (1.5).We also prove inequality (1.5) with weights n 2k , for positive integers k.The constant obtained is asymptotically sharp as k Ñ 8.
As a side product, we discovered a surprising connection between functional and combinatorial identities.Using purely analytic methods, we managed to prove a non-trivial combinatorial identity, whose appearance in the context of discrete Hardy-type inequalities seems mysterious.This connection will be explained in Sections 3 and 6.We hope that the analytic method presented here might lead to the discovery of new combinatorial identities.
The paper is structured as follows: In Section 2, we state the main results of the paper.In Section 3, we prove auxiliary results using which we prove our main results in Sections 4 and 5.In Section 6, we prove a combinatorial identity using lemmas proved in Section 3. Finally we conclude the paper with an appendix A.
1.1 Remark.For the convenience of reader we would recommend that reader should read Sections 4 and 5 before reading Section 3 to get a better understanding of ideas involved and origin of the lemmas proved in Section 3.

Hardy inequalities
2.1 THEOREM (Improved weighted Hardy inequalities).Let u P C c pZq, the space of finitely supported functions, and also assume up0q " 0. Then for k P N, we have where the non-negative constants γ k i are given by Here Γpxq denotes the Gamma function and `x y ˘:" Moreover, the constant p2k ´1q 2 {4 is sharp.
We would like to mention that inequality (2.3) was proved in paper [22] with the weight pn´1{2q α and α P p0, 1q.Note that the above inequalities reduce to corresponding Hardy inequalities on non-negative integers N 0 , when we restrict ourselves to functions u taking value zero on negative integers.
Using the method used in the proofs of above Hardy inequalities, we also managed to prove higher-order versions of the Hardy inequality, in-particular we prove a discrete Rellich inequality which has been missing from the current literature.

Higher Order Hardy inequalities
for all u P C c pN 0 q with upiq " 0 for 0 ď i ď 2m ´1, and for all u P C c pN 0 q with upiq " 0 for 0 ď i ď 2m.Here Du and ∆u denotes the first and second order difference operators on N 0 respectively.
Theorem 2.3 is a discrete analogue of inequalities of Birman (1.2).Inequality (2.4) for m " 1 gives Rellich inequality: 2.4 COROLLARY (Rellich inequality).Let u P C c pN 0 q and up0q " up1q " 0. Then we have (2.6) 2.5 Remark.It is worthwhile to notice that in Theorem 2.3 the number of zero conditions on the function u equals the order of the operator.Whether the number of zero conditions are optimal or not is not clear to us.Furthermore, we don't believe the constants obtained in Theorem 2.3 are sharp.There seems to be a lot of room for the improvement in the constants, though it is not clear how to get better explicit bounds.
Finally, we obtain explicit constants in weighted versions of higher order Hardy Inequalities.For functions u : Z Ñ R we define the first and second order difference operators on Z analogously: Dupnq :" upnq ´upn ´1q and ∆upnq :" 2upnq ´upn ´1q ´upn `1q.

THEOREM (Power weight higher order Hardy inequalities).
Let m ě 1 and u P C c pZq with up0q " 0. Let Du and ∆u denote the first and second order difference operators on Z. Then for k ě 2m and where Cpkq is given by Cpkq :" kpk ´1qpk ´3{2q 2 . (2.9) 2.7 Remark.By taking n β as test functions in the inequalities (2.7) and (2.8) it can be easily seen that the sharp constants in these inequalities are of the order Opk 4m q and Opk 4m`2 q respectively.Therefore constants obtained in Theorem 2.6 are asymptotically sharp as k Ñ 8.
3 Some Auxiliary Results 3.1 LEMMA.Let u P C 8 pr´π, πsq.Furthermore, assume that derivatives of u satisfy d k up´πq " d k upπq for all k P N 0 .For every k P N we have where and Proof.Using the Leibniz product rule for the derivative we get Integrating both sides, we obtain Let 0 ď i ă j and Ipi, jq :"Re ş π ´π d i upxqd j upxqd k´i sinpx{2qd k´j sinpx{2q.Applying integration by parts iteratively, we get Ipi, jq " Re ż π ´π d i upxqd j upxqd k´i sinpx{2qd k´j sinpx{2q " where C i,j σ is given by and w ij pxq :" d k´i sinpx{2qd k´j sinpx{2q.1 Using (3.6) in (3.5), we see that since the derivatives which appear in the expression of Ipi, jq are of order between i and t i`j 2 u.Observing that the terms which contributes to D i are of the form Ipi ´m, i `nq with the condition m ď n, we get the following expression for D i pxq: where C i,i i pxq :" 1 2 |d k´i sinpx{2q| 2 .
It can be checked that for non-negative integers l, d l w ij pxq P tsin 2 px{2q, cos 2 px{2q, cos x, sin xu (with some multiplicative constant).Thus D i pxq is a linear combination of sin 2 px{2q, cos 2 px{2q, cos x and sin x.Namely, we have Note that sin 2 px{2q can appear in the expression of D i iff w i´m,i`n is a multiple of sin 2 px{2q and m " n.Further, observing that w i´m,i`m is a multiple of sin 2 px{2q iff k ´i `m is even, we get where Similarly, cos 2 px{2q can appear in the expression of D i iff w i´m,i`n is a multiple of cos 2 px{2q and m " n, and w i´m,i`m is a multiple of cos 2 px{2q iff k ´i `m is odd.Therefore we have Let us compute the coefficient of sin x in D i .Observe that sin x can appear in D i in two different ways; first, when either w i´m,i`n is a multiple of sin 2 px{2q or cos 2 px{2q and n ´m is odd; secondly, when w i´m,i`n is a multiple of sin x and n ´m is even.Further, observing that w i´m,i`n is a multiple of sin 2 px{2q or cos 2 px{2q iff n ´m is even and w i´m,i`n is a multiple of sin x iff n ´m is odd implies that C i 4 " 0.
After computing C i 1 , C i 2 and C i 4 , it's not hard to see that (3.10) Simplifying further, we find that Simplifying further we obtain Using the expression of C i 3 from (3.11), we get (3.12) In the last step we used Chu-Vandermonde Identity: `m`n r ˘" r ř i"0 `m i ˘`n r´i ˘with as change of variable.
3.2 LEMMA.Let u be a function satisfying the hypothesis of Lemma 3.1.Furthermore, assume that u has zero average, that is ş π ´π udx " 0. Then we have Proof.Let wpxq :" 1  4 secpx{2q.Expanding the square we obtain Fix ǫ ą 0. Doing integration by parts, we obtain Using ´w2 `pw sinpx{2qq 1 " 1 16 sec 2 px{2q ě 1{16 above, we obtain Using periodicity of u along with the first order taylor expansion of u around π and ´π, one can easily conclude that B.T. goes to 0 as ǫ goes to 0. Now taking limit ǫ Ñ 0 on both sides of (3.17) and using dominated convergence theorem, we obtain i , which will be proved in Section 6.
The next two lemmas are weighted versions of Lemmas 3.2 and 3.4 and will be used in proving the higher order Hardy inequalities.
3.5 LEMMA.Let u be a function satisfying the hypotheses of Lemma 3.1.Furthermore, assume that ş π ´π u sin 2k´2 px{2qdx " 0. For k ě 1, we have Proof.Let w :" Proof.We begin with the observation that although f " u sinpx{2q does not satisfy the hypothesis d k f p´πq " d k f pπq of Lemma 3.1, identity (3.1) still holds for f .In the proof of Lemma 3.1, the periodicity of derivatives is only used in the derivation of (3.6); to make sure that no boundary term appears while doing integration by parts.The key observation is that d i f p´πq " ´di f pπq, which imply that d i f p´πqd j f p´πq " d i f pπqd j f pπq.This makes sure no boundary terms appears while performing integration by parts in (3.6) for the function f .First using identity (3.1) for u sinpx{2q and then for u, along with non-negativity of the constants α k i and β k i (will be proved in Section 6) we obtain Last inequality uses Lemma 3.2.

Proof of Hardy inequalities
Proof of Theorem 2.1.Let u P ℓ 2 pZq, we define its Fourier transform Fpuq P L 2 pp´π, πqq as follows: Fpuqpxq :" p2πq ´1 where γ k i :" 4α k k´i `1 4 β k k´i`1 .In the last step we used the classical Hardy inequality.In Section 6 we simplify the expressions of α k i and β k i , which will complete the proof of Theorem 2.1.
Proof of Corollary 2.2.Assuming γ k i ě 0 (which will be proved in Section 6) Theorem 2.1 immediately implies It can be checked that ξ k k´1 " ´kpk`1q.Using this in the expression of γ k 1 , we find that γ k 1 " p2k´1q 2

4
. Next, we prove the sharpness of the constant γ k 1 .Let C be a constant such that for all u P C c pZq.
Let N P N, β P R and α ě 0 be such that 2β `2k ´2 ă ´1.Consider the following family of finitely supported functions on Z.
Some basic estimates: Using the above in (4.7), we get Using estimates (4.6) and (4.8) in (4.5) and taking limit N Ñ 8, we get Finally, taking limit β Ñ 1´2k 2 on the both sides, we obtain This proves the sharpness of γ k 1 .
Let u P C c pN 0 q with upiq " 0 for all 0 ď i ď 2m ´1.We define v P C c pZq as vpnq :" It is quite straightforward to check that the condition (5.3) is trivially satisfied.Now applying inequality (5.2) to the above defined function v, we obtain ( This proves the inequality (2.4).Inequality (2.5) can be proved in a similar way, by following the proof of (2.4) step by step.
Proof of Theorem 2.6.First we prove inequality (2.7).We begin by proving the result for m " 1 and then apply the result for m " 1 iteratively to prove it for general m.Using inversion formula and integration by parts, we obtain upnqn k´2 " p2πq (5.7) In the last line we used α k k´1 " kpk ´1q and β k k " 1 (see (3.2)-(3.4)).Now applying the inequality (5.7) inductively completes the proof of inequality (2.7).For the proof of inequality (2.8), we first apply inequality (2.3) and then inequality (2.7).

Combinatorial identity
In this section, we prove a combinatorial identity using the Lemma 3.1.This develops a very nice connection between combinatorial identities and functional identities.We believe that the method we present here can be used to prove new combinatorial identities which might be of some value.From the above expressions, it is quite straightforward that the above constants are nonnegative, thus justifying the assumptions used in the proofs of Lemma 3.4, lemma 3.6 and Corollary 2.2.Finally, the expression of γ k i along with (4.4) completes the proof of Theorem 2.1.

A Appendix
We prove identity (3.6) with w ij replaced with an arbitrary smooth 2π periodic funcition.
The base cases j ´i P t1, 2, 3u can be checked by hand (it's a consequence of iterative integration by parts).

6. 1
THEOREM.Let k P N and 0