Abstract
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(with weights \(n^\alpha \)) which are asymptotically sharp as \(\alpha \rightarrow \infty \). As a by-product of this work we derive a combinatorial identity using purely analytic methods, which suggests a plausible correlation between combinatorial and functional identities.
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1 Introduction
The classical Hardy inequality on the positive half-line reads as
for \(u \in C_0^\infty (0, \infty )\), 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 \(k \in {\mathord {{\mathbb {N}}}}\) and \(u \in C_0^\infty (0, \infty )\), then
where \(u^{(k)}\) denotes the \(k^{th}\) derivative of u and \((2k-1)!! = (2k-1)(2k-3)(2k-5) \dots 3 \cdot 1\). 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 \({\mathord {{\mathbb {N}}}}_0\) denotes the set of non-negative integers and \(u: {\mathord {{\mathbb {N}}}}_0 \rightarrow {\mathord {{\mathbb {R}}}}\) be a finitely supported function with \(u(0)=0\). Let \(Du(n):= u(n)-u(n-1)\) denote the first order difference operator on \({\mathord {{\mathbb {N}}}}_0\). Then
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, 14,15,16, 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 \(n^\alpha \):
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 \(\alpha \) is a non-negative even integer. This problem has been studied previously: in [21], (1.4) was proved when \( \alpha \in (0,1)\) and recently it was extended to \(\alpha > 5\) 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 c(k) in the following Rellich inequality for finitely supported functions on non-negative integers:
where the second order difference operator on \({\mathord {{\mathbb {N}}}}_0\), called \({\textbf {Laplacian}}\) is given by
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 \(k \rightarrow \infty \).
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 Sects. 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 Sect. 2, we state the main results of the paper. In Sect. 3, we prove auxiliary results using which we prove our main results in Sects. 4 and 5. In Sect. 6, we prove a combinatorial identity using lemmas proved in Sect. 3. Finally we conclude the paper with an appendix 1.
Remark 1.1
For the convenience of reader we would recommend that reader should read Sects. 4 and 5 before reading Sect. 3 to get a better understanding of ideas involved and origin of the lemmas proved in Sect. 3.
2 Main Results
2.1 Hardy Inequalities
Theorem 2.1
(Improved weighted Hardy inequalities) Let \(u \in C_c(\mathbb {Z})\), the space of finitely supported functions, and also assume \(u(0)=0\). Then for \(k \in {\mathord {{\mathbb {N}}}}\), we have
where the non-negative constants \(\gamma _i^k\) are given by
Here \(\Gamma (x)\) denotes the Gamma function and \({x \atopwithdelims ()y}:= \frac{\Gamma (x+1)}{\Gamma (x-y+1) \Gamma (y+1)}\) denotes the binomial coefficient.
Dropping the remainder terms in inequality (2.1) gives the following weighted Hardy inequalities:
Corollary 2.2
(Weighted Hardy inequalities) Let \(u \in C_c(\mathbb {Z})\) and \(u(0)=0\). Then for \(k \in {\mathord {{\mathbb {N}}}}\), we have
Moreover, the constant \((2k-1)^2/4\) is sharp.
We would like to mention that inequality (2.3) was proved in paper [22] with the weight \((n-1/2)^\alpha \) and \(\alpha \in (0,1)\). Note that the above inequalities reduce to corresponding Hardy inequalities on non-negative integers \(\mathbb {N}_0\), 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.
2.2 Higher Order Hardy Inequalities
Theorem 2.3
(Higher order Hardy inequalities) Let \(m\in \mathbb {N}\). Then we have
for all \(u \in C_c(\mathbb {N}_0)\) with \(u(i)=0\) for \(0\le i \le 2\,m-1\), and
for all \(u \in C_c(\mathbb {N}_0)\) with \(u(i)=0\) for \(0\le i \le 2\,m\). Here Du and \(\Delta u\) denotes the first and second order difference operators on \({\mathord {{\mathbb {N}}}}_0\) respectively.
Theorem 2.3 is a discrete analogue of inequalities of Birman (1.2). Inequality (2.4) for \(m=1\) gives Rellich inequality:
Corollary 2.4
(Rellich inequality) Let \(u \in C_c(\mathbb {N}_0)\) and \(u(0)=u(1)=0\). Then we have
Remark 2.5
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 \(u: {\mathord {{\mathbb {Z}}}}\rightarrow {\mathord {{\mathbb {R}}}}\) we define the first and second order difference operators on \({\mathord {{\mathbb {Z}}}}\) analogously: \(Du(n):= u(n)-u(n-1)\) and \(\Delta u(n):= 2u(n)-u(n-1)-u(n+1)\).
Theorem 2.6
(Power weight higher order Hardy inequalities) Let \(m \ge 1\) and \(u \in C_c(\mathbb {Z})\) with \(u(0)=0\). Let Du and \(\Delta u\) denote the first and second order difference operators on \({\mathord {{\mathbb {Z}}}}\). Then
for \(k \ge 2m\) and
for \(k \ge 2m+1\), where C(k) is given by
Remark 2.7
By taking \(n^\beta \) 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 \(O(k^{4m})\) and \(O(k^{4m+2})\) respectively. Therefore constants obtained in Theorem 2.6 are asymptotically sharp as \(k \rightarrow \infty \).
3 Some Auxiliary Results
Lemma 3.1
Let \(u \in C^\infty ([-\pi , \pi ])\). Furthermore, assume that derivatives of u satisfy \(d^k u(-\pi ) = d^ku(\pi )\) for all \(k \in \mathbb {N}_0\). For every \(k \in \mathbb {N}\) we have
where
and
Proof
Using the Leibniz product rule for the derivative we get
Integrating both sides, we obtain
Let \(0 \le i<j\) and \(I(i,j):=\)Re \(\int _{-\pi }^{\pi } d^i u(x) \overline{d^ju(x)} d^{k-i}\sin (x/2) d^{k-j}\sin (x/2)\). Applying integration by parts iteratively, we get
where \(C_{\sigma }^{i,j}\) is given by
and \(w_{ij}(x):= d^{k-i}\sin (x/2) d^{k-j}\sin (x/2)\).Footnote 1
Using (3.6) in (3.5), we see that
since the derivatives which appear in the expression of I(i, j) are of order between i and \(\lfloor \frac{i+j}{2}\rfloor \). Observing that the terms which contributes to \(D_i\) are of the form \(I(i-m,i+n)\) with the condition \(m\le n\), we get the following expression for \(D_i(x)\):
where \(C_i^{i,i}(x):= \frac{1}{2}|d^{k-i}\sin (x/2)|^2\).
It can be checked that for non-negative integers l, \(d^l w_{ij}(x) \in \{\sin ^2(x/2), \cos ^2(x/2), \cos x, \sin x \}\) (with some multiplicative constant). Thus \(D_i(x)\) is a linear combination of \(\sin ^2(x/2), \cos ^2(x/2), \cos x\) and \(\sin x\). Namely, we have
Note that \(\sin ^2(x/2)\) can appear in the expression of \(D_i\) iff \(w_{i-m,i+n}\) is a multiple of \(\sin ^2(x/2)\) and \(m=n\). Further, observing that \(w_{i-m,i+m}\) is a multiple of \(\sin ^2(x/2)\) iff \(k-i+m\) is even, we get
where \(\begin{array}{rl}\delta _i := 1&{} \hbox { if }k-i\hbox { is even}\\ 0 &{} \hbox { if }k-i\hbox { is odd}\\ \end{array}\)
Similarly, \(\cos ^2(x/2)\) can appear in the expression of \(D_i\) iff \(w_{i-m,i+n}\) is a multiple of \(\cos ^2(x/2)\) and \(m=n\), and \(w_{i-m,i+m}\) is a multiple of \(\cos ^2(x/2)\) 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 \(\sin ^2(x/2)\) or \(\cos ^2(x/2)\) 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 \(\sin ^2(x/2)\) or \(\cos ^2(x/2)\) iff \(n-m\) is even and \(w_{i-m,i+n}\) is a multiple of \(\sin x\) iff \(n-m\) is odd implies that \(C_4^i=0\).
After computing \(C_1^i, C_2^i\) and \(C_4^i\), it’s not hard to see that
Simplifying further, we find that \(D_i(x) = (C_2^i+C_3^i) + (C_1^i - C_2^i -2C_3^i)\sin ^2(x/2)\). Next we simplify the constants \((C_2^i+C_3^i)\) and \((C_1^i - C_2^i -2C_3^i)\). Let
and consider
Simplifying further we obtain
Using the expression of \(C_3^i\) from (3.11), we get
In the last step we used Chu-Vandermonde Identity: \({m+n \atopwithdelims ()r } = \sum \limits _{i=0}^r {m \atopwithdelims ()i}{n \atopwithdelims ()r-i}\) with as change of variable. \(\square \)
Lemma 3.2
Let u be a function satisfying the hypothesis of Lemma 3.1. Furthermore, assume that u has zero average, that is \(\int _{-\pi }^\pi u dx = 0\). Then we have
Remark 3.3
Inequality (3.13) is an improvement of well known Poincaré-Friedrichs inequality in dimension one [10, Theorem 258]:
since \(\sin ^2(x/2) \le 1\).
Proof
Let \( w(x):= \frac{1}{4}\sec (x/2)\). Expanding the square we obtain
Fix \(\epsilon >0\). Doing integration by parts, we obtain
where the boundary term B.T. is given by
Therefore we have
Using \(-w^2 + (w\sin (x/2))' = \frac{1}{16} \sec ^2(x/2) \ge 1/16\) above, we obtain
Using periodicity of u along with the first order taylor expansion of u around \(\pi \) and \(-\pi \), one can easily conclude that B.T. goes to 0 as \(\epsilon \) goes to 0. Now taking limit \(\epsilon \rightarrow 0\) on both sides of (3.17) and using dominated convergence theorem, we obtain
\(\square \)
Lemma 3.4
Let u be a function satisfying the hypotheses of Lemma 3.2. For \(k \in \mathbb {N}\), the following holds
where \(\alpha _i^k\) and \(\beta _i^k\) are as defined in (3.2) and (3.3) respectively.
Proof
Let \(f = d^{i-1} u\). Applying Lemma 3.2 to f we get
Using (3.19) in (3.1) and using \(\alpha _k^k =0\) gives the desired estimate (3.18). Note that in proving (3.18) we have assumed the non-negativity of the constants \(\beta _i^k\), which will be proved in Sect. 6. \(\square \)
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.
Lemma 3.5
Let u be a function satisfying the hypotheses of Lemma 3.1. Furthermore, assume that \(\int _{-\pi }^\pi u \sin ^{2k-2}(x/2) dx = 0\). For \(k \ge 1\), we have
Proof
Let \(w:= \frac{1}{4}\sin ^{k-1}(x/2)\sec (x/2)\). Expanding the square, we obtain
Now integrating over \((-\pi + \epsilon , \pi -\epsilon )\) for a fixed \(\epsilon >0\), we get
Finally, using integrating by parts, we obtain
where the boundary term B.T. is given by
Now using \((w\sin ^{k}(x/2))'- w^2 = \frac{1}{16} \sin ^{2k-2}(x/2) \big (\sec ^2(x/2) + 4k-4\big ) \ge \frac{4k-3}{16} \sin ^{2k-2}(x/2)\), we arrive at
Now taking limit \(\epsilon \rightarrow 0\) on both sides of (3.23) and using dominated convergence theorem, we obtain
\(\square \)
Lemma 3.6
Suppose u satisfies the hypotheses of Lemma 3.2. Further, assume that u has zero average. For \(k \ge 2\), we have
Proof
We begin with the observation that although \(f=u \sin (x/2)\) does not satisfy the hypothesis \(d^k f(-\pi ) = d^k f(\pi )\) 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 \(d^i f(-\pi ) = -d^i f (\pi )\), which imply that \(d^i f(-\pi ) \overline{d^j f (-\pi )} = d^i f(\pi ) \overline{d^j f (\pi )}\). 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 \(u\sin (x/2)\) and then for u, along with non-negativity of the constants \(\alpha _i^k\) and \(\beta _i^k\) (will be proved in Sect. 6) we obtain
The last inequality uses Lemma 3.2. \(\square \)
4 Proof of Hardy Inequalities
Proof of Theorem 2.1
Let \(u \in \ell ^2(\mathbb {Z})\), we define its Fourier transform \({\mathcal {F}}(u)\in L^2((-\pi , \pi ))\) as follows:
Let \(1 \le j \le k\). Using the inversion formula for Fourier transform and integration by parts, we get
Applying Parseval’s Identity gives us
Similarly one gets the following identity
Finally, applying Lemma 3.4 on F(u) and then using (4.2), (4.3) we get
where \(\gamma _i^k:= 4\alpha _{k-i}^k + \frac{1}{4}\beta _{k-i+1}^k\). In the last step we used the classical Hardy inequality. In Sect. 6 we simplify the expressions of \(\alpha _i^k\) and \(\beta _i^k\), which will complete the proof of Theorem 2.1. \(\square \)
Proof of Corollary 2.2
Assuming \(\gamma _i^k \ge 0\) (which will be proved in Sect. 6) Theorem 2.1 immediately implies
It can be checked that \(\xi _{k-1}^k = -k(k+1)\). Using this in the expression of \(\gamma _1^k\), we find that \(\gamma _1^k = \frac{(2k-1)^2}{4}\). Next, we prove the sharpness of the constant \(\gamma _1^k\). Let C be a constant such that
for all \(u \in C_c(\mathbb {Z})\).
Let \(N \in \mathbb {N}\), \(\beta \in \mathbb {R}\) and \(\alpha \ge 0\) be such that \(2\beta + 2k-2 <-1\). Consider the following family of finitely supported functions on \(\mathbb {Z}\).
Clearly we have
and
Some basic estimates:
Using the above in (4.7), we get
Using estimates (4.6) and (4.8) in (4.5) and taking limit \(N \rightarrow \infty \), we get
Finally, taking limit \(\beta \rightarrow \frac{1-2k}{2}\) on the both sides, we obtain
This proves the sharpness of \(\gamma _1^k\). \(\square \)
5 Proof of Higher Order Hardy Inequalities
Proof of Theorem 2.3
First we prove inequality (2.4) and then inequality (2.5).
Let \(m \in \mathbb {N}\), \(v \in C_c(\mathbb {Z})\) with \(v(0)=0\) and
Using the inversion formula for the Fourier transform, we obtain
Therefore we have \(\mathcal {F}(\Delta v) = 4 \sin ^2(x/2) \mathcal {F}(v)\). Applying this formula iteratively, we obtain \(\mathcal {F}(\Delta ^m v) = 4^m \sin ^{2\,m}(x/2)\mathcal {F}(v)\). Using Parseval’s, identity we get
Using Lemma 3.5 iteratively we obtain
under the assumption that
for \(1 \le k \le 2m\).
Next we compute the inverse Fourier transform of \(\sin ^{2(2m-k)}(x/2)\) to simplify the condition (5.1). Consider
Using the above expression, condition (5.1) becomes
So finally we arrive at the following inequality
provided \(v \in C_c(\mathbb {Z})\) with \(v(0) =0 \) satisfies
for \(1 \le k \le 2m\).
Let \(u \in C_c(\mathbb {N}_0)\) with \(u(i)=0\) for all \(0\le i \le 2\,m-1\). We define \(v \in C_c(\mathbb {Z})\) as
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. \(\square \)
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
Applying Parseval’s identity gives us
Similarly, one gets the following identity
Now applying Lemma 3.6 and then using equations (5.5) and (5.6), we get
In the last line we used \(\alpha _{k-1}^k = k(k-1)\) and \(\beta _k^k = 1\) (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). \(\square \)
6 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.
Theorem 6.1
Let \(k \in \mathbb {N}\) and \( 0 \le i \le k\). Then
Proof
Using \(\sin ^2(x/2) = (1-\cos x)/2\), identity (3.1) can be re-written as
Let \(u = e^{in(x/2)}\sin (x/2)\). Then some straightforward calculations give us the following identities for \(m \ge 0\)
Using equations (6.3) - (6.5) in (6.2), we obtain
The last step uses
and
Therefore, for \(n \in \mathbb {N}\), we have
which implies the identity (6.1). \(\square \)
Remark 6.2
Using identity (6.1), expressions of \(\alpha _i^k, \beta _i^k\) defined by (3.2), (3.3) respectively become
and \(\gamma _i^k:= 4\alpha _{k-i}^k + \frac{1}{4}\beta _{k-i+1}^k\) becomes
From the above expressions, it is quite straightforward that the above constants are non-negative, thus justifying the assumptions used in the proofs of Lemma 3.4, lemma 3.6 and Corollary 2.2. Finally, the expression of \(\gamma _i^k\) along with (4.4) completes the proof of Theorem 2.1.
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Acknowledgements
It is a pleasure to thank Professor Ari Laptev for his input and encouragement and also for his comments on early drafts of this paper. The author would also like to thank Ashvni Narayanan for proof reading the document. Finally, we thank an anonymous reviewer for their thorough reading and many helpful suggestions. The author is supported by President’s Ph.D. Scholarship, Imperial College London.
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The author has been supported by President’s PhD Scholarship, Imperial College London.
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Appendix
Appendix
We prove identity (3.6) with \(w_{ij}\) replaced with an arbitrary smooth \(2\pi \) periodic funcition.
Lemma A.1
Let \(u, w \in C^\infty [-\pi , \pi ]\) such that their derivatives satisfy \(d^k u(-\pi ) = d^k u(\pi )\) and \(d^k w(-\pi ) = d^k w(\pi )\), for all \(k \in {\mathord {{\mathbb {N}}}}_0\). Then for non-negative integers \(0 \le i<j\) we have
where \(C_{\sigma ,w}^{i,j}\) is given by
Proof
We prove the result using induction on the parameter \(k:= j-i\). Let us assume that (A.1) is true for all \(0 \le i < j\) such that \(3 \le j-i \le k \). Consider non-negative integers \(i<j\) such that \(j-i = k+1\). Then integration by parts yields
Further using induction hypothesis we get
where \(\delta (\text {odd numbers}):= 0\) and \(\delta (\text {even numbers}):= 1\). Using identity \({n \atopwithdelims ()r} + {n \atopwithdelims ()r-1} = {n+1 \atopwithdelims ()r}\) we obtain
It can also be checked that \(-C_{i, w'}^{i, j-1} = C_{i, w}^{i, j} = \frac{1}{2}d^{j-1} w\) as well as \(-C_{\lfloor (i+j)/2 \rfloor ,w}^{i+1, j-1} = C_{\lfloor (i+j)/2 \rfloor ,w}^{i, j} = (-1)^{j-i} w\) (for even \(i+j\)). These observations along with (A.3) and (A.2) proves (A.1) for \(3 \le j-i = k+1\).
The base cases \(j-i \in \{1, 2, 3\}\) can be checked by hand (it’s a consequence of iterative integration by parts). \(\square \)
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Gupta, S. One-Dimensional Discrete Hardy and Rellich Inequalities on Integers. J Fourier Anal Appl 30, 15 (2024). https://doi.org/10.1007/s00041-024-10070-6
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DOI: https://doi.org/10.1007/s00041-024-10070-6