Hardy and Rellich inequality on lattices

In this paper, we study the asymptotic behaviour of the sharp constant in discrete Hardy and Rellich inequality on the lattice $\mathbb{Z}^d$ as $d \rightarrow \infty$. In the process, we proved some Hardy-type inequalities for the operators $\Delta^m$ and $\nabla(\Delta^m)$ for non-negative integers $m$ on a $d$ dimensional torus. It turns out that the sharp constant in discrete Hardy and Rellich inequality grows as $d$ and $d^2$ respectively as $ d \rightarrow \infty$.


Introduction and Main Results
In this paper, we are interested in the asymptotic behaviour of sharp constants (largest constant for which an inequality holds true) in the following inequalities as d → ∞: and e j is the j th canonical basis of R d .
Inequalities (1.1) and (1.2) are discrete analogues of Hardy and Rellich inequalities respectively on R d (see [GL] and references therein), and will be referred to as discrete Hardy and Rellich inequalities. The discrete Hardy inequality (1.1) has been considered in the past in works [KL,RS1,RS2]and more generally for graphs in [KPP1]. To our best knowledge [KPP3] is the only paper where (1.2) has been studied in the past in the context of graphs. In particular, they prove inequality (1.2) with a weight that grows like |n| −4 as |n| → ∞ for d ≥ 5.
Although Hardy and Rellich type inequalities in the continuum are very well studied and the literature on them is enormous(for instance see book by Balinsky et al. [BEL]), very little is known about their discrete counterparts. In fact, the sharp constant in (1.1) and (1.2) is only known when d = 1. Computation of sharp constant in (1.1) goes back to G.H. Hardy [HLP](see [KPP2] for a more recent proof and improvement of (1.1) for d = 1). For (1.2), sharp constant was recently obtained in d = 1 [GKS]. In fact, authors in [GKS] proved an improvement of sharp Rellich inequality (1.2) by adding lower order terms in the RHS of (1.2).
In this work, we compute the behaviour of sharp constants in these inequalities for large dimensions. In fact, we will also find the asymptotic behaviour of sharp constants in the higher order versions of inequalities (1.1) and (1.2). In the following theorems C c (Z d ) denotes the space of finitely supported functions on Z d . The main results of the paper can be summarized as follows: Theorem 1.2. Let k ≥ 1 and let u ∈ C c (Z d ) with u(0) = 0. Let C 2 (k, d) be the sharp constant in the following inequality: Remark 1.3. We would like to point out that the sharp constants in continuous analogues of (1.3) and (1.4) on R d grows as d 4k+2 and d 4k as d → ∞ respectively, see [DH], for the computation of sharp constants.
Remark 1.4. Note that putting k = 0 in (1.3) and k = 1 in (1.4) give the discrete Hardy (1.1) and Rellich inequality (1.2) respectively. (1.5) Remark 1.7. We would like to mention that in [KPP1] authors proved (1.5) with C H (d)|n| −2 replaced by an optimal Hardy weight w(n) (see [KPP1] for the definition of optimal Hardy weight), which behaves as w(n) = 4 −1 (d − 2) 2 |n| −2 + lower order terms, under the limit |n| → ∞ . It is interesting to note that the constant in the leading term of optimal Hardy weight grows as d 2 as compared to linear growth of sharp constant in (1.5).
Remark In the next section we convert inequalities (1.3), (1.4) into continuous Hardy type inequalities on a torus. In section 3, we will prove various higher order Hardy-type inequalities on the torus, which we believe are completely new. Finally in section 4 we will use the results proved in sections 2 and 3 to prove the main results of the paper.

Equivalent integral inequalities
In this section, we will convert the inequalities (1.3) and (1.4) into some equivalent integral inequalities on the torus. In this paper, Q d := (−π, π) d denotes the open square in R d .
Definition 2.1. Let ψ : Q d → C be a map. Then we say it is 2π-periodic in each variable if for all 1 ≤ i ≤ d.
Lemma 2.2. Let k ≥ 0 be an integer. Let u ∈ C c (Z d ) with u(0) = 0. There exists ψ ∈ C ∞ (Q d ), all of whose derivatives are 2π-periodic in each variable and which has zero average, such that Proof. We will first prove the result for k = 0 and then extend the proof to general k.
Let u ∈ l 2 (Z d ). We define its Fourier transform u as Let u j (n) := n j |n| 2 u(n) for n = 0 and u j (0) = 0. Then Parseval's identity gives us Using the inversion formula for Fourier transform we get Applying Parseval's identity and summing w.r.t. to j, we obtain The inversion formula for Fourier transform along with integration by parts gives us This further implies that there exists a smooth function ψ such that u j (x) = ∂ x j ψ(x), whose average is zero. It is easy to see that periodicity of u j along with its zero average imply that ψ is also 2π periodic in each variable. This proves the result for k = 0.
, all of whose derivatives are 2π-periodic in each variable and which has zero average, such that Proof. The proof of this Lemma follows the proof of Lemma 2.2 step by step but we include the proof here for the sake of completeness.

Hardy-type inequalities on a torus
In this section, we will prove Hardy-type inequalities for the operators ∆ m and ∇(∆ m ) for non-negative integers m on the torus Q d . We begin by proving a weighted Hardy inequality for the gradient.
In the next lemma, we prove a two-parameter family of inequalities from which we will derive a weighted Rellich and Hardy-Rellich type inequalities on the torus.
Next, we derive a Rellich inequality on the torus Q d from Lemma 3.4.
Theorem 3.8 (Weighted Rellich inequality). Let ψ ∈ C ∞ (Q d ) all of whose derivatives are 2π-periodic in each variable. Further assume that ψ has zero average. Let k be a non-positive integer. Then, for d > −2k + 4, we have where ω(x) := j sin 2 (x j /2), non-negative constants C 1 (α, d), C 2 (α, d) are given by , Remark 3.9. Note that R(k, d) ∼ d 2 as d → ∞, since H(k, d) ∼ d, HR(k, d) ∼ d and In the next theorem we apply weighted Hardy and Rellich inequality (3.1), (3.24) respectively to prove a Hardy type inequality for the operators ∆ m and ∇(∆ m ).
Theorem 3.11. Let ψ ∈ C ∞ (Q d ) all of whose derivatives are 2π-periodic in each variable. Further assume that ψ has zero average. Let k be a non-positive integer and m be a non-negative integer.
(1) If d > −2k + 4m, then  Corollary 3.12. Let ψ ∈ C ∞ (Q d ) all of whose derivatives are 2π-periodic in each variable. Further assume that ψ has zero average. Let m be a non-negative integer.