Improved higher order Poincar\'e inequalities on the hyperbolic space via Hardy-type remainder terms

The paper deals about Hardy-type inequalities associated with the following higher order Poincar\'e inequality: $$ \left( \frac{N-1}{2} \right)^{2(k -l)} := \inf_{ u \in C_{c}^{\infty} \setminus \{0\}} \frac{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^2 \ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}} }\,, $$ where $0 \leq l<k$ are integers and $\mathbb{H}^{N}$ denotes the hyperbolic space. More precisely, we improve the Poincar\'e inequality associated with the above ratio by showing the existence of $k$ Hardy-type remainder terms. Furthermore, when $k = 2$ and $l = 1$ the existence of further remainder terms are provided and the sharpness of some constants is also discussed. As an application, we derive improved Rellich type inequalities on upper half space of the Euclidean space with non-standard remainder terms.


Introduction
Let H N denote the hyperbolic space and let k, l be non-negative integers such that l < k. The following higher order Poincaré inequality [22,Lemma 2.4] holds if j is an even integer , if j is an odd integer and ∇ H N denotes the Riemannian gradient while ∆ j H N denotes the j−th iterated Laplace-Beltrami operator. The present paper takes the origin from the basic observation that the inequality in (1.1) is strict for u = 0, namely the following infimum is never achieved It becomes then a natural problem to look for possible remainder terms for (1.1). In this direction, when k = 1 and l = 0, a remainder term of Sobolev type has been determined in [25]. The aim of our study is to deal with Hardy remainder terms, namely to determine improved Hardy inequalities for higher order operators, where the improvement is meant with respect to the higher order Poincaré inequality (1.1). More precisely, settled r := ̺(x, x 0 ), where ̺ denotes the geodesic distance and x 0 ∈ H N denotes the pole, we wish to answer the question Does there exist positive constants C and γ such that the following Poincaré-Hardy inequality holds for all u ∈ H k (H N )?
The literature on improved Hardy and Rellich inequalities in the Euclidean setting dates back to the seminal works of Brezis-Vazquez [10] and Brezis-Marcus [9]. Without claiming of completeness, we also recall [1,3,4,5,13,16,17,18,19,26,29,31] and references therein. The reason of such a great interest is surely do to the fact that Hardy inequalities and their improved versions have various applications in the theory of partial differential equations and nonlinear analysis, see for istance [10,33,34]. Further generalizations to Riemannian manifolds are quite recent and a subject of intense research after the work of Carron [11]. We enlist few important recent works [7,8,12,14,20,21,23,28,32] and references therein. Most of these works deals with classical Hardy inequalities and their improvement on Riemannian manifolds. Namely, differently from (1.2), the optimal Hardy constant is taken as fixed and one looks for bounds of the constant in front of other remainder terms. The main motivation of our study initiated in [2] on improved Poincaré inequalities comes from a paper of Devyver-Fraas-Pinchover [14], which deals with optimal Hardy inequalities for general second order operators. In particular, the existence of at least one Hardy-type remainder term for (1.1) with k = 1 and l = 0 follows as an application of their results. Nevertheless, their weight is given in terms of the Green's function of the associated operator and does not imply the validity of an inequality like (1.2). See [2] for further details. The same can be said for the inequality in [6,Example 5.3] where N = 3. The above mentioned goal was achieved in [2] where, developing a suitable construction of super solution, the following inequality was shown • Case k = 1 and l = 0. For N > 2 and for all u ∈ C ∞ 0 (H N ) there holds where the constants N −1 2 2 and 1 4 are sharp. Unfortunately, the super solution construction applied in the proof of (1.3) seems not applicable to the higher order case. Nevertheless, by exploiting a completely different technique based on spherical harmonics, in [2] the following second order analogue of (1.3) was obtained • Case k = 2 and l = 0. For N > 4 and for all u ∈ C ∞ 0 (H N ) there holds It is clear that (1.3) and (1.4) do not give a complete proof of (1.2). The aim of the present paper is either to generalize to the higher order (1.3) and (1.4) and to investigate all the remaining cases when l = 0. A first step in this direction is represented by the proof of the validity of (1.2) when k = 2 and l = 1. This case is not covered by (1.3) and (1.4) and its proof requires some effort. A clever transformation which uncovers the Poincaré term and spherical harmonics technique are the main tools applied, see Sections 2 and 5. Also we note that when k = 2 and l = 1 further singular remainder terms, involving hyperbolic functions, are provided and some optimality issues are proved. Namely, we have The constant N −1 2 2 is sharp by construction and sharpness of the other constants is discussed in Section 2.
Theorem 1.1 turns out to be one of the key ingredients in our strategy to get the arbitrary case, i.e. inequality (1.2) for every l < k. Furthermore, from Theorem 1.1 we derive improved Rellich type inequalities on upper half space of the Euclidean space having their own interest. See Corollary 2.2 for the details. The technique adopted relies on the so-called "Conformal Transformation" to the Euclidean space.
As concerns the general case l < k, a fine combination of the previous results and some technical inequalities allow us to finally derive the following family of inequalities Theorem 1.2. (Case 0 ≤ l < k) Let k, l be integers such that 0 ≤ l < k and let N > 2k. There exist k positive constants α j k,l = α j k,l (N ) such that the following inequality holds . Furthermore, the constant N −1 2 2(k−l) is sharp and the leading terms as r → 0 and r → +∞, namely α 1 k,l and α k k,l , are given explicitly in Theorems 3.1 and 4.1 below. In view of possible applications to differential equations, we point out that the strategy of our proofs basically allows to determine explicitly all the constants α j k,l in Theorem 1.2. Nevertheless, for the sake of simplicity, we prefer to focus on the leading terms α 1 k,l and α k k,l . This choice is also justified by the fact that our interest is devoted to the non-Euclidean behavior of inequalities and the constant highlighting this aspect is exactly α 1 k,l , i.e. the constant in front of the leading term as r → +∞. As a matter of example, here below we specify our family of inequalities for some particular choices of k and l. If k = 2m for some positive integer m, there holds If k = 2m for some positive integer m, there holds where we use the convention 0 j=1 = 1. If k = 2m + 1 for some positive integer m, there holds . The article is organized as follows. Section 2 is devoted to the precise statement and discussion of results for the case k = 2 and l = 1. The complete proof of the results discussed in Section 2 is postponed to Section 5. Section 3 and Section 4 are devoted to discussions and proofs of the results for 0 = l < k and for 0 = l < k. The statements of the results given in these sections will contain the precise constants for the leading terms mentioned in the statement of Theorem 1.2.
2. Case k = 2 and l = 1 We start by restating Theorem 1.1 in its complete form. The proof of the results given in this section will be postponed to Section 5.
The constant N −1 2 2 is sharp by construction, namely cannot be replaced by a larger one. Furthermore, the is sharp in the sense that no inequality of the form Remark 2.1. Inequality (2.1) does not follow directly from (1.3) and (1.4) but requires an independent proof which is achieved by means of a suitable modification of the proof of (1.4) as given in [2]. As already remarked in the Introduction, the main tools exploited are a suitable transformation which uncovers the Poincaré term and spherical harmonic analysis. Recently, spherical harmonics technique has been successfully exploited in the context of Weighted Calderón-Zygmund and Rellich inequalities [27]. Remark 2.3. It's worth noting that, as happens for inequality (1.4), the constants appearing in front of the terms 1/r 4 and 1/ sinh 4 r are jointly sharp. In the sense that the inequality This follows by noting that sinh r ∼ r as r → 0 and that (2.1) yields is the best constant (namely, the larger) for the standard N dimensional Euclidean Rellich inequality, both on the whole R N or in any open set containing the origin.
Consider the upper half space model for H N , namely R N + = {(x, y) ∈ R N −1 ×R + } endowed with the Riemannian metric δij y 2 . We set It is readily seen that d ∼ log(1/y) as y → 0. By exploiting the transformation (2.4)

Furthermore, we have:
• no inequality of the form Similar conclusions hold for the constants (N 2 −2N −9) 3. Case k arbitrary and l = 0 In this section we restate and prove Theorem 1.2 for 0 = l < k.
. Furthermore, the leading terms a r → 0 and r → ∞ are explicitly given by where we use the conventions: 0 j=1 = 0 and 0 j=1 = 1. Proof. Here and after, for shortness we will write ∆ H N = ∆. In the proof we will repeatedly exploit the following inequality from [32,Theorem 4.4]: for all u ∈ C ∞ 0 (H N ) and 0 ≤ β < N − 4. We prove separately the case k even and k odd. First we assume k = 2m even. If m = 1, (3.1) follows directly from (1.4). When m = 2, by (1.4) and (3.2) with N > 8, we have Next we assume (3.1) holds for k = 2m with m > 2, namely where, for 2 ≤ i ≤ 2m − 1, the c i 2m are suitable positive constants and N > 4m. Inequality (3.3) yields Next, by (3.2), for N > 4m + 4 we have where, for 2 ≤ i ≤ 2m + 1,c i 2m are suitable positive constants. The above inequalities and (1.4), finally yield 2m are suitable positive constants. By induction, this completes the proof of (3.1) for k even.

Case k > l > 0 arbitrary
In this section we restate and prove Theorem 1.2 for l > 0, the case l = 0 has already been dealt with in Section 3.
for all u ∈ C ∞ 0 (H N ). Furthermore, the leading terms a r → 0 and r → ∞, namely α 1 k,l and α k k,l are explicitly given as follows  The proof is achieved by considering separately four cases. In each proof we will exploit the following technical lemma whose proof can be obtained by induction, iterating (3.2). Notice that, except for the main statements, for shortness we will always write ∆ H N = ∆.
for all u ∈ C ∞ 0 (H N ). Furthermore, the leading terms a r → 0 and r → ∞ are explicitly given by where, for any γ and β positive integers, , d γ and e γ are the constants defined in Theorem 3.1.

4.2.
Case k = 2m even and l = 2h + 1 odd. Theorem 4.4. Let m, h be integers such that 0 ≤ h < m and N > 4m. There exist 2m positive constants α i =ᾱ i (N, m, h) such that the following inequality holds for all u ∈ C ∞ 0 (H N ). Furthermore, the leading terms a r → 0 and r → ∞ are explicitly given by where a 0 = 1 and, for any γ and β positive integers, , d γ and e γ are the constants defined in Theorem 3.1.
Proof. Let 0 < h < m − 1, by applying first (3.1) with k = 2(m − h − 1), then (2.1) and finally (4.1) with γ = h, h + 1 and β = 2, 4, 2i, we deduce Hence, with an argument similar to that applied in the proof of Theorem 4.3, it's readily deduced the existence of 2m positive constantsᾱ i = α i (N, m, h) such that (4.3) holds. Furthermore, When h = 0 the above computations may be slightly modified to show the validity of (4.3). Furthermore, by setting a 0 = 0, the leading terms are still given as above.
Theorem 4.5. Let m, h be integers such that 0 < h ≤ m and N > 4m + 2. There exist 2m + 1 positive constants δ i = δ i (N, m, h) such that the following inequality holds

4)
for all u ∈ C ∞ 0 (H N ). Furthermore, the leading terms as r → 0 and r → ∞ are explicitly given by: where d 0 = 0 and, for any γ and β positive integers, , d γ and e γ are the constants defined in Theorem 3.1.
If 0 < h < m, from (4.2) and (4.1) we readily get When h = m the same proof may be adopted without applying (4.2). In this case, the leading terms are defined as above by assuming d 0 = 0.
Theorem 4.6. Let m, h be integers such that 0 ≤ h < m and N > 4m + 2. There exist 2m + 1 positive constants δ i =δ i (N, m, h) such that the following inequality holds , d γ and e γ are the constants defined in Theorem 3.1.

Proof. From (1.3) we know
Now, by applying (4.3) and (4.1) we deduce We note that the spherical harmonic P n of order n is the restriction to S N −1 of a homogeneous harmonic polynomial of degree n. Now we recall the following In the sequel we will also exploit the following 1-dimensional Hardy-type inequality from [2]: Proof of Theorem 2.1.
The proof is divided in several steps.
Then the following relation holds for x = (r, σ) ∈ (0, ∞) × S N −1 , Now by rearranging the terms above we conclude the proof of Step 1.
Step 2. In this step we compute, is the best constant.
Next we turn to the optimality issues. Assume by contradiction that the following inequality holds . The above inequality, jointly with (1.1) with k = 1, l = 0 and Hardy-Maz'ya inequality: