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On some strong Poincar\'e inequalities on Riemannian models and their improvements

We prove second and fourth order improved Poincar\'e type inequalities on the hyperbolic space involving Hardy-type remainder terms. Since theirs l.h.s. only involve the radial part of the gradient or of the laplacian, they can be seen as stronger versions of the classical Poincar\'e inequality. We show that such inequalities hold true on model manifolds as well, under suitable curvature assumptions and sharpness of some constants is also discussed.


Introduction
Let M be a Cartan-Hadamard manifold of dimension N (namely, a manifold which is complete, simply-connected, and has everywhere non-positive sectional curvature). It is well known that if the sectional curvatures of M are bounded above by a strictly negative constant, then M admits a Poincaré type, or L 2 -gap, inequality, namely there exists Λ > 0 such that Recently, there has been a constant effort to improve the Poincaré inequality in terms of adding possibly optimal Hardy weights, i.e. adding on the r.h.s. a term of the form M W u 2 dv g with W ≥ 0 "as large as possible", see [11] for a general treatment of Hardy weights for second-order elliptic operators. Starting from the works [2] and [4], where a Poincaré-Hardy inequality was shown with sharp constants, further generalisation to p-Laplacian and higher order case have been obtained in [6] and [5], respectively. We also refer to [7] for more general improvements and the study of extremals. The kind of weights obtained in these papers, which are singular at a fixed point of M , makes this subject a sort of lateral branch of that very rich field of research originated from the seminal paper [8], and dealing with possible improvements (not only of L 2 type) of the classical Hardy inequality on bounded euclidean domains or on curved spaces, see e.g. [1,3,9,10,14,15,16,17,19] and reference therein.
The most studied example of M is clearly the hyperbolic space H N , where Λ = N −1 2 2 . In this setting the following improved Poincarè inequality was established in [4] for all u ∈ C ∞ c (H N ): where N ≥ 3, r := ρ(x, o) denotes the geodesic distance function and o ∈ H N is fixed. Moreover, all constants in (1.1) are proved to be sharp in a suitable sense.
In view of the work [21], where a similar question has been recently posed in the context of Hardy inequalities, one may wonder whether inequality (1.1) still holds if we replace |∇ H N u| 2 with its radial part ( ∂u ∂r ) 2 . Since, by Gauss's lemma one has |∇ H N u| 2 ≥ ( ∂u ∂r ) 2 , giving an answer to this question is by no means obvious and it represents the first goal of this paper. Indeed, we prove that the following stronger version of (1.1) holds for all u ∈ C ∞ c (H N ) and N ≥ 3. Clearly, (1.2) reproves inequality (1.1) but we derive it by using a different technique: here the main tool exploited is spherical harmonics decomposition technique, while the proof of (1.1) was based on finding a ground state and on criticality theory, see [4,Theorem 2.1]. Furthermore, we argue quite differently also in proving the optimality of the constants appearing in (1.2). Notice that the sharpness of all constants in (1.2) can be derived by combining Gauss's lemma with the sharpness of the corresponding constants in (1.1). Nevertheless, in this paper we provide and alternative, and more direct, proof of the sharpness of the dominating term at infinity of (1.2), namely of the constant 1 4 , which is based on the delicate construction of a suitable minimising sequence. This argument may have its own interest in the study of related partial differential equations, furthermore it can be carried over to more general Riemannian models having negative sectional curvatures bounded above, see Theorem 2.1 below.
It is known from [24] that in the Hyperbolic space there also hold the following second order Poincarè inequalities: for all u ∈ C ∞ c (H N ) with N ≥ 3 and with l = 0 or l = 1. We refer to [20] for a proof of the sharpness of the above constants. Now, in view of (1.3) and for what previously stated in the first order, it is natural to think about possible extensions of (1.2) to the second order. In this respect the following inequality from [21, Theorem 5.2] turns out to be meaningful: where ∆ r,H N denotes the radial part of the Laplace-Beltrami operator ∆ H N on H N , see also Lemma 5.1 below. Clearly, inequality (1.4) suggests that a possible stronger version of (1.3) might involve the operator ∆ r,H N . Motivated by this, in the present paper we prove the following improved Poincarè type inequality: We notice that the interest of (1.5) is not only due to the fact that it looks likes the proper second order analogue of (1.2), but also to the fact that it improves (1.3) from several points of view. In fact, on one hand, by combining (1.2) and a related weighted inequality, from (1.5) we derive the inequality: for all u ∈ C ∞ c (H N ) with N ≥ 5; in view of (1.4), (1.6) reads as a strong (and improved) version of (1.3) with l = 0. On the other hand, a clever exploitation of (1.5), jointly with the spherical harmonics decomposition technique, yields the following improved version of (1.3) with l = 1: for all u ∈ C ∞ c (H N ) with N ≥ 5. We note that, despite their similarity, (1.5) and (1.7) are not mutually implicated.
We complete this short discussion by mentioning that inequality (1.6) can be compared with the following improved Poincarè inequality proved in [4]: was proved to be sharp. In view of (1.4), (1.6) has a l.h.s. with a stronger impact than that of (1.8), furthermore, since 9 16 < (N −4) 2

16
for N ≥ 8, inequality (1.6) gives a better improvement than (1.8) of the Poincarè inequality (1.3) near the pole, when N ≥ 8. However, for all dimensions, the r.h.s. of (1.8) performs better at infinity. Also we mention that, unfortunately, neither (1.6) or (1.8) solve the problem of sharpness of the constant in front of 1 r 4 which is still open. We notice that within the paper we prove and state all results on general model manifolds satisfying suitable curvature bounds and having H N as a remarkable particular case, see Section 2 below.
The paper is organized as follows: we state our main theorems in Section 2 while the rest of the paper deals with the proofs of the results. More precisely, in Section 2 we first briefly recall some useful facts concerning to Riemannian model manifolds and we introduce some curvature bounds related to the models on which we will settle our analysis. Then, we state our first and second order improved Poincaré inequalities. In Section 3 we give the proof of our improved strong Poincaré inequality in the first order, namely inequality (1.2) but on more general Riemannian models. Section 4 contains the proof of inequalities related to the radial part of Laplace-Beltrami operator and to the radial part of gradient, namely the analogous of inequality (1.5) on model manifolds. In Section 5 we first prove (1.4) on general models and then we derive our improved strong Poincaré inequality in the second order, which reads as (1.6) on H N . Finally, in Section 6 we derive our improved version of (1.3) with l = 1 on model manifolds, i.e. (1.7) on H N .

Preliminaries and Statement of Main results
This section is devoted to the statements of the main results. Before stating our theorems let us briefly recall some of the known facts concerning Riemannian model manifolds.
These conditions on ψ ensure that the manifold is smooth and the metric at the pole o is given by the euclidean metric [18,Chapter 1,3.4]. The coordinate r, by construction, represents the Riemannian distance from the pole o, see e.g. [12,13,18] for further details. In particular, all the assumptions above are satisfied by ψ(r) = r and by ψ(r) = sinh(r): in the first case M coincides with the euclidean space R N , in the latter with the hyperbolic space H N .
It is known that there exist an orthonormal frame {F j } j=1,...,N on (M, g) where F N corresponds to the radial coordinate, and F 1 , . . . , F N −1 to the spherical coordinates, for which F i ∧F j diagonalize the curvature operator R : The quantities then coincide with the sectional curvatures w.r.t. planes containing the radial direction and, respectively, orthogonal to it. In what follows we will sometimes need to assume that (2.4) K rad π,r ≤ −1 in (0, +∞) . Since, by the Sturm-Comparison Theorem, the above condition also implies that H tan π,r ≤ −1, with (2.4) we basically require the boundedness from above of both sectional curvatures. See also Lemma 3.1 in the following. Finally, a further crucial quantity in our statements will be (2.5) Λ rad π,r := −2K rad π,r − (N − 3)H tan π,r , Λ rad π,r is related to the bottom of the spectrum of the laplacian, see inequality (2.8) below; in particular, when ψ(r) = sinh(r), then Λ rad π,r = N − 1 hence (N −1) 4 Λ rad π,r coincides with the bottom of the spectrum of the laplacian in H N .
Next, we recall that the Riemannian Laplacian of a scalar function u on M is given by where ∆ S N −1 is the Riemannian Laplacian on the unit sphere S N −1 . In particular, for radial functions, namely functions depending only on r, ∆ g u reads where from now on a prime will denote, for radial functions, derivative w.r.t r.
We are now ready to state our results.
with Λ rad π,r as defined in (2.5). Moreover, if condition (2.4) holds and furthermore for some C > 0 and a ≥ 0, then the constant 1 4 (2.10) hence 1/4 is no more the sharp constant in front of the term u 2 r 2 . As concerns condition (2.9), it is needed to show that the quotient (2.10) is finite for our minimizing sequence, see Section 3. Nevertheless, we notice that this condition is not "too restrictive", in the sense that it allows unbounded curvatures from below as it happens, for instance, if ψ(r) = r e r 2 for which (2.9) holds with C = 2 and a = 1.
We point out that when (2.4) holds inequality (2.8) implies the following more explicit inequality: A remarkable particular case to which Theorem 2.1 applies is when M = H N , i.e. ψ(r) = sinh(r). In this case all constants in (2.8) are proved to be sharp.    Next we state a Rellich type improvement for the second order Poincaré inequality (1.3) with ℓ = 0 but on more general model manifolds. The proof comes by exploiting either inequality (1.4) on Riemannian models (see Lemma 5.1 in the following) and inequality (2.12), the first brought the restriction N ≥ 5 and the latter yields the curvature conditions (2.4) and (2.13) below. In particular, it gives rise to the interesting fact that the constant appearing in front of the Rellich term u 2 r 4 can be larger than 9 16 .
We conclude by stating a Hardy-type improvement of the second order Poincaré inequality (1.3) with ℓ = 1 on model manifolds. Here the main tools exploited in the proofs are spherical harmonics decomposition and reduction of dimension technique. The latter yields the restriction N ≥ 5, while conditions (2.4) and (2.13) come again from inequality (2.12) that we apply for each component of the decomposition.

Proofs of
Next we discuss some basic facts on spherical harmonics. More details can be found in [22]. A spherical harmonic P n of order n is the restriction to S N −1 of a homogeneous harmonic polynomial of degree n. Moreover, it satisfies −∆ S N −1 P n = λ n P n for all n ∈ N 0 where λ n = (n 2 + (N − 2)n) are the eigenvalues of Laplace Beltrami operator −∆ S N −1 on S N −1 with corresponding eigenspace dimension c n . We note that λ n ≥ 0, λ 0 = 0, c 0 = 1, c 1 = N and We are now ready to begin the proof.
Proof of (2.8). The proof is divided into two steps.
Step 1. Let u ∈ C ∞ c (M ) and define the transformation v(x) = ψ(r) An easy calculation gives ∂v ∂r = ∂ ∂r ψ(r) By arranging the terms we obtain: Step 2. Now, expanding v in terms of spherical harmonics: We will evaluate each term of (3.3) separately. Using the integration by parts formula and the orthonormal properties of {P n } i.e S N −1 P n P m dσ = δ nm , we find Next, by using the 1-dimensional Hardy inequality: Finally, inserting the equalities: into (3.9) and recalling (2.5) we obtain the desired inequality (2.8).
Optimality : We set If (2.4) holds, then Λ rad π,r > 0 and, by combining density arguments with Fatou's Lemma, we infer that (2.8) holds in H 1 (M ) and, in turn, that C M ≥ 1 4 . So it remains to show that C M ≤ 1 4 and this will be done by giving a proper minimizing sequence. Again we divide the proof in some steps.
Step 1. Let us define the sequence {φ n } for n ∈ N as follows (3.11) φ n (r) = where α > 1 + a and a ≥ 0 is as given in (2.9). Clearly, φ n ∈ L 1 loc (0, +∞) and its weak derivative writes Next we recall that, from the proof of Proposition 4.1 in [4], the function u 0 (r) := r Using polar coordinates and by exploiting (2.9), it follows that u 0 φ n and u 0 φ 2 n both belong to H 1 (M ) for all n ≥ 3 and for α > 1 + a. In particular, (2.9) assures (u 0 φ n ) ′ , (u 0 φ 2 n ) ′ ∈ L 2 (M ). Indeed, for r ≥ n, we have that and this term turns out to be integrable at infinity for α > 1 + a, if ψ satisfies the above condition; the term (u 0 φ 2 n ) ′ can be treated similarly.
Step 2. Let R > n, by multiplying the equation (3.13) by u 0 φ 2 n and integrating by parts, we obtain where we have exploited the fact that φ n is supported outside B 1 (0), hence no problem of integrability arises at r = 0. Next we note that where in the above we have exploited the fact that |∂B R (0)| = ω N ψ N −1 (R). On the other hand, for r > n, we have that (2.4) and (3.1)).
As for the terms involving the curvatures, we first note that, if (2.4) holds then Λ rad π,r > 0 (see the proof of Corollary 2.1 below) and, by density arguments, (2.8) yields Hence, since u 0 φ n ∈ H 1 (M ), we deduce that where the boundedness of the latter term and, in turn, of the first, follows by (3.14). Since for all terms listed above we have integrability at infinity whenever α > 1 + a, by Lebesgue Theorem, we can pass to the limit in (3.15) as R → +∞ and we conclude that Step 3. Next we observe that Note that, by the above discussion, the first two terms are well defined. Furthermore, for r ≥ n, we have that By using this into (3.16) we have and, by considering the quotient in (3.10) with u = u 0 φ n , we obtain Then, from definition of C M and the above, we infer Step 4. We estimate each term of the r.h.s. of (3.17). Note that ω N denotes the N dimensional measure of unit sphere, hence a finite number.
First we estimate the denominator Finally, using integration by parts, α > 1, (3.1) and sinh r ≥ r, we obtain Exploiting the spherical harmonic decomposition, we write u(x) := u(r, σ) = ∞ n=0 a n (r)P n (σ) and we obtain Recalling that {P n } is orthonormal and by using polar coordinates and integrating by parts, we get Now, exploiting Theorem 2.1 for each a ′ n , we get and, in turn, we infer We consider each term of the r.h.s. of (6.2) separately. First we use (2.12) for each a n (r) and we get Then, we exploit the equation −∆ S N −1 P n = λ n P n , the orthonormal properties of the {P n } and by parts formula to obtain ψ ′ ψ a ′ n a n ψ 2 ψ N −1 dr