Hardy-Poincar\'e, Rellich and Uncertainty principle inequalities on Riemannian manifolds

We continue our previous study of improved Hardy, Rellich and Uncertainty principle inequalities on a Riemannian manifold $M$, started in \cite{Kombe-Ozaydin}. In the present paper we prove new weighted Hardy-Poincar\'e, Rellich type inequalities as well as improved version of our Uncertainty principle inequalities on a Riemannian manifold $M$. In particular, we obtain sharp constants for these inequalities on the hyperbolic space $\mathbb{H}^n$.


Introduction
The classical Hardy, Rellich and Heisenberg-Pauli-Weyl (uncertainty principle) inequalities play important roles in many questions from spectral theory, harmonic analysis, partial differential equations, geometry as well as quantum mechanics. In order to motivate our work, we present these three classical (sharp) inequalities on the Euclidean space R n . The Hardy inequality states that for n ≥ 3 where φ ∈ C ∞ 0 (R n ). Here the constant ( n−2 2 ) 2 is sharp, in the sense that Another inequality involving second order derivatives is the Rellich inequality [20]: where φ ∈ C ∞ 0 (R n ), n ≥ 5 and the constant n 2 (n−4) 2 16 is again sharp.( There are also versions for lower dimensions under additional hypotheses.) The classical Heisenberg-Pauli-Weyl inequality, a precise mathematical formulation of the uncertainty principle of quantum mechanics, states that: for all f ∈ L 2 (R n ). Here the constant n 2 4 is sharp and also it is well-known that equality is attained in (1.3) if and only if f is a Gaussian (i.e. f (x) = Ae −α|x| 2 for some A ∈ R, α > 0).
These inequalities have been extensively studied in the Euclidean setting and now the literature on this topic is quite vast and rich, encompassing many generalizations and refinements, e.g. [2], [10], [6], [1], [3], [12], [8], [13] and references therein. Many new developments are still forthcoming. For instance, Tertikas and Zographopoulos [22] give a sharp Rellich-type inequality and its improved versions which involves both first and second order derivatives: where φ ∈ C ∞ 0 (R n ), n ≥ 5 and the constant n 2 4 is sharp. On the other hand the Euclidean results mentioned above continues to be a source of inspiration for the problem of finding analogues inequalities in the setting of Riemannian manifolds. There has been continuously growing literature in this direction, e.g. [7], [9], [14], [4], [18], [23], [17], [19], and the references therein. For instance, in an interesting paper Carron [7] obtained the following weighted L 2 -Hardy inequality on a complete non-compact Riemannian manifold M: , α ∈ R, C > 1, C + α − 1 > 0 and the weight function ρ satisfies |∇ρ| = 1 and ∆ρ ≥ C ρ in the sense of distribution. For complete non-compact Riemannian manifolds, under the same geometric assumptions on the weight function ρ we obtained in [17] an L p version of (1.5) (where 1 < p < ∞ and C + 1 + α − p > 0): as well as a Rellich-type inequality (where α < 2, C + α − 3 > 0): where ∆ is the Laplace-Beltrami operator on M.
We also found an L p Heisenberg-Pauli-Weyl uncertainty principle type inequality (for a complete noncompact Riemannian manifold) and an L 2 version with a (nonnegative) remainder term. In the specific case when the manifold M is the hyperbolic space H n , we obtained sharp constants for the Hardy and Rellich-type inequalities, and explicit (not sharp) constants for the Heisenberg-Pauli-Weyl uncertainty inequalities.
In the present paper we continue our investigation on Hardy, Rellich and Heisenberg-Pauli-Weyl type inequalities. The plan of the paper is as follows. In Section 2 we first prove a new form of weighted Hardy-Poincaré type inequality and then we prove various improved versions of the weighted Hardy inequality (1.5)( in the sense that nonnegative terms are added in the right hand side of (1.5)). We note that these improved inequalities are the main tool in proving improved Rellich type inequalities. In Section 3 we first prove a weighted analogue of (1.4) and then obtain improved versions. Section 4 is devoted to the study of Heisenberg-Pauli-Weyl (uncertainty principle) type inequalities where we obtain better constants than those of [17] and prove sharp analogue of the classical uncertainty principle inequality (1.3) on the Hyperbolic space H n . In each section we first prove inequalities in the context of a general complete Riemannian manifold. Then, turning our attention to hyperbolic space H n , we consider specific weight functions and obtain inequalities with explicit and usually sharp constants.

Weighted Hardy-Poincaré type inequalities
Throughout this paper, M denotes a complete noncompact Riemannian manifold endowed with a metric g. We denote by dV , ∇, and ∆ respectively the Riemannian volume element, the Riemannian gradient and the Laplace-Beltrami operator on M.
We begin this section by proving a new form of the Hardy-Poincaré type inequality for a complete noncompact Riemannian manifold M with a weight function ρ modelled on the distance from a point. (In this context the hypotheses |∇ρ| = 1 and ∆ρ ≥ C ρ seem to be geometrically quite natural.) One advantage of this set-up is that it implies and thus provides another (shorter) proof of (1.6) above (Theorem 2.1 in [17]) as explained in the Remark below.
Multiplying both side of (2.2) by ρ α |φ| p and integrating over M yields As an immediate consequence of divergence theorem we have An application of Hölder's and Young's inequality yields for any ǫ > 0. Therefore Note that the function ǫ −→ ǫ −p C + α + 1 − (p − 1)ǫ −p/(p−1) attains the maximum for , and this maximum is equal to C+α+1 p p . Now we obtain the desired inequality: Remark. Applying the Cauchy-Schwarz inequality to |∇ρ · ∇φ|, replacing α with α − p and using |∇ρ| = 1 yields the weighted L p -Hardy inequality (1.6).
We will give a sharp version of Theorem 2.1 in the hyperbolic space H n . Recall that the hyperbolic space H n (n ≥ 2) is a complete simple connected Riemannian manifold having constant sectional curvature equal to −1. There are several models for H n and we will use the Poincaré ball model B n in this paper. The Poincaré ball model for the hyperbolic space is: give an orthonormal basis of the tangent space at , thus the hyperbolic gradient and the Laplace Beltrami operator are: ∆ H n u = λ −n div(λ n−2 ∇u); where ∇ and div denote the Euclidean gradient and divergence in R n , respectively.
The hyperbolic distance d H n (x, y) between x, y ∈ B n in the Poincare ball model is given by the formula: .
From this we immediately obtain for x ∈ B n , which is the distance from x ∈ B n to the origin. Moreover, the geodesic lines passing through the origin are the diameters of B n along with open arcs of circles in B n perpendicular to the boundary at ∞, ∂B n = S n−1 = {x ∈ R n : |x| = 1}. The hyperbolic volume element is given by : where dx denotes the Lebesgue measure in B n and dσ is the normalized surface measure on S n−1 .
A hyperbolic ball in B n with center 0 and hyperbolic radius R ∈ (0, ∞) is defined by and note that B R (0) is also Euclidean ball with center 0 and radius S = tanh R 2 ∈ (0, 1).
Note that we have the following two relations for the distance function d = log( 1+|x| 1−|x| ) We are now ready to give a sharp version of Theorem 2.1 above in the hyperbolic space H n . Here ρ is chosen to be the distance function from the origin in the Poincaré ball model for the hyperbolic space H n .
Then we have: Proof. The inequality follows from Theorem 2.1. We show that n+α p p is the best constant in (2.4): It is clear that If we pass to the inf in (2.5) we get that n+α p p ≤ C H . We only need to show that C H ≤ n+α p p and for this we use the following family of radial functions where ǫ > 0. Notice that φ ǫ (d) can be approximated by smooth functions with compact support in H n . A direct computation shows that Let us denote by B 1 = {x ∈ H n : d ≤ 1} the unit ball with respect to the distance d. Hence and then we have On the other hand It is clear that n+α p + ǫ p ≥ C H and letting ǫ −→ 0 we obtain n+α p p ≥ C H . Therefore We now prove an improved L 2 weighted Hardy inequality involving two weight functions ρ and δ modeled on distance functions from a point and distance to the boundary of a domain Ω with smooth boundary.
Multiplying both sides of (2.8) by the ρ α and applying integration by parts over M gives Choosing β = 1 − α − C 2 gives the following We now focus on the second term on the right-hand side of this inequality. Let us define a new variable ϕ(x) := δ(x) −1/2 ψ(x) where δ(x) is a nonnegative function and δ(x) ∈ C 2 0 (M). It is clear that Since −div(ρ 1−C ∇δ) ≥ 0 and ψ = ρ C+α−1 2 φ then we get Substituting (2.11) into (2.10) gives the desired inequality: Our next goal is to find model functions which satisfies the assumption of the above theorem. A straightforward computation shows that δ = log( R ρ ) satisfies the differential inequality −div(ρ 1−C ∇δ) ≥ 0. As a consequence of Theorem 2.3 we have the following weighted L 2 -Hardy-type inequality on the hyperbolic space H n which has a logarithmic remainder term. The sharpness of the constant ( n+α−2 2 ) 2 follows as in [17] Theorem 3.1.
Let B R = {x ∈ B n | d < R} be a hyperbolic ball with center 0 and hyperbolic radius R. It is clear that δ := R − d is the distance function of the point x ∈ B R to the boundary of B R and satisfies the differential inequality in Theorem 2.3. Therefore we have: Corollary 2.2. Let B R be a hyperbolic ball with center 0 and hyperbolic radius R. Let d = log( 1+|x| 1−|x| ) and δ := R − d, α ∈ R, n + α − 2 > 0. Then we have: for all φ ∈ C ∞ 0 (B R ) and the constant n+α−2 2 2 is sharp.
Hardy-Sobolev-Poincaré inequalities. The following sharp form of the Sobolev inequality on the hyperbolic space H n is due to [16]. It states that for all φ ∈ C ∞ 0 (H n ): (2.14) n is the sharp constant for the Sobolev inequality on R n , |S n | is the volume of the n-dimensional unit sphere in R n+1 and the constant B n = n(n−2) 4 is sharp for n ≥ 4. Recently, sharp form of the inequality (2.13) in three dimensional hyperbolic space H n has been proved by Benguria, Frank and Loss [5].
The Sobolev inequality (2.14) and Hardy inequality [17] yield the following Hardy-Sobolev inequality in H n .
Before we state and prove our next theorem, we first recall the (Euclidean) weighted Sobolev inequality of Fabes-Kenig-Serapino [11] which plays an important role in our proof. They proved the following inequality : where B r is a ball in R n , φ ∈ C ∞ 0 (B r ), w(B r ) = Br w(x)dx, 1 < p < ∞, 1 ≤ k ≤ n n−1 + ǫ, ǫ > 0 and the weight function w belongs to Muckenhoupt's class A p . In particular, if the weight function w belongs to Muckenhoupt's class A 2 then k can be taken equal to n n−1 + ǫ and this is sharp. Recall that a weight function w belongs to Muckenhoupt's class where the supremum is taken over all balls B in R n (see [21]).
Motivated by the classical work of Brezis and Vázquez [6], our next theorem shows that sharp weighted Hardy inequality on the hyperbolic space H n can be improved by a weighted Sobolev term.
It is easy to see that |∇d| 2 = λ 2 and integrating (2.17) over B n , we get Applying integration by parts to the middle integral on the right-hand side of (2.18), we obtain One can show that A direct computation shows that ∆d = λ 2 r + n − 1 r λ and ∇d · ∇λ = λ 3 r. Substituting these above (2.21) We can easily show that λr 2 + 1 λr ≥ 1 d .

Now we substitute (2.22) into (2.19) and we get
Note that the function β −→ −β 2 − β(α + n − 2) attains the maximum for β = 2−α−n 2 , and this maximum is equal to ( n+α−2 2 ) 2 . Therefore we have the following inequality Using the fact d ≤ λr we get Notice that the weight function r 2−n is in the Muckenhoupt A 2 class. We now apply weighted Sobolev inequality (2.15) to the second integral term on the right hand side of (2.23) and obtain where q > 2 and c 1 = 1 q . This completes the proof.

Rellich-type inequalities
In this section we prove weighted Rellich-type inequality and its improved versions which connects first to the second order derivatives. The following is the weighted analogue of (1.4) in the setting of Riemannian manifold M.
Theorem 3.1. Let M be a complete Riemannian manifold of dimension n > 1. Let ρ be a nonnegative function on M such that |∇ρ| = 1 and ∆ρ ≥ C ρ in the sense of distribution where C > 1. Then the following inequality is valid: for all compactly supported smooth function φ ∈ C ∞ 0 (M \ ρ −1 {0}), 7−C 3 < α < 2 Proof. A straightforward computation shows that Then we have: Proof. The inequality follows from Theorem 3.1. To show that the constant ( n−α 2 ) 2 is sharp, we use the following family of functions as in [17]: Notice that φ ǫ (d) can be well approximated by smooth functions with compact support in H n and direct computation shows that (n−α) 2 4 is the best constant in (3.9): The following inequality is an improved version of the Rellich-type inequality (3.1) for bounded domains. Let Ω be a bounded domain with smooth boundary ∂Ω in a complete Riemannian manifold of dimension n > 1. Let ρ be a nonnegative function on M such that |∇ρ| = 1, ∆ρ ≥ C ρ and −div(ρ 1−C ∇δ) ≥ 0 in the sense of distribution, where C > 1. Then the following inequality is valid: for all compactly supported smooth function φ ∈ C ∞ 0 (M \ ρ −1 {0}), 7−C 3 < α < 2 and K(C, α) = (C+1−α)(C+3α−7) 16 .
Proof. The proof is similar to the proof of Theorem 3.1. The only difference is that we apply improved Hardy-type inequality (2.7) to the first term on the right hand side of (3.6).
The following corollaries are the direct consequences of the Theorem 3.3.
Proof. The inequality follows from Theorem 4.1. In order to achieve equality, inspired by the Euclidean case, we consider hyperbolic analogues of Gaussians: φ(x) = Ae −αd 2 where A ∈ R and α > 0. A straightforward but tedious calculation shows that φ(x) = Ae −αd 2 is the minimizer where α = ( n−1 n−2 ) n − 1 + 2π C n−2 Cn and C n = H n e −αd 2 dV .
Remark. Note that even though φ(x) = Ae −αd 2 does not have a compact support, it can be approximated by such functions yielding that (4.3) is sharp.
There is a natural link between Hardy, Heisenberg-Pauli-Weyl and Rellich type inequalities. For instance, using the Rellich-type inequality II (3.1) we have the following second order Heisenberg-Pauli-Weyl inequality.  As an immediate consequence of the Theorem 4.3 we have the following second order Heisenberg-Pauli-Weyl inequality with an explicit constant on the hyperbolic space H n .