Sharp subcritical and critical Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}$\end{document} Hardy inequalities on the sphere

We obtain sharp inequalities of Hardy type for functions in the Sobolev space W1,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W^{1,p}$\end{document} on the unit sphere Sn−1 in Rn. We achieve this in both the subcritical and critical cases. The method we use to show optimality takes into account all the constants involved in our inequalities. We apply our results to obtain lower bounds for the the first eigenvalue of the p-Laplacian on the sphere.


Introduction
Hardy inequalities have been studied extensively on various types of manifolds (see for example [3,4,8,11], the references therein, and the recent cited papers that are too many to mention). To our best knowledge, Xiao [15] was the first to look at Hardy inequalities in the particular case of the Euclidean sphere. He developed the idea used in [9] to obtain sharp inequalities of Hardy and Rellich type on Riemannian manifolds. The Laplacian of the geodesic distance on the sphere changes sign (see formula (3.13) of Lemma 3.4 below). This makes known results on compact manifolds not easy to apply directly. Xiao [15] obtained L 2 inequalities of the Hardy type on the sphere S n , n ≥ 3. These results were complemented in [1] in the limiting case where optimal L 2 inequalities of the Hardy type were proved on S 2 . The results in [15] were also extended in [12] to L p (S n ), 1 < p < n, n ≥ 3.
In [1,12,15], the singularity is assumed to be at either the north or south pole so that the geodesic distance will be simply the polar angle. Hence, if the singularity is not polar, we must rotate the local axes in order to apply these inequalities. However, we should not need to rotate the axes. It is not physically plausible as we could be dealing with a punctured sphere missing a closed connected piece, or a sphere with a crack missing an open simple curve. This motivates us to look for L p Hardy inequalities in which the singularity is the geodesic distance from an arbitrary point.
The general geodesic distance was very recently considered in [2,16]. The proofs in [2,16] are based on a formula for the Laplacian of the geodesic distance. No reference was provided for that formula, and no proof of it was given either. We also noted that the definition of the geodesic distance on S n adopted in [2,16] is not specified. Such a definition is important to understand the set up of the inequalities. This is also technically important since the singularities in the inequalities involve trigonometric functions. That in turn necessitates determining whether the range of the geodesic distance is [-π 2 , π 2 ] or [0, π].
The results in [2] are supposed to generalize the L 2 Hardy inequality presented in [16] to an L p inequality on S n where 1 < p < n and n ≥ 3. We revisit the proof presented in [2] in [17], where we additionally prove the limiting case L n Hardy-type inequalities on the sphere S n , n ≥ 2, with optimal coefficients, considering the general geodesic distance and adopting Xiao's method.
When it comes to the sharpness of the coefficients, all the results in [1,2,12,15,16] are based on the same principle that we find insufficient. The method implemented is also unnecessarily involved at times. Inequalities of Hardy type obtained in [1,2,12,15,16] on S n take the generic form where u ∈ C ∞ (S n ), and f is a continuous function of the geodesic distance ρ. Sharpness of the constants A n,p , B n,p and C n,p is claimed to be proved by showing that However, the latter does not prove that the constants B n,p and C n,p are both the smallest possible.
We prove sharp L p Hardy inequalities on the sphere S n in R n+1 in both the subcritical and critical exponent cases. We follow a method of proof different from that used in [1,2,12,15,16]. The method we adopt is fairly simple and requires fewer computations. Before delving into the derivation of the inequalities, we use explicit formulas for the geodesic distance, the surface gradient and the Laplace-Beltrami operator on the n-dimensional sphere to demonstrate some basic properties of the geodesic distance on which we rely heavily in obtaining our results.
In addition to proving (1.1), we show the optimality of all the constants in our inequalities by proving that To achieve this, we exploit a formula for integration over spheres (see (2.5) below) to calculate the ratios above for explicit functions in the appropriate Sobolev space. Finally, inspired by the interesting applications given in [2], we use our inequalities to obtain the lower bounds for the first eigenvalue of the p-Laplacian on the sphere.

The Sobolev space W 1,p (S n-1 )
It is useful to define the weak Laplace-Beltrami gradient of a function f ∈ L 1 (S n-1 ). Let f ∈ C ∞ (S n-1 → R). Then, by the divergence theorem, we have for any vector field V ∈ C ∞ (S n-1 → T(S n-1 )), where T(S n-1 ) is the tangent bundle on the smooth manifold S n-1 . Therefore, f is weakly differentiable if there exists a vector field Such a vector field f , if it exists, is called the weak surface gradient of f . The weak surface gradient is unique up to a set of measure zero. As shown in ( [5], Proposition 3.2, page 15) The definition (2.6) is equivalent to defining W 1,p (S n-1 ) as the completion of the space C ∞ (S n-1 ) in the usual Sobolev norm.
In the next section, we show some interesting properties of the geodesic distance on the sphere that carry on to all dimensions.

The gradient and Laplacian of the geodesic distance on the sphere
The geodesic distance d on the sphere S n-1 has a gradient and Laplacian analogous to those of the Euclidean metric. We demonstrate that |∇ S n-1 d| S n-1 = 1 and that S n-1 d = (n -2) cos d/ sin d, in any dimension n ≥ 2. Unlike with the Euclidean distance, the Laplacian of the geodesic distance d changes sign on the sphere. We start with showing that the Kronecker delta. Lemma 3.1 Let n ≥ 2 and let ∇ S n-1 be the gradient on the unit sphere S n-1 in R n . Then, Proof Lemma 3.1 is trivial in the dimension n = 2 and similarly easily verifiable when n = 3 by the computation cos θ 1 cos θ 2 θ 1 -sin θ 2 θ 2 , m = 2; cos θ 1 sin θ 2 θ 1 + cos θ 2 θ 2 , m = 3.
Suppose n ≥ 4. Again, the identity (3.1) is easy to prove when m = 1, 2, and so is the identity (3.2) when 1 ≤ , m ≤ 2. Observe that, for all n ≥ 4, Fix m ≥ 3. We obtain (3.1) from the calculation and the orthonormality of the set { θ j } n-1 j=1 along with the identity Indeed, one can write Now, we turn to the identity (3.2). Assume, losing no generality, that 1 ≤ < m. Then, tedious yet straightforward computation uncovers that and when 2 ≤ ≤ m -1 we have The next lemma shows that the components x m are eigenfunctions of the Laplace-Beltrami operator (2.2).
where are the differential operators Then, to prove (3.5), it suffices to establish that Straightforward calculations affirm (3.6) when m = 1. We prove (3.6) by induction. Assume (3.6) holds true for some 1 ≤ m ≤ n -2. Let us define Consequently, what remains to prove is Calculating further, we find which is easy to verify. Having proved (3.6), we can exploit its validity for m = n -1 in particular to prove (3.7). Write x n = x n-1 δ n-1 with δ n-1 ( n-1 ) := sin θ n-1 / cos θ n-1 . Arguing as above, we discover that This reduces (3.7) to which is simple to check.
Proof Using Lemma 3.1, we obtain This shows (3.10). We also obtain (3.11) as a direct consequence of Lemma 3.2, since λ( n-1 , n-1 ) is a linear combination of eigenfunctions of S n-1 that all correspond to the eigenvalue -(n -1).

Subcritical L p Hardy inequalities
Let S n-1 be the unit sphere in R n , n ≥ 4. Let 1 < p < n -1 and consider the following nonlinear positive functionals on W 1,p (S n-1 − → R): Define also the constant α n,p := np -1 p .
which exists for p < n -1.
We show that the functionals T p , T p , S p , and S p are all well defined and related by the following L p inequalities of Hardy type: Then, u sin d ∈ L p (S n-1 ), when 1 < p < n -1, and u | tan d| ∈ L p (S n-1 ), when 2 ≤ p < n -1. Moreover, Proof Let us start with the inequality (4.1). Using a density argument, we may assume u ∈ C ∞ (S n-1 ). Recalling the identities (3.12) and (3.13) in Lemma 3.4, we can compute S n-1 sin d = -sin d + (n -2) Integrating both sides of (4.3) against |u| p / sin p-1 d over S n-1 , then employing the divergence theorem, we obtain (n -2) Observe that we simplified the latter integral using the fact |∇ S n-1 d| = 1. So far, it suffices to require that p > 1 to make sense of the gradient of |u| p . Invoking Hölder's inequality then applying Young's inequality and using (3.12) once more, we can bound with β > 0 as yet undetermined. Inserting the estimate (4.5) into the inequality (4.4) then rearranging gives (np -1) Note here that Remark 4.1 justifies this manipulation of the terms of (4.4). We proceed from (4.6) by simply replacing the factor cos 2 d by 1 -sin 2 d in the first integral to obtain The optimal value of β for (4.7) is easily determined through finding the maximum point Hence, the inequality (4.7) takes the form (4.1).
With the exception of some technical details, the proof of (4.2) is similar to that of (4.1). Instead of using (4.3), we capitalize on (3.13). Let 2 ≤ p < n -1. Integration by parts on S n-1 yields S n-1 |u| p | tan d| p dσ n-1 Observe that the restriction 2 ≤ p < n-1 is necessary to make sense of ∇ S n-1 | cos d| p-2 cos d.
It also guarantees the convergence of the integral S n-1 1 | tan d| p 1 cos 2 d dσ n-1 . This is inferred by formula (2.5) that asserts Since |∇ S n-1 d| = 1, then, applying Hölder's inequality followed by Young's inequality analogously to (4.5) gives for any β > 0. We can also split Returning to (4.8) with (4.9) and (4.10) we deduce that (4.11) The optimal value of β for (4.11) is α p-1 p n,p . This proves the inequality (4.2). Remark 4.2 The values n 2 ≤ p < n -1 can be admitted in (4.14) if the supremum is taken over nontrivial functions in L p (S n-1 ) with a weak gradient in the weighted space L p (S n-1 ; | cos d( n-1 , n-1 )| p dσ n-1 ).
An important consequence of (4.14) is that, in any dimension n ≥ 4, and for every 1 < p < n -1, we can find u ∈ C ∞ (S n-1 ) such that α p n,p S p (u) > G p (u) + α p-1 n,p S p (u).
It similarly follows from (4.17) that the inequality α p n,p T p (u) ≤ F p (u) + α p-1 n,p T p (u), p > 2 does not hold true on C ∞ (S n-1 ), n ≥ 4. More interestingly:

Theorem 4.3 The inequality
is generally false on W 1,p (S n-1 ) for every 1 < p < n -1, n ≥ 4. In particular, there exists u ∈ H 1 (S n-1 ) such that for every n > 4.

Critical L p Hardy inequalities
Let n ≥ 2 and define the following nonlinear positive functionals on W 1,n (S n − → R): Remark 5.3 The inequality (5.2) is proved in [17] using a different method.

A lower bound for the first eigenvalue of the p-Laplacian on the sphere
Let M be a compact connected manifold without boundary. The p-Laplacian on M is the operator given by p u := -div g (|∇ g u| p-2 ∇ g u), where ∇ g is the gradient induced by the Riemannian metric g on M and div g is the adjoint of ∇ g for the L 2 -norm induced by the metric g on the space of differential forms. The p-Laplacian is associated with the p-energy functional E p (u) := M |∇ p u| p d g , where d g is the Riemannian volume element induced by g.