Sharp subcritical and critical $L^{p}$ Hardy inequalities on the sphere

We prove sharp inequalities of Hardy type for functions in the Sobolev space $W^{1,p}$ on the unit sphere $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$. 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.


Introduction
Since the authors in [5] introduced an idea to obtain sharp inequalities of Hardy and Rellich type on noncompact Riemannian manifolds, there has been interest in establishing the corresponding inequalities on the sphere.Obviously, the method given in [5] does not apply directly to the compact manifold S n .Another technical difficulty comes from the fact that the Laplacian of the geodesic distance on the sphere changes sign.Xiao [8] was the first to make progress on this problem.He obtained L 2 inequalities of the Hardy type on the sphere S n , n ≥ 3.These results were complemented in [2] in the limiting case where optimal L 2 inequalities of the Hardy type were proved on S 2 .Xiao's results were also extended to L p (S n ), 1 < p < n, n ≥ 3 in [7].
In [2,7,8], the singularity is assumed to be at either the north or south pole so that the geodesic distance will be simply the polar angle.So, if the singularity is not polar, we must rotate the local axes in order to apply these inequalities.But 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 is very recently considered in [1,9].The proofs in [1,9] are based on a formula for the Laplacian of the geodesic distance.Inconveniently, no reference was provided for that formula, and no proof of it was given either.More inconveniently, the definition of the geodesic distance on S n adopted in [1,9] is not specified.Such 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].In [9], an L 2 Hardy inequality is proved in high dimensions using Xiao's method.The results in [1] are supposed to generalize the L 2 Hardy inequality presented in [9] to an L p inequality on S n where 1 < p < n and n ≥ 3. Unfortunately, the proof presented in [1] requires revision.We discuss that in detail in [3], where we additionally prove 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,[7][8][9] 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,[7][8][9] 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 But 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,[7][8][9].The method we adopt is fairly simpler and require less 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.
Besides proving (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 (8) below) to calculate the ratios above for explicit functions in the appropriate Sobolev space.

Preliminaries
Let n ≥ 2 and define Then any point on the unit sphere S n−1 in R n has the spherical coordinates parametrization (x m (Θ n−1 )) n m=1 , where The gradient ∇ S n−1 on the sphere S n−1 is then given by where θ j is an orthonormal set of tangential vectors with θ j pointing in the direction of increase of θ j .Moreover, the Laplace-Beltrami operator ∆ S n−1 is given by Identifying each point (x m (Θ n−1 )) n m=1 ∈ S n−1 with its parameters Θ n−1 , we can express the geodesic distance where

A useful formula for integration over
where . (See [6], Appendix D).

The Sobolev space
It is useful to define the weak Laplace -Beltrami gradient of a function Then, by the divergence theorem, we have )) 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 Ψ f ∈ L 1 (S n−1 → T (S n−1 )) such that Such vector field Ψ 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 ( [4], Proposition 3.2., page 15) The definition ( 7) 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.

A formula for integration on the sphere
where In the next section, we show 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 Unlike with the Euclidean distance, the laplacian of the geodesic distance d changes sign on the sphere.We start with showing that x Proof.Lemma 1 is trivial in the dimension n = 2 and similarly easily verifiable when n = 3 by the computation Suppose n ≥ 4. Again, the identity ( 9) is easy to prove when m = 1, 2, and so is the identity (10) when 1 ≤ ℓ, m ≤ 2. Observe that, for all n ≥ 4, Fix m ≥ 3. We get (9) 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 (10).Assume, losing no generality, that 1 ≤ ℓ < m.Then, tedious yet straightforward computation uncovers that and when 2 ≤ ℓ ≤ m − 1 we have The next lemma shows that the components x m are eigenfunctions of the Laplace -Beltrami operator (3): Proof.Write where ∆ ℓ are the differential operators Then, to prove (12), it suffices to establish that Straightforward calculations affirm (13) when m = 1.We prove (13) by induction.Assume (13) holds true for some 1 Consequently, what remains to prove is Calculating further, we find Therefore, (15) is equivalent to which is easy to verify.Having proved (13), we can exploit its validity for m = n − 1 in particular to prove (14).Write Arguing as above, we discover that This reduces (14) to which is simple to check.
be the function defined in (5).Then Proof.Using Lemma 1, we obtain This shows (17).We also get (18) as a direct consequence of Lemma 2, since λ(Θ n−1 , Φ 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 Remark 1. Formula (8) makes it clear that the integrals dσ n−1 are convergent when u is continuous.Indeed, recalling that We show that the functionals T p , T p , S p , S p are all well-defined and related by the following L p inequalities of Hardy type: Proof.Let us start with the inequality (22).Using a density argument, we may assume u ∈ C ∞ (S n−1 ).Recalling the identities ( 19) and (20) in Lemma 4, we can compute Integrating both sides of (24) 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 (19) once more, we can bound with β > 0 as yet undetermined.Plugging the estimate (26) into the inequality (25) then rearranging gives Note here that Remark 1 justifies this manipulation of the terms of (25).We proceed from (27) by simply replacing the factor cos 2 d by 1 − sin 2 d in the first integral of to get The optimal value of β for ( 28 With the exception of some technical details, the proof of ( 23) is similar to that of (22).Instead of using (24), we capitalize on (20).Let 2 ≤ p < n − 1. Integration by parts on S n−1 yields 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 This is inferred by formula (8) that asserts Since |∇ S n−1 d| = 1, then, applying Hölder's inequality followed by Young's inequality analogously to (26) gives for any β > 0. We can also split Returning to (29) with ( 30) and (31) we deduce that The optimal value of β for (32) is α p−1 p n,p .This proves the inequality (23).
It similarly follows from (38) that the inequality does not hold true on C ∞ (S n−1 ), n ≥ 4.More interestingly, Theorem 7. 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.