The classical Rellich inequality states that , for and all ,
The constant is optimal and never archived. Under additional conditions there are also versions for lower dimensions. There has been a lot of research concerning the Rellich inequality on the Euclidean space due to its applications to spectral theory, harmonic analysis, geometry and partial differential equations. We see [2,3,4,5,6] and the references therein.
The validity of the Rellich inequality on a manifold and its best constants allows people to obtain qualitative properties on the manifold. For complete noncompact Riemannian manifolds, under some geometric assumptions on the weight function , Kome and Özaydin  proved that for (where )
Particularly, they also obtained the improved versions of a Rellich-type inequality which involves both first and second order derivatives in the Poincaré conformal disc model ()
where and is the geodesic distance. Furthermore, the constant is sharp. Along this line, we refer to [4,7,8,9,10,11] and so on.
However, there are not many literatures discussing the Rellich inequality on the sphere so far. See [12,13,14] for details. In  Xiao derived the following inequality
for , where is the geodesic distance from p to x on and C is some positive constant. Moreover, the constant is sharp.
In this short note we will obtain another type of Rellich inequality on the sphere and also give the corresponding sharp constant. Our main theorem is as follows:
Let () be the n-sphere with sectional curvature 1 and p be a fixed point in . Then for any function ,
where and the constant is sharp.
In Euclidean spaces (resp. a Riemannian manifold, the Poincaré conformal disc model), the Laplacian of the distance function (resp. some weighted function) equals to (resp. is not less than , ). Thus the Rellich inequality certainly contains the term (resp. , ). Since on the sphere the Laplacian of the distance function is when d is smooth (see  p. 207), the terms and are naturally involved. So, it is a bit different in form from that in Euclidean spaces and some other type of Rellich inequalities. It is interesting that, even though the coefficient is replaced by an arbitrary number, the constant is still sharp. To prove the result, we give some modifications in constructing the auxiliary function, and then do calculations in two hemispheres by using the antipodal points. The remainder of the approaches used are similar to Xiao’s paper . See also in [7,16].
2. The Proof of the Main Result
Proofof Theorem 1.
Denote by the distance function from the fixed point . Let f be a smooth function in , where q is the antipodal point of p. Then
where we have used in the sphere. To estimate , we put Then
In what follows, we show the constant is sharp. The skill is borrowed from  (see also ). Let be a smooth function such that and
Let . For sufficient small , Set
Observe that can be approximated by smooth functions on the sphere .
Let q be the antipodal point of p. Then and for any point we have . Since the constructed function possesses a fair degree of bilateral symmetry on the sphere, it is easier to compute in the following by using the antipodal points p and q.
Next we are to estimate . When , the distance function is smooth, and thus
and when , one can get the same formula as above by letting . Therefore,
and thus by Minkowski inequality,
A straightforward calculation yields
Since can be approximated by smooth functions on the sphere , then, by (5)–(7), it holds that
Letting , we have
This completes the proof. □
This project is supported by AHNSF (1608085MA03), NNSFC(11471246) and KLAMFJPU (No. SX201805).
Conflicts of Interest
The authors declare no conflict of interest.
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