Hardy type inequalities on the sphere

In this paper, we consider the 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. By the divergence theorem, we establish the 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. Furthermore, we also obtain their best constants. Our results can be regarded as the extension of Xiao’s (J. Math. Inequal. 10:793-805, 2016).


Introduction
The classical Hardy inequality states that for N ≥  and p >  with u ∈ C ∞  (R N ) \ {} and | N-p p | p the best constant. In recent years, many papers have been dedicated to improved versions of the above inequality because of its application to singular problems. We see [-] and the references therein. Hardy inequalities are a subfamily of the Caffarelli-Kohn-Nirenberg inequalities. In a Riemannian manifold, the knowledge of the validity of these inequalities and their best constants allows us to obtain qualitative properties on the manifold [-].
Recently, Carron [] studied the weighted L  -Hardy inequalities on a Riemannian manifold under some geometric assumptions on the weighted function ρ and obtained the following inequality: where the weighted function ρ satisfies |∇ρ| =  and ρ ≥ C ρ .

Our main results
Our main result is the following L p -Hardy inequality on the sphere.
where d(x, q) is the geodesic distance of x and q. Moreover, ( N-p p ) p is the best constant.
Remark  When p = , inequality () was obtained by Xiao []. When  < p < , dV cannot control the right-hand side of () since sin p- d(x, q) is large enough when x is close to q. Therefore, we use S N |f | p dV as the left-hand side of the inequality instead of S N |f | p Although our approach is similar to Xiao's [], the appearance of general p makes the calculation more complicated, especially for the existence of the constant C in Theorem .

Preliminaries and notations
Let S N = {x = (x  , x  , . . . , x n+ ) ∈ R N+ ; |x| = } be the unit sphere of dimension N . Let (θ  , θ  , . . . , θ N ) be the angular variables on S N . For simplicity, we define θ N = θ , where x N+ = |x| cos θ N . By polar coordinates associated with θ , we get where dσ is the canonical measure of the unit sphere S N- . We say that a function f on S N is an angular function if f depends only on θ . In this case, See []. For more basic properties on the sphere, we refer to [].

The proof of Theorem 1
Now we give the proof of Theorem . Let f = ρ γ ϕ, ρ = sin θ , γ = -N-p p , by the calculation that appeared in [], one has then integration by parts gives Since The previous inequality can be written as follows: In order to get our result, we rewrite the above inequality as As we know, for p > , and for p = , Also, by a similar calculation, one has for p >  Therefore, there exist a constant θ  >  small enough and θ  < π close to π such that for any θ ∈ (, θ  ] ∪ [θ  , π), one has that Thus from inequality (), one has which is exactly inequality ().
While for  < p < , we get from inequality () that By a similar calculation, we get that for  < p <  we get that there exists a constant θ  >  small enough such that