Sharp L^p Hardy type and uncertainty principle inequalities on the sphere

Abstract

This paper studies L p-version of the Hardy type inequalities on the geodesic sphere of constant sectional curvature and establishes that the corresponding constant is sharp. Furthermore, the inequalities obtained are used to derive an uncertainty principle inequality and another inequality involving the first nonzero eigenvalue of the p-Laplacian on the sphere.

A sharp Lp-Hardy type inequality on the n-sphere

“…In this paper, we present a more general version of L p Hardy inequalities on the unit n-sphere and show that the associated constant is the best possible, which is a generalization of those in [23] and [22].…”

“…is sharp. Based on the results above, Abolarinwa-Apata [1], Abolarinwa-Rauf-Yin [23], and Sun-Pan [24] further gave some L p -Hardy inequalities on the sphere.…”

“…The L p -Hardy inequalities obtained in [1,23,24] are divided into two cases: 1 p < 2 and p 2. They declare that if 1 p < 2, then…”

“…To prove the result, we follow the arguments in [22] (see also [23]) with some modifications. First, we construct an auxiliary function by utilizing the symmetry of the sphere, and then using the antipodal points.…”

“…When α = 0 the corresponding inequality is obtained in [23], where it is divided into two cases: 1 p < 2 and p 2 due to some technical reasons. In fact, we can combine them into a unified inequality as above, and thus Corollary 1 is in a form more concise than that in [23]. [22], the coefficient in the right-hand side is n(n − 2)/4.…”