The Hardy-Rellich inequality and uncertainty principle on the sphere

2012/12/17 by Dai, Feng, Xu, Yuan
#33C45 #33C55 #42B10 #42C10 #43A75 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1212.3887

Abstract

Let Δ₀ be the Laplace-Beltrami operator on the unit sphere \mathbbS^d-1 of ℝ^d. We show that the Hardy-Rellich inequality of the form ∫_\mathbbS^d-1 | f (x)|² dσ(x) ≤ c_d min_e∈ \mathbbS^d-1 ∫_\mathbbS^d-1 (1- ⟨ x, e ⟩) |(-Δ₀)^(1)/(2)f(x) |² dσ(x) holds for d =2 and d ≥ 4 but does not hold for d=3 with any finite constant, and the optimal constant for the inequality is c_d = 8/(d-3)² for d =2, 4, 5 and, under additional restrictions on the function space, for d≥ 6. This inequality yields an uncertainty principle of the form min_e∈\mathbbS^d-1 ∫_\mathbbS^d-1 (1- ⟨ x, e ⟩) |f(x)|² dσ(x) ∫_\mathbbS^d-1 |∇₀ f(x) |² dσ(x) ≥ c'_d on the sphere for functions with zero mean and unit norm, which can be used to establish another uncertainty principle without zero mean assumption, both of which appear to be new. This paper is published in Constructive Approximation, 40(2014): 141-171. An erratum is now appended.

Related

• Uncertainty Principles on weighted spheres, balls and simplexes
• Uncertainty principle on weighted spheres, balls and simplexes
• Revisiting the Rellich inequality
• Hardy-Poincar 'e, Rellich and Uncertainty principle inequalities on\n Riemannian manifolds
• Optimal critical exponent L^p inequalities of Hardy type on the sphere via Xiao's method