The Hardy-Rellich inequality and uncertainty principle on the sphere

TL;DR

This paper establishes Hardy-Rellich inequalities and uncertainty principles on the sphere, identifying dimensions where the inequalities hold with finite constants and deriving new uncertainty relations for functions on the sphere.

Contribution

It proves Hardy-Rellich inequalities on the sphere for specific dimensions and derives new uncertainty principles, including cases with zero mean and without, which were previously unknown.

Findings

Hardy-Rellich inequality holds for d=2 and d≥4 but not for d=3.

Optimal constant c_d=8/(d-3)^2 for certain dimensions.

New uncertainty principles on the sphere for functions with zero mean and without zero mean.

Abstract

Let $\Delta_0$ be the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$. We show that the Hardy-Rellich inequality of the form $$ \int_{\mathbb{S}^{d-1}} \left | f (x)\right|^2 d\sigma(x) \leq c_d \min_{e\in \mathbb{S}^{d-1}} \int_{\mathbb{S}^{d-1}} (1- \langle x, e \rangle) \left |(-\Delta_0)^{\frac{1}{2}}f(x) \right |^2 d\sigma(x) $$ holds for $d =2$ and $d \ge 4$ 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 $d\ge 6$. This inequality yields an uncertainty principle of the form $$ \min_{e\in\mathbb{S}^{d-1}} \int_{\mathbb{S}^{d-1}} (1- \langle x, e \rangle) |f(x)|^2 d\sigma(x) \int_{\mathbb{S}^{d-1}}\left |\nabla_0 f(x)\right |^2 d\sigma(x) \ge c'_d $$ 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.