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
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
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.
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Acknowledgments
The work of the first author was supported in part by NSERC Canada under Grant RGPIN 311678-2010. The work of the second author was supported in part by NSF Grant DMS-1106113 and a Grant from the Simons Foundation (# 209057 to Y. Xu)
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Communicated by Pencho Petrushev.
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Dai, F., Xu, Y. The Hardy–Rellich Inequality and Uncertainty Principle on the Sphere. Constr Approx 40, 141–171 (2014). https://doi.org/10.1007/s00365-014-9235-5
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DOI: https://doi.org/10.1007/s00365-014-9235-5