Advances in Operator Theory

$L^p$-Hardy-Rellich and uncertainty principle inequalities on the sphere

Abimbola Abolarinwa and Timothy Apata

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Abstract

In this paper, we study the Hardy-Rellich type inequalities and uncertainty principle on the geodesic sphere. Firstly, we derive $L^p$-Hardy inequalities via divergence theorem, which are in turn used to establish the $L^p$-Rellich inequalities. We also establish Heisenberg uncertainty principle on the sphere via the Hardy-Rellich type inequalities. The best constants appearing in the inequalities are shown to be sharp.

Article information

Source
Adv. Oper. Theory, Advance publication (2018), 18 pages.

Dates
Received: 1 January 2018
Accepted: 15 April 2018
First available in Project Euclid: 10 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1525917618

Digital Object Identifier
doi:10.15352/aot.1712-1282

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

Keywords
Hardy inequalities Rellich inequalities uncertainty principle geodesic sphere compact manifold

Citation

Abolarinwa, Abimbola; Apata, Timothy. $L^p$-Hardy-Rellich and uncertainty principle inequalities on the sphere. Adv. Oper. Theory, advance publication, 10 May 2018. doi:10.15352/aot.1712-1282. https://projecteuclid.org/euclid.aot/1525917618


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References

  • E. Berchio, L. D'Ambrosio, D. Ganguly, and G. Grillo, Improved $L^p$-Poincaré inequalities on the hyperbolic space, Nonlinear Anal. 157 (2017), 146–166.
  • H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Comp. Madrid 10 (1997), no. 2, 443–469.
  • G. Carron, Inégalités de Hardy sur les vari ét és riemanniennes non-compactes, J. Math. Pures Appl., 76 (1997), no. 10, 883–891.
  • I. Chavel, Eigenvalues in Riemannian geometry. New York: Academic Press, 1984.
  • I. Chavel, Riemannian geometry: a modern introduction, second edition Cambridge Tracts in Mathematics, 108. Cambridge University Press, Cambridge, 2006.
  • F. Dai and Y. Xu, The Hardy-Rellich inequality and uncertainty principle inequalities on the sphere, http://arxiv.org/abs/1212.3887v3 [math.CA], 2014.
  • E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in $L^p(\Omega)$, Math. Z. 227 (1998), no. 3, 511–523.
  • M. M. Fall and F. Mahmoudi, Weighted Hardy inequality with higher dimensional singularity on the boundary, Calc. Var. Partial Differential Equations 50 (2014), no. 3-4, 779–798.
  • G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), no. 3, 207–233.
  • S. Gallot, D, Hulin, and J. Lafontaine, Riemannian geometry, Universitext. Springer-Verlag, Berlin, 1987.
  • N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 2, 313–356.
  • S. S. Goh and T. N. Goodman, Uncertainty principles and asymptotic behavior, Appl. Comput. Harmon. Anal. 16 (2004), no. 1, 69–89.
  • I. Kombe, Hardy, Rellich and uncertainty principle inequalities on Carnot groups., 2006.
  • I. Kombe and M. Özaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6191–6203.
  • I. Kombe and M. Özaydin, Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds, Trans. Amer. Math. Soc. 365 (2013), no. 10, 5035–5050.
  • I. Kombe and A. Yener, Weighted Hardy and Rellich type inequalities on Riemannian manifolds, Math. Nachr. 289 (2016), no. 8-9, 994–1004.
  • P. Lindqvist, On the equation $div(|\nabla u|^{p-2} \nabla u) + \lambda |u|^{p-2} u = 0$, Proc. Amer. Math. Soc. 109 (1990), no. 1, 157–164.
  • F. J. Narcowich and J. D. Ward, Nonstationary wavelets on the $m$-sphere for scattered data, Appl. Comput. Harmon. Anal. 3 (1996), no. 4, 324–336.
  • A. Sitaram, M.Sundari, and S. Thangavelu, Uncertainty principles on certain Lie groups, Proc. Indian Acad. Sci. Math. Sci. 105 (1995), no. 2, 135–151.
  • X. Sun and F. Pan, Hardy type inequalities on the sphere, J. Ineq. Appl. 2017, Paper No. 148, 8 pp.
  • Y. Xiao, Some Hardy inequalities on the sphere, J. Math. Inequal. 10 (2016), no. 3, 793–805.
  • Q. Yang, Best constants in the Hardy–Rellich type inequalities on the Heisenberg group, J. Math. Anal. Appl. 342 (2008), no. 1, 423–431.
  • Q. Yang, D. Su, and Y. Kong, Hardy inequalities on Riemannian manifolds with negative curvature, Commun. Contemp. Math. 16 (2014), no. 2, 1350043, 24 pp.