On a class of sharp multiplicative Hardy inequalities

Guzu, D.; Hoffmann-Ostenhof, T.; Laptev, A.

doi:10.1090/spmj/1659

On a class of sharp multiplicative Hardy inequalities
HTML articles powered by AMS MathViewer

by D. Guzu, T. Hoffmann-Ostenhof and A. Laptev

St. Petersburg Math. J. 32, 523-530

DOI: https://doi.org/10.1090/spmj/1659

Published electronically: May 11, 2021

PDF | Request permission

Abstract:

A class of weighted Hardy inequalities is treated. The sharp constants depend on the lowest eigenvalues of auxiliary Schrödinger operators on a sphere. In particular, for some block radial weights these sharp constants are given in terms of the lowest eigenvalue of a Legendre type equation.

References

A. A. Balinsky, W. D. Evans, and R. T. Lewis, On the number of negative eigenvalues of Schrödinger operators with an Aharonov-Bohm magnetic field, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2014, 2481–2489. MR 1862664, DOI 10.1098/rspa.2001.0851
E. B. Davies, A review of Hardy inequalities, The Maz′ya anniversary collection, Vol. 2 (Rostock, 1998) Oper. Theory Adv. Appl., vol. 110, Birkhäuser, Basel, 1999, pp. 55–67. MR 1747888
Jean Dolbeault, Maria J. Esteban, and Ari Laptev, Spectral estimates on the sphere, Anal. PDE 7 (2014), no. 2, 435–460. MR 3218815, DOI 10.2140/apde.2014.7.435
Jean Dolbeault, Maria J. Esteban, Ari Laptev, and Michael Loss, Spectral properties of Schrödinger operators on compact manifolds: rigidity, flows, interpolation and spectral estimates, C. R. Math. Acad. Sci. Paris 351 (2013), no. 11-12, 437–440. MR 3090125, DOI 10.1016/j.crma.2013.06.014
Jean Dolbeault, Maria J. Esteban, Ari Laptev, and Michael Loss, Magnetic rings, J. Math. Phys. 59 (2018), no. 5, 051504, 10. MR 3805403, DOI 10.1063/1.5022121
Thomas Hoffmann-Ostenhof and Ari Laptev, Hardy inequalities with homogeneous weights, J. Funct. Anal. 268 (2015), no. 11, 3278–3289. MR 3336725, DOI 10.1016/j.jfa.2015.03.016
Matthias Keller, Yehuda Pinchover, and Felix Pogorzelski, Optimal Hardy inequalities for Schrödinger operators on graphs, Comm. Math. Phys. 358 (2018), no. 2, 767–790. MR 3774437, DOI 10.1007/s00220-018-3107-y
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
V. G. Maz′ya and T. O. Shaposhnikova, Theory of multipliers in spaces of differentiable functions, Uspekhi Mat. Nauk 38 (1983), no. 3(231), 23–86 (Russian). MR 704635
Leszek Skrzypczak and Cyril Tintarev, On Caffarelli-Kohn-Nirenberg inequalities for block-radial functions, Potential Anal. 45 (2016), no. 1, 65–81. MR 3511804, DOI 10.1007/s11118-016-9535-4
Elliott H. Lieb and Robert Seiringer, The stability of matter in quantum mechanics, Cambridge University Press, Cambridge, 2010. MR 2583992