Hardy inequalities for Landau Hamiltonian and for Baouendi-Grushin operator with Aharonov-Bohm type magnetic field. Part I


  • Ari Laptev
  • Michael Ruzhansky
  • Nurgissa Yessirkegenov




In this paper we prove the Hardy inequalities for the quadratic form of the Laplacian with the Landau Hamiltonian type magnetic field. Moreover, we obtain a Poincaré type inequality and inequalities with more general families of weights. Furthermore, we establish weighted Hardy inequalities for the quadratic form of the magnetic Baouendi-Grushin operator for the magnetic field of Aharonov-Bohm type. For these, we show refinements of the known Hardy inequalities for the Baouendi-Grushin operator involving radial derivatives in some of the variables. The corresponding uncertainty type principles are also obtained.


Adimurthi, Chaudhuri, N., and Ramaswamy, M., An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc. 130 (2002), no. 2, 489–505. https://doi.org/10.1090/S0002-9939-01-06132-9

Adimurthi, Ratnakumar, P. K., and Sohani, V. K., A Hardy-Sobolev inequality for the twisted Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 147 (2017), no. 1, 1–23. https://doi.org/10.1017/S0308210516000081

Adimurthi and Sandeep, K., Existence and non-existence of the first eigenvalue of the perturbed Hardy-Sobolev operator, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 5, 1021–1043. https://doi.org/10.1017/S0308210500001992

Aermark, L. and Laptev, A., Hardy inequalities for a magnetic Grushin operator with Aharanov-Bohm type magnetic field, Algebra i Analiz 23 (2011), no. 2, 1–8, English translation: St. Petersburg Math. J. 23 (2012), no. 2, 203–208. https://doi.org/10.1090/S1061-0022-2012-01193-3

D'Ambrosio, L., Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc. 132 (2004), no. 3, 725–734. https://doi.org/10.1090/S0002-9939-03-07232-0

D'Ambrosio, L., Some Hardy inequalities on the Heisenberg group, Differ. Equ. 40 (2004), no. 4, 552–564. https://doi.org/10.1023/B:DIEQ.0000035792.47401.2a

Dou, J., Guo, Q., and Niu, P., Hardy inequalities with remainder terms for the generalized Baouendi-Grushin vector fields, Math. Inequal. Appl. 13 (2010), no. 3, 555–570. https://doi.org/10.7153/mia-13-38

Fischer, V. and Ruzhansky, M., Quantization on nilpotent Lie groups, Progress in Mathematics, vol. 314, Birkhäuser/Springer, 2016. https://doi.org/10.1007/978-3-319-29558-9

Folland, G. B. and Stein, E. M., Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982.

Garofalo, N., Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. Differential Equations 104 (1993), no. 1, 117–146. https://doi.org/10.1006/jdeq.1993.1065

Garofalo, N. and Lanconelli, E., Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 2, 313–356.

Ghoussoub, N. and Moradifam, A., Bessel pairs and optimal Hardy and Hardy-Rellich inequalities, Math. Ann. 349 (2011), no. 1, 1–57. https://doi.org/10.1007/s00208-010-0510-x

Ioku, N., Ishiwata, M., and Ozawa, T., Hardy type inequalities in $L^p$ with sharp remainders, J. Inequal. Appl. (2017), paper No. 5, 7 pp. https://doi.org/10.1186/s13660-016-1271-1

Kombe, I., Hardy, Rellich and uncertainty principle inequalities on Carnot groups, preprint arXiv:math/0611850v1, 2006.

Kombe, I., Hardy and Rellich-type inequalities with remainders for Baouendi-Grushin vector fields, Houston J. Math. 41 (2015), no. 3, 849–874.

Laptev, A. and Weidl, T., Hardy inequalities for magnetic Dirichlet forms, in “Mathematical results in quantum mechanics (Prague, 1998)”, Oper. Theory Adv. Appl., vol. 108, Birkhäuser, Basel, 1999, pp. 299–305.

Machihara, S., Ozawa, T., and Wadade, H., Hardy type inequalities on balls, Tohoku Math. J. (2) 65 (2013), no. 3, 321–330. https://doi.org/10.2748/tmj/1378991018

Niu, P., Chen, Y., and Han, Y., Some Hardy-type inequalities for the generalized Baouendi-Grushin operators, Glasg. Math. J. 46 (2004), no. 3, 515–527. https://doi.org/10.1017/S0017089504002034

Ozawa, T., Ruzhansky, M., and Suragan, D., $L^p$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups, Q. J. Math. 70 (2019), no. 1, 305–318. https://doi.org/10.1093/qmath/hay040

Ruzhansky, M. and Suragan, D., Anisotropic $L^2$-weighted Hardy and $L^2$-Caffarelli-Kohn-Nirenberg inequalities, Commun. Contemp. Math. 19 (2017), no. 6, 1750014, 12 pp. https://doi.org/10.1142/S0219199717500146

Ruzhansky, M. and Suragan, D., Layer potentials, Kac's problem, and refined Hardy inequality on homogeneous Carnot groups, Adv. Math. 308 (2017), 483–528. https://doi.org/10.1016/j.aim.2016.12.013

Ruzhansky, M. and Suragan, D., Local Hardy and Rellich inequalities for sums of squares of vector fields, Adv. Differential Equations 22 (2017), no. 7-8, 505–540.

Ruzhansky, M. and Suragan, D., On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and $p$-sub-Laplacian inequalities on stratified groups, J. Differential Equations 262 (2017), no. 3, 1799–1821. https://doi.org/10.1016/j.jde.2016.10.028

Ruzhansky, M., Suragan, D., and Yessirkegenov, N., Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for $L^p$-weighted Hardy inequalities, Trans. Amer. Math. Soc. Ser. B 5 (2018), 32–62. https://doi.org/10.1090/btran/22

Ruzhansky, M., Suragan, D., and Yessirkegenov, N., Sobolev type inequalities, Euler-Hilbert-Sobolev and Sobolev-Lorentz-Zygmund spaces on homogeneous groups, Integral Equations Operator Theory 90 (2018), no. 1, Art. 10, 33 pp. https://doi.org/10.1007/s00020-018-2437-7

Shen, S.-F. and Jin, Y.-Y., Rellich type inequalities related to Grushin type operator and Greiner type operator, Appl. Math. J. Chinese Univ. Ser. B 27 (2012), no. 3, 353–362. https://doi.org/10.1007/s11766-012-2908-6

Solomyak, M., A remark on the Hardy inequalities, Integral Equations Operator Theory 19 (1994), no. 1, 120–124. https://doi.org/10.1007/BF01202293



How to Cite

Laptev, A., Ruzhansky, M., & Yessirkegenov, N. (2019). Hardy inequalities for Landau Hamiltonian and for Baouendi-Grushin operator with Aharonov-Bohm type magnetic field. Part I. MATHEMATICA SCANDINAVICA, 125(2), 239–269. https://doi.org/10.7146/math.scand.a-114892