Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Hardy inequalities for a magnetic Grushin operator with Aharonov-Bohm type magnetic field

Authors: L. Aermark and A. Laptev
Translated by: the authors
Original publication: Algebra i Analiz, tom 23 (2011), nomer 2.
Journal: St. Petersburg Math. J. 23 (2012), 203-208
MSC (2010): Primary 35P15; Secondary 81Q10
Published electronically: January 23, 2012
MathSciNet review: 2841670
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A version of the Aharonov-Bohm magnetic field for a Grushin sub-elliptic operator is introduced; then its quadratic form is shown to satisfy an improved Hardy inequality.

References [Enhancements On Off] (What's this?)

  • 1. M. Sh. Birman, On the spectrum of singular boundary value problems, Mat. Sb. (N. S.) 55 (1961), no. 2, 125-174; English transl., Eleven Papers on Analysis, Amer. Math. Soc. Transl. (2), vol. 53, Amer. Math. Soc., Providence, RI, 1966, pp. 23-60. MR 0142896 (26:463)
  • 2. L. D'Ambrosio, Some Hardy inequalities on the Heisenberg group, Differ. Uravn. 40 (2004), no. 4, 509-521; English transl., Differ. Equ. 40 (2004), no. 4, 552-564. MR 2153649 (2007d:26016)
  • 3. -, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc. 132 (2004), no. 3, 725-734. MR 2019949 (2005c:35050)
  • 4. E. B. Davies, A review of Hardy inequalities, The Maz'ya Anniversary Collection. Vol. 2 (Rostock, 1998), Oper. Theory Adv. Appl., vol. 110, Birkhäuser-Verlag, Basel, 1999, pp. 55-67. MR 1747888 (2001f:35166)
  • 5. J. Dou, Q. Guo, and P. Niu, Hardy inequalities with remainder terms for the generalized Baouendi-Grushin vector fields, Math. Inequal. Appl. 13 (2010), no. 3, 555-570. MR 2662838 (2011e:35250)
  • 6. N. Garofalo, 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. MR 1224123 (94i:35037)
  • 7. N. Garofallo and E. Lanconelli, Frequency functions on Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 2, 313-356. MR 1070830 (91i:22014)
  • 8. V. V. Grushin, A certain class of hypoelliptic operators, Mat. Sb. (N. S.) 83 (1970), no. 3, 456-473; English transl., Math. USSR-Sb. 12 (1970), 458-476. MR 0279436 (43:5158)
  • 9. I. Kombe, Hardy, Rellich and uncertainty principle inequalities on Carnot groups, Preprint arXiv:math/0611850.
  • 10. A. Laptev and T. Weidl, Hardy inequalities for magnetic Dirichlet forms, Mathematical Results in Quantum Mechanics (Prague, 1998), Oper. Theory Adv. Appl., vol. 108, Birkhäuser, Basel, 1999, pp. 299-305. MR 1708811 (2001d:35146)
  • 11. V. G. Maz'ya, Sobolev spaces, Leningrad. Univ., Leningrad, 1985; English transl., Springer-Verlag, Berlin, 1985. MR 0807364 (87g:46055); MR 0817985 (87g:46056)
  • 12. P. Niu, Y. Chen, and Y. Han, Some Hardy-type inequalities for the generalized Baouendi-Grushin operators, Glasg. Math. J. 46 (2004), no. 3, 515-527. MR 2094807 (2006a:35040)
  • 13. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser., vol. 43, Princeton Univ. Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35P15, 81Q10

Retrieve articles in all journals with MSC (2010): 35P15, 81Q10

Additional Information

L. Aermark
Affiliation: Stockholm University, SE-106 91 Stockholm, Sweden

A. Laptev
Affiliation: Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom

Keywords: Hardy inequalities
Received by editor(s): November 28, 2010
Published electronically: January 23, 2012
Dedicated: Dedicated to the memory of M. Sh. Birman whose enormous scientific achievements continue to guide many generations of mathematicians. The strong school in Spectral Theory that he developed is renowned all over the world. All his pupils and colleagues remember Professor Birman as a wonderful person who was always ready to help. His warm and generous support certainly aided the second author of this paper to survive as a mathematician.
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society