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Hardy-type inequalities on H-type groups and anisotropic Heisenberg groups

Abstract

The author obtains some weighted Hardy-type inequalities on H-type groups and anisotropic Heisenberg groups. These inequalities generalize some recent results due to N. Garofalo, E. Lanconelli, I. Kombe and P. Niu et al.

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References

  1. [1]

    Garofalo, N. and Lanconelli, E., Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble), 40(2), 1990, 313–356.

  2. [2]

    Kombe, I., Sharp Hardy type inequalities on the Carnot group. arXiv:math.FA/0501522

  3. [3]

    Niu, P., Zhang, H. and Wang, Y., Hardy type and Rellich type inequalities on the Heisenberg group, Proc. Amer. Math. Soc., 129(12), 2001, 3623–3630.

  4. [4]

    D’Ambrozio, L., Some Hardy inequalities on the Heisenberg group, Differ. Uravn., 40(4), 2004, 509–521.

  5. [5]

    Garcia Azorero, J. P., Peral Alnoso, I., Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Eqs., 144(2), 1998, 441–476.

  6. [6]

    Goldstein, J. A. and Kombe, I., Nonlinear degenerate parabolic equations on the Heisenberg group, Int. J. Evol. Eqn., 1(1), 2005, 1–22.

  7. [7]

    Sun, M. and Yang, X., Quasi-convex Functions in Carnot Groups, Chin. Ann. Math., 28B(2), 2007, 235–242.

  8. [8]

    D’Ambrozio, L., Hardy-type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 4(3), 2005, 451–486.

  9. [9]

    Han, J., Niu, P. and Han, Y., Some Hardy-type inequalities on groups of Heisenberg type (in Chinese), J. Systems Sci. Math. Sci., 25(5), 2005, 588–598.

  10. [10]

    Kaplan, A., Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc., 258(1), 1980, 147–153.

  11. [11]

    Cowling, M., Dooley, A., Korányi, A. and Ricci, F., An approach to symmetric spaces of rank one via groups of Heisenberg type, J. Geom. Anal., 8(2), 1998, 199–237.

  12. [12]

    Cowling, M., Dooley, A., Korányi, A. et al, H-type groups and Iwasawa decompositions Adv. Math., 87(1), 1991, 1–41.

  13. [13]

    Kaplan, A., Lie groups of Heisenberg type, Conference on Differential Geometry on Homogeneous Spaces (Turin, 1983), Rend. Sem. mat. Univ. Politec. Torino, Special Issue, 1983, 117–130.

  14. [14]

    Korányi, A., Geometric properties of Heisenberg-type groups, Adv. in Math., 56(1), 1985, 28–38.

  15. [15]

    Capogna, L., Danielli, D. and Garofalo, N., Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math., 118(6), 1996, 1153–1196.

  16. [16]

    Jin, Y. and Zhang, G., Fundamental solutions for a class of degenerate p-Laplacian operators and applications to Hardy type inequalities, preprint, 2006.

  17. [17]

    Tyson, J. T., Sharp weighted Young’s inequalities and Moser-Trudinger inequalities on the Heisenberg groups and Grushin spaces, Potential Anal., 24(4), 2006, 357–384.

  18. [18]

    Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, 1982.

  19. [19]

    Chang, D. and Greiner, P., Harmonic analysis and sub-Riemannian geometry on Heisenberg groups, Bull. Inst. Math. Acad. Sinica, 30(3), 2002, 153–190.

  20. [20]

    Chang, D. and Tie, J., A note on Hermite and subelliptic operators, Acta Math. Sin., Engl. Ser., 21(4), 2005, 803–818.

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Author information

Correspondence to Yongyang Jin.

Additional information

Project supported by the China State Scholarship (No. 2003833095) and the Department of Education of Zhejiang Province (No. 20051495).

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Jin, Y. Hardy-type inequalities on H-type groups and anisotropic Heisenberg groups. Chin. Ann. Math. Ser. B 29, 567–574 (2008). https://doi.org/10.1007/s11401-006-0291-4

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Keywords

  • Hardy-type inequalities
  • H-type groups
  • Anisotropic Heisenberg groups

2000 MR Subject Classification

  • 26D10
  • 22E30
  • 46E35