Skip to main content Accessibility help
×
Home

Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups

  • Yongyang Jin (a1) and Genkai Zhang (a2)

Abstract

Let $\mathbb{G}$ be a step-two nilpotent group of $\text{H}$ -type with Lie algebra $\mathfrak{G}\,=\,V\,\oplus \,\text{t}$ . We define a class of vector fields $X\,=\,\left\{ {{X}_{j}} \right\}$ on $\mathbb{G}$ depending on a real parameter $k\,\ge \,1$ , and we consider the corresponding $p$ -Laplacian operator ${{L}_{p,\,k}}u\,=\,di{{v}_{X}}\left( {{\left| {{\nabla }_{X}}u \right|}^{p-2}}{{\nabla }_{X}}u \right)$ . For $k\,=\,1$ the vector fields $X\,=\,\left\{ {{X}_{j}} \right\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$ ; for $\mathbb{G}$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator ${{L}_{p,\,k}}$ and as an application, we get a Hardy type inequality associated with $X$ .

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups
      Available formats
      ×

Copyright

Corresponding author

Footnotes

Hide All

The first author’s research was supported by NNSF of China (10871180), NSF of Zhejiang province (Y6090359, Y6090383) and Department of Education, Zhejiang (Z200803357) The second author was supported by the Swedish Research Council and a STINT Institutional Grant.

Footnotes

References

Hide All
[1] Beals, R., Gaveau, B., and Greiner, P., Uniforms hypoelliptic Green's functions. J. Math. Pures Appl. 77(1998), no. 3, 209–248.
[2] Capogna, L., Danielli, D., and Garofalo, N., Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations. Amer. J. Math. 118(1996), no. 6, 1153–1196. doi:10.1353/ajm.1996.0046
[3] D’Ambrozio, L., Some Hardy inequalities on the Heisenberg group. Differ. Uravn. 40(2004), no. 4, 509–521, 575.
[4] D’Ambrozio, L., Hardy-type inequalities related to degenerate elliptic differential operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4(2005), no. 3, 451–486.
[5] Folland, G. B., A fundamental solution for a subelliptic operator. Bull. Amer. Math. Soc. 79(1973), 373–376. doi:10.1090/S0002-9904-1973-13171-4
[6] Folland, G. B., Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13(1975), no. 2, 161–207. doi:10.1007/BF02386204
[7] Folland, G. B. and Stein, E. M., Hardy spaces on homogeneous groups. Mathematical Notes 28. Princeton University Press, Princeton, NJ, 1982.
[8] Garćıa Azorero, J. P. and Peral Alonso, I., Hardy inequalities and some critical elliptic and parabolic problems. J. Differential Equations 144(1998), no. 2, 441–476. doi:10.1006/jdeq.1997.3375
[9] 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.
[10] Goldstein, J. A. and Kombe, I., Nonlinear degenerate parabolic equations on the Heisenberg group. Int. J. Evol. Equ. 1(2005), no. 1, 1–22.
[11] Goldstein, J. A. and Zhang, Q. S., On a degenerate heat equation with a singular potential. J. Funct. Anal. 186(2001), no. 2, 342–359. doi:10.1006/jfan.2001.3792
[12] Greiner, P. C., A fundamental solution for a nonelliptic partial differential operator. Canad. J. Math. 31(1979), no. 5, 1107–1120.
[13] Heinonen, J. and Holopainen, I., Quasiregular maps on Carnot groups. J. Geom. Anal. 7(1997), no. 1, 109–148.
[14] Hörmander, L., Hypoelliptic second order differential equations. Acta Math. 119(1967), 147–171. doi:10.1007/BF02392081
[15] Jin, Y., Hardy-type inequalities on H-type groups and anisotropic Heisenberg groups. Chin. Ann. Math. 29(2008), no. 5, 567–574.
[16] Kaplan, A., Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans. Amer. Math. Soc. 258(1980), no. 1, 147–153. doi:10.2307/1998286
[17] Kohn, J. J., Hypoellipticity and loss of derivatives. Ann. of Math. 162(2005), no. 2, 943–986.With an appendix by Derridj, M. and Tartakoff, D. S.. doi:10.4007/annals.2005.162.943
[18] Nagel, A., Stein, E. M., and S.Wainger. Balls and metrics defined by vector fields. I. Basic properties. Acta Math. 155(1985), no. 1-2, 103–147. doi:10.1007/BF02392539
[19] Niu, P., Zhang, H., and Y.Wang, Hardy type and Rellich type inequalities on the Heisenberg group. Proc. Amer. Math. Soc. 129(2001), no. 12, 3623–3630. doi:10.1090/S0002-9939-01-06011-7
[20] Rothschild, L. P. and Stein, E. M., Hypoelliptic differential operators and nilpotent groups. Acta Math. 137(1976), no. 3-4,247–320. doi:10.1007/BF02392419
[21] Sánchez-Calle, A., Fundamental solutions and geometry of the sum of squares of vector fields. Invent. Math. 78(1984), no. 1, 143–160. doi:10.1007/BF01388721
[22] Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series 43. Princeton University Press, Princeton, NJ, 1993.
[23] Zhang, H. and Niu, P., Hardy-type inequalities and Pohozaev-type identities for a class of p-degenerate subelliptic operators and applications. Nonlinear Anal. 54(2003), 1, 165–186, 2003. doi:10.1016/S0362-546X(03)00062-2
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Related content

Powered by UNSILO

Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups

  • Yongyang Jin (a1) and Genkai Zhang (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.