DOI QR코드

DOI QR Code

A COUNTEREXAMPLE FOR IMPROVED SOBOLEV INEQUALITIES OVER THE 2-ADIC GROUP

Chamorro, Diego

  • Received : 2010.11.08
  • Published : 2013.04.30

Abstract

On the framework of the 2-adic group $\mathcal{Z}_2$, we study a Sobolev-like inequality where we estimate the $L^2$ norm by a geometric mean of the BV norm and the $\dot{B}_{\infty}^{-1,{\infty}}$ norm. We first show, using the special topological properties of the $p$-adic groups, that the set of functions of bounded variations BV can be identified to the Besov space ˙$\dot{B}_1^{1,{\infty}}$. This identification lead us to the construction of a counterexample to the improved Sobolev inequality.

Keywords

Sobolev inequalities;p-adic groups

References

  1. D. Chamorro, Some functional inequalities on polynomial volume growth Lie groups, Canad. J. Math. 64 (2012), no. 3, 481-496. https://doi.org/10.4153/CJM-2011-050-4
  2. A. Cohen, W. Dahmen, I. Daubechies, and R. De Vore, Harmonic Analysis of the space BV, Rev. Mat. Iberoamericana 19 (2003), no. 1, 235-263.
  3. P. Gerard, Y. Meyer, and F. Oru, Inegalites de Sobolev Precisees, Equations aux Derivees Partielles, Seminaire de l'Ecole Polytechnique, expose $n^{\circ}$ IV (1996-1997).
  4. K. Ikeda, T. Kim, and T. K. Shiratani, On p-adic bounded functions, Mem. Fac. Sci. Kyushu Univ. Ser. A 46 (1992), no. 2, 341-349.
  5. L. C. Jang, T. Kim, J.-W. Son, and S.-H. Rim, On p-adic bounded functions. II, J. Math. Anal. Appl. 264 (2001), no. 1, 21-31. https://doi.org/10.1006/jmaa.2000.6826
  6. T. Kim, q-Volkenborn integration. Russ. J. Math. Phys. 9 (2002), no. 3, 288-299.
  7. N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta-functions, GTM 58. Springer Verlag, 1977.
  8. M. Ledoux, On improved Sobolev embedding theorems, Math. Res. Lett. 10 (2003), no. 5-6, 659-669. https://doi.org/10.4310/MRL.2003.v10.n5.a9
  9. E. M. Stein, Topics in Harmonic Analysis, Annals of mathematics Studies, 63. Princeton University Press, 1970.
  10. P. Strzelecki, Gagliardo-Nirenberg inequalities with a BMO term, Bull. Lond. Math. Soc. 38 (2006), no. 2, 294-300. https://doi.org/10.1112/S0024609306018169
  11. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, World Scientific, Singapore, 1994.
  12. Y. Amice, Les nombres p-adiques, Presses Universitaires de France, Paris, 1975.
  13. J. Bergh and J. Lofstrom, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, 223. Springer Verlag, 1976.
  14. D. Chamorro, Improved Sobolev Inequalities and Muckenhoupt weights on stratified Lie groups, J. Math. Anal. Appl. 377 (2011), no. 2, 695-709. https://doi.org/10.1016/j.jmaa.2010.11.047