Advertisement

Archiv der Mathematik

, Volume 111, Issue 2, pp 203–214 | Cite as

A note on BV capacities on Grushin spaces

  • Guoliang Li
  • Yu Liu
Article
  • 70 Downloads

Abstract

In this paper we formulate some basic properties for the BV capacity and Hausdorff capacity on the Grushin space \(\mathbb {G}^n_{\alpha }\) and develop the sharp BV isocapacity inequalities on \(\mathbb {G}^n_{\alpha }\) under a dimensional condition.

Keywords

Isoperimetric inequality Sub-Riemannian manifolds Grushin spaces BV capacity 

Mathematics Subject Classification

32U20 53C17 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. R. Adams, A note on Choquet integral with respect to Hausdorff capacity, In: Function Spaces and Applications, 115-124, Lund 1986, Lecture Notes in Math., 1302, Springer-Verlag, Berlin, 1988.Google Scholar
  2. 2.
    L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces. Calculus of variations, nonsmooth analysis and related topics, Set-Valued Anal. 10 (2002), 111–128.MathSciNetzbMATHGoogle Scholar
  3. 3.
    B. Franchi and E. Lanconelli, Une métrique associée à une classe d’opérateurs elliptiques dégénérés, Conference on linear partial and pseudodifferential operators, (Torino, 1982), Rend. Sem. Mat. Univ. Politec. Torino 1983, Special Issue, 105–114.Google Scholar
  4. 4.
    B. Franchi, BV spaces and rectifiability for Carnot–Carathéodory metrics: an introduction, In: NAFSA 7–Nonlinear Analysis, Vol. 7, 72–132, Czech. Acad. Sci., Prague, 2003.Google Scholar
  5. 5.
    N. Garofalo and D. M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot–Carathéodory spaces and the existence of minimal surfaces, Commun. Pure Appl. Math. 49 (1996), 1081–1144.CrossRefzbMATHGoogle Scholar
  6. 6.
    J. Heinonen, Lectures on Analysis on Metric Spaces, Springer-Verlag, New York, 2001.CrossRefzbMATHGoogle Scholar
  7. 7.
    H. Hakkarainen and J. Kinnunen, The BV capacity in metric spaces, Manuscripta Math. 132 (2010), 51–73.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Y. Liu, BV capacity on generalized Grushin plane, J. Geom. Anal. 27 (2017), 409–441.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Yu Liu and J. Xiao, Functional capacities on the Grushin space \(\mathbb{G}^n_{\alpha }\), Annali di Matematica (2017).  https://doi.org/10.1007/s10231-017-0699-3
  10. 10.
    J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen, Lebesgue points and capacities via boxing inequality in metric spaces, Indiana Univ. Math. J. 57 (2008), 401–430.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M. Miranda, Jr., Functions of bounded variation on “good” metric spaces, J. Math. Pures Appl. 82 (2003), 975–1004.Google Scholar
  12. 12.
    V. Franceschi and R. Monti, Isoperimetric in \(H\)-type groups and Grushin spaces, Rev. Mat. Iberoam. 32 (2016), 1227–1258.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    F. Montefalcone, Sets of finite perimeter associated with vector fields and polyhedral approximation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 14 (2003), 4 (2004), 279–295.Google Scholar
  14. 14.
    R. Monti and D. Morbidelli, Isoperimetric inequality in the Grushin plane, J. Geom. Anal. 14 (2004), 355–368.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    R. Monti and F. S. Cassano, Surface measures in Carnot–Carathéodory spaces, Calc. Var. 13 (2001), 339–376.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    J. Xiao, The sharp Sobolev and isoperimetric inequalities split twice, Adv. Math. 211 (2007), 417–435.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsUniversity of Science and TechnologyBeijingPeople’s Republic of China

Personalised recommendations