Skip to main content
SpringerLink
Log in

Modules of systems of measures on polarizable Carnot groups

  • Published:
Arkiv för Matematik

Abstract

The paper presents a study of Fuglede’s \(p\)-module of systems of measures in condensers in polarizable Carnot groups. In particular, we calculate the \(p\)-module of measures in spherical ring domains, find the extremal measures, and finally, extend a theorem by Rodin to these groups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aikawa, H. and Ohtsuka, M., Extremal length of vector measures, Ann. Acad. Sci. Fenn. Math. 24 (1999), 61–88.

    MathSciNet  MATH  Google Scholar 

  2. Balogh, Z. M. and Tyson, J. T., Polar coordinates in Carnot groups, Math. Z. 241 (2002), 697–730.

    Article  MathSciNet  MATH  Google Scholar 

  3. Brakalova, M. A., Boundary extension of \(\mu \)–homeomorphisms, in Complex and Harmonic Analysis, pp. 231–247, DEStech, Lancaster, PA, 2007.

    Google Scholar 

  4. Brakalova, M., Markina, I. and Vasil’ev, A., Extremal functions for modules of systems of measures, arXiv:1409.1626 [math.CA], 2014.

  5. Calin, O., Chang, D. C. and Greiner, P., Geometric Analysis on the Heisenberg Group and Its Generalizations, AMS/IP Studies in Advanced Mathematics 40, Amer. Math. Soc./International Press, Providence, RI/Somerville, MA, 2007.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Chang, D. C. and Markina, I., Anisotropic quaternion Carnot groups: geometric analysis and Green function, Adv. in Appl. Math. 39 (2007), 345–394.

    Article  MathSciNet  MATH  Google Scholar 

  8. Federer, H., Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften 153, Springer, New York, 1969.

    MATH  Google Scholar 

  9. Folland, G. B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161–207.

    Article  MathSciNet  MATH  Google Scholar 

  10. Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Groups, Mathematical Notes 28, Princeton Univ. Press, Princeton, NJ, 1982.

    MATH  Google Scholar 

  11. Franchi, B., Serapioni, R. and Serra Cassano, F., Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups, Comm. Anal. Geom. 11 (2003), 909–944.

    Article  MathSciNet  MATH  Google Scholar 

  12. Franchi, B., Marchi, M., Serapioni, R. P. and Serra Cassano, F., Differentiability and approximate differentiability for intrinsic Lipschitz functions in Carnot groups and a Rademacher theorem, Anal. Geom. Metr. Spaces 2 (2014), 258–281.

    MathSciNet  MATH  Google Scholar 

  13. Fuglede, B., Extremal length and functional completion, Acta Math. 98 (1957), 171–219.

    Article  MathSciNet  MATH  Google Scholar 

  14. Gehring, F. W., Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961), 499–519.

    Article  MathSciNet  MATH  Google Scholar 

  15. Gehring, F. W., Extremal length definitions for the conformal capacity of rings in space, Michigan Math. J. 9 (1962), 137–150.

    Article  MathSciNet  MATH  Google Scholar 

  16. Gutlyanskii, V.Ya., Martio, O., Sugawa, T. and Vuorinen, M., On the degenerate Beltrami equation, Trans. Amer. Math. Soc. 357 (2005), 875–900.

    Article  MathSciNet  MATH  Google Scholar 

  17. Gutlyanskii, V.Ya., Sakan, K. and Sugawa, T., On \(\mu \)-conformal homeomorphisms and boundary correspondence, Complex Var. Elliptic Equ. 58 (2013), 947–962.

    Article  MathSciNet  MATH  Google Scholar 

  18. Heinonen, J., Calculus on Carnot groups, in Fall School in Analysis, Juväskylä, 1994, Ber. Univ. Juväskylä Math. Inst. 68, pp. 1–31, 1995.

    Google Scholar 

  19. Heinonen, J. and Holopainen, I., Quasiregular maps on Carnot groups, J. Geom. Anal. 7 (1997), 109–148.

    Article  MathSciNet  MATH  Google Scholar 

  20. Hesse, J., A \(p\)-extremal length and \(p\)-capacity equality, Ark. Mat. 13 (1975), 131–144.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Karmanova, M. and Vodop’yanov, S., Carnot-Carathéodory spaces, coarea and area formulas, in Analysis and Mathematical Physics, Trends in Mathematics, pp. 233–335, Birkhäuser, Basel, 2009.

    Chapter  Google Scholar 

  23. Korányi, A. and Reimann, H. M., Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), 309–338.

    Article  MathSciNet  MATH  Google Scholar 

  24. Korányi, A. and Reimann, H. M., Horizontal normal vectors and conformal capacity of spherical rings in the Heisenberg group, Bull. Sci. Math. (2) 111 (1987), 3–21.

    MathSciNet  MATH  Google Scholar 

  25. Krivov, V. V., Some properties of moduli in space, Dokl. Akad. Nauk SSSR 154 (1964), 510–513.

    MathSciNet  MATH  Google Scholar 

  26. Loewner, Ch., On the conformal capacity in space, J. Math. Mech. 8 (1959), 411–414.

    MathSciNet  MATH  Google Scholar 

  27. Magnani, V., The coarea formula for real-valued Lipschitz maps on stratified groups, Math. Nachr. 278 (2005), 1689–1705.

    Article  MathSciNet  MATH  Google Scholar 

  28. Markina, I., On coincidence of \(p\)-module of a family of curves and \(p\)-capacity on the Carnot group, Rev. Mat. Iberoam. 19 (2003), 143–160.

    Article  MathSciNet  MATH  Google Scholar 

  29. Markina, I., Extremal widths on homogeneous groups, Complex Var. Theory Appl. 48 (2003), 947–960.

    MathSciNet  MATH  Google Scholar 

  30. Markina, I., \(p\)-module of vector measures in domains with intrinsic metric on Carnot groups, Tohoku Math. J. (2) 56 (2004), 553–569.

    Article  MathSciNet  MATH  Google Scholar 

  31. Markina, I., Modules of vector measures on the Heisenberg group, in Complex Analysis and Dynamical Systems II, Contemp. Math. 382, pp. 291–304, Amer. Math. Soc., Providence, RI, 2005.

    Chapter  Google Scholar 

  32. Mitchell, J., On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), 35–45.

    MathSciNet  MATH  Google Scholar 

  33. Montefalcone, F., 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), 279–295. 2004.

    MathSciNet  MATH  Google Scholar 

  34. Ohtsuka, M., Extremal Length and Precise Functions, GAKUTO International Series. Mathematical Sciences and Applications 19, Gakkōtosho, Tokyo, 2003, with a preface by Fumi-Yuki Maeda

    MATH  Google Scholar 

  35. Pansu, P., Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), 1–60 (in French).

    Article  MathSciNet  MATH  Google Scholar 

  36. Platis, I. D., Modulus of surface families and the radial stretch in the Heisenberg group, arXiv:1310.4292.

  37. Platis, I. D., Modulus of revolution rings in the Heisenberg group, arXiv:1504.05099.

  38. Rodin, B., The method of extremal length, Bull. Amer. Math. Soc. 80 (1974), 587–606.

    Article  MathSciNet  MATH  Google Scholar 

  39. Rodin, B. and Warschawski, S. E., Extremal length and the boundary behavior of conformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 467–500.

    Article  MathSciNet  MATH  Google Scholar 

  40. Rodin, B. and Warschawski, S. E., Extremal length and univalent functions. I. The angular derivative, Math. Z. 153 (1977), 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  41. Šabat, B. V., The modulus method in space, Dokl. Akad. Nauk SSSR 130 (1960), 1210–1213. (in Russian).

    MathSciNet  Google Scholar 

  42. Shanmugalingam, N., Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), 243–279.

    Article  MathSciNet  MATH  Google Scholar 

  43. Shlyk, V. A., Capacity of a condenser and the modulus of a family of separating surfaces, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 185 (1990), 168–182 (in Russian).

    MathSciNet  MATH  Google Scholar 

  44. Shlyk, V. A., On the equality between \(p\)-capacity and \(p\)-modulus, Sibirsk. Mat. Zh. 34 (1993), 216–221 (in Russian).

    MathSciNet  MATH  Google Scholar 

  45. Väisälä, J., On quasiconformal mapping in space, Ann. Acad. Sci. Fenn. Ser. AI 296 (1961), 1–36.

    MATH  Google Scholar 

  46. Vodopyanov, S. K., Potential theory on homogeneous groups, Mat. Sb. 180 (1989), 57–77.

    Google Scholar 

  47. Ziemer, W. P., Extremal length and conformal capacity, Trans. Amer. Math. Soc. 126 (1967), 460–473.

    Article  MathSciNet  MATH  Google Scholar 

  48. Ziemer, W. P., Extremal length and \(p\)-capacity, Michigan Math. J. 16 (1969), 43–51.

    Article  MathSciNet  MATH  Google Scholar 

  49. Ziemer, W. P., Extremal length as a capacity, Michigan Math. J. 17 (1970), 117–128.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Fordham University, 407 John Mulcahy Hall, Bronx, NY, 10458-5165, USA

    M. Brakalova

  2. Department of Mathematics, University of Bergen, P.O. Box 7800, 5020, Bergen, Norway

    I. Markina & A. Vasil’ev

Authors
  1. M. Brakalova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Markina
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Vasil’ev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Vasil’ev.

Additional information

M. Brakalova was partially supported by a Faculty Research Grant, Fordham University, USA. I. Markina and A. Vasil’ev were partially supported by the grants of the Norwegian Research Council #239033/F20 and #213440/BG, as well as by EU FP7 IRSES program STREVCOMS, grant no. PIRSES-GA-2013-612669.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brakalova, M., Markina, I. & Vasil’ev, A. Modules of systems of measures on polarizable Carnot groups. Ark Mat 54, 371–401 (2016). https://doi.org/10.1007/s11512-016-0242-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11512-016-0242-6

Search

Navigation