Skip to main content

Semmes family of curves and a characterization of functions of bounded variation in terms of curves

Abstract

On metric spaces supporting a geometric version of a Semmes family of curves, we provide a Reshetnyak-type characterization of functions of bounded variation in terms of the total variation on such a family of curves. We then use this characterization to obtain a Federer-type characterization of sets of finite perimeter, that is, we show that a measurable set is of finite perimeter if and only if the Hausdorff measure of its measure theoretic boundary is finite. We present a construction of a geometric Semmes family of curves in the first Heisenberg group.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    We thank Stephen Semmes for pointing out that the initial idea of such a family of curves can be found in the works of Mary Weiss on lacunary series.

References

  1. 1.

    Aalto, D., Kinnunen, J.: The discrete maximal operator in metric spaces. J. Anal. Math. 111, 369–390 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Ambrosio, L.: Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces. Adv. Math. 159, 51–67 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Ambrosio, L.: Fine properties of sets of finite perimeter in doubling metric measure spaces. Set-Valued Anal. 10, 111–128 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Ambrosio, L., Di Marino, S.: Equivalent definitions of \(BV\) space and of total variation on metric measure spaces. J. Funct. Anal. 266(7), 4150–4188 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Ambrosio, L., Di Marino, S., Savaré, G.: On the duality between \(p\)-modulus and probability measures. http://arxiv.org/pdf/1311.1381 1–40 (2014)

  6. 6.

    Ambrosio, L. , Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, pp. xviii+434. The Clarendon Press, Oxford University Press, New York (2000)

  7. 7.

    Ambrosio, L., Miranda, M., Jr., Pallara, D.: Special functions of bounded variation in doubling metric measure spaces. In: De Giorgi, E. (ed.) Calculus of variations: topics from the mathematical heritage of, 1–45, Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta (2004)

  8. 8.

    Ambrosio, L., Tilli, P.: Topics on analysis in metric spaces. Oxford Lecture Series in Mathematics and its Applications, vol. 25, pp. viii+133. Oxford University Press, Oxford (2004)

  9. 9.

    Bellaïche, A., Risler, J.-J.: Sub-Riemannian geometry. Progress in Mathematics, vol. 144, pp. viii+393. Birkhäuser Verlag, Basel (1996)

  10. 10.

    Björn, A., Björn, J.: Nonlinear potential theory on metric spaces. EMS Tracts in Mathematics, vol. 17, pp. xii+403. European Mathematical Society (EMS), Zürich (2011)

  11. 11.

    Bourdon, M., Pajot, H.: Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings. Proc. Am. Math. Soc. 127, 2315–2324 (1999)

  12. 12.

    Buckley, S.: Is the maximal function of a Lipschitz function continuous? Ann. Acad. Sci. Fenn. Math. 24, 519–528 (1999)

    MathSciNet  Google Scholar 

  13. 13.

    Evans, L.C., Gariepy, R.: Measure theory and fine properties of functions. Studies in Advanced Mathematics, pp. viii+268. CRC Press, Boca Raton (1992)

  14. 14.

    Federer, H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153, pp. xiv+676. Springer, New York (1969)

  15. 15.

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

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Hajłasz, P.: Sobolev spaces on metric-measure spaces. Contemp. Math. 338, 173–218 (2003)

    Article  Google Scholar 

  17. 17.

    Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688), x+101 (2000)

  18. 18.

    Heikkinen, T., Koskela, P., Tuominen, H.: Sobolev type spaces from generalized Poincaré inequalities. Studia Math. 181, 1–16 (2007)

  19. 19.

    Heinonen, J.: Lectures on analysis on metric spaces, Universitext, pp. x+140 pp. Springer, New York (2001)

  20. 20.

    Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.: Sobolev classes of Banach space-valued functions and quasiconformal mappings. J. Anal. Math. 85, 87–139 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Kinnunen, J., Korte, R., Shanmugalingam, N., Tuominen, H.: A characterization of Newtonian functions with zero boundary values. Calc. Var. Partial Differ. Equ. 43(3–4), 507–528 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Korte, R., Lahti, P.: Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 31(1), 129–154 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Koskela, P., Shanmugalingam, N., Tuominen, H.: Removable sets for the Poincaré inequality on metric spaces. Indiana Univ. Math. J. 49, 333–352 (2000)

  24. 24.

    Lahti, P., Tuominen, H.: A pointwise characterization of functions of bounded variation on metric spaces. Ric. Mat. 63(1), 47–57 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Miranda, M., Jr.: Functions of bounded variation on ”good” metric spaces. J. Math. Pures Appl. 82(9), 975–1004 (2003)

  26. 26.

    Ohtsuka, M.: Extremal length and precise functions, GAKUTO International Series. Mathematical Sciences and Applications vol. 19, pp. vi+343. Gakkōtosho Co., Ltd. Tokyo (2003)

  27. 27.

    Royden, H.L.: Real analysis, 3rd edn. Macmillan Publishing Company, New York, pp. xx+444 (1988)

  28. 28.

    Rudin, W.: Real and complex analysis, 3rd edn. McGraw-Hill Book Co., New York, pp. xiv+416 (1987)

  29. 29.

    Semmes, S.: Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities. Selecta Math. (N.S.) 2, 155–296 (1996)

  30. 30.

    Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16(2), 243–279 (2000)

  31. 31.

    Väisälä, J.: Lectures on \(n\)-dimensional quasiconformal mappings. Lecture Notes in Mathematics, vol. 229, pp. xiv+144. Springer, Berlin-New York (1971)

  32. 32.

    Ziemer, W.P.: Weakly differentiable functions. Graduate Texts in Mathematics, vol. 120, pp. xvi+308, Springer, New York (1989)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Nageswari Shanmugalingam.

Additional information

We thank Riikka Kangaslampi for illuminating conversations on the geometry of geodesics in the Bourdon-Pajot spaces. We also thank Luigi Ambrosio and Juha Kinnunen for their encouragement with this project, and the anonymous referee for suggestions that helped improve the exposition of the paper. R. K. was partially supported by Academy of Finland, Grant #250403. P. L. was supported by the Finnish Academy of Science and Letters, the Vilho, Yrjö and Kalle Väisälä Foundation. N. S. was partially supported by NSF Grants DMS-0355027 and DMS-1200915.

Communicated by L. Ambrosio.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Korte, R., Lahti, P. & Shanmugalingam, N. Semmes family of curves and a characterization of functions of bounded variation in terms of curves. Calc. Var. 54, 1393–1424 (2015). https://doi.org/10.1007/s00526-015-0829-y

Download citation

Mathematics Subject Classification

  • Primary 31E05
  • Secondary 26A45
  • 26B30
  • 30L99