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Pointwise properties of functions of bounded variation in metric spaces

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Abstract

We study pointwise properties of functions of bounded variation on a metric space equipped with a doubling measure and a Poincaré inequality. In particular, we obtain a Lebesgue type result for \(BV\) functions. We also study approximations by Lipschitz continuous functions and a version of the Leibniz rule. We give examples which show that our main result is optimal for \(BV\) functions in this generality.

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References

  1. Ambrosio, L.: Fine properties of sets of finite perimeter in doubling metric measure spaces. Set Valued Anal. 10(2–3), 111–128 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)

    Google Scholar 

  3. Ambrosio, L., Miranda M. Jr., Pallara, D.: Special functions of bounded variation in doubling metric measure spaces. Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi, pp. 1–45 (2004)

  4. Camfield, C.: Comparison of BV norms in weighted Euclidean spaces and metric measure spaces, PhD Thesis, University of Cincinnati (2008). http://etd.ohiolink.edu/view.cgi?acc_num=ucin1211551579

  5. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)

  6. Federer, H.: Geometric Measure Theory. Grundlehren 153, Springer, Berlin (1969)

  7. Hajłasz, P., Kinnunen, J.: Hölder quasicontinuity of Sobolev functions on metric spaces. Rev. Mat. Iberoamericana 14(3), 601–622 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688), 1–101 (2000)

    Google Scholar 

  9. Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York (2001)

    Book  Google Scholar 

  10. Kinnunen, J., Latvala, V.: Lebesgue points for Sobolev functions on metric spaces. Rev. Mat. Iberoamericana 18, 685–700 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kinnunen, J., Korte, R., Shanmugalingam, N., Tuominen, H.: Lebesgue points and capacities via boxing inequality in metric spaces. Indiana Univ. Math. J. 57(1), 401–430 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Miranda, M. Jr.: Functions of bounded variation on “good” metric spaces. J. Math. Pure Appl. (9) 82(8), 975–1004 (2003)

    Google Scholar 

  13. Mäkäläinen, T.: Adams inequality on metric measure spaces. Rev. Mat. Iberoamericana 25(2), 533–558 (2009)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Vol’pert, A.I.: Spaces BV and quasilinear equations. Mat. Sb. (N.S.) 73(115), 255–302 (1967) (Russian); Mathematics of the USSR-Sbornik 2(2), 225–267 (1967) (English)

  16. Vol’pert, A.I., Hudjaev, S.I.: Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics. Martinus Nijhoff, Dordrecht (1985)

    MATH  Google Scholar 

  17. Ziemer, W.P.: Weakly differentiable functions. Graduate Texts in Mathematics, vol. 120. Springer, Berlin (1989)

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Correspondence to Juha Kinnunen.

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Part of this research was conducted during the visit of the third author to Aalto University; she wishes to thank that institution for its kind hospitality. The research is supported by the Academy of Finland.

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Kinnunen, J., Korte, R., Shanmugalingam, N. et al. Pointwise properties of functions of bounded variation in metric spaces. Rev Mat Complut 27, 41–67 (2014). https://doi.org/10.1007/s13163-013-0130-6

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  • DOI: https://doi.org/10.1007/s13163-013-0130-6

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