Abstract
We study basic properties of the BV-capacity and Sobolev capacity of order one in a complete metric space equipped with a doubling measure and supporting a weak Poincaré inequality. In particular, we show that the BV-capacity is a Choquet capacity and the Sobolev 1-capacity is not. However, these quantities are equivalent by two sided estimates and they have the same null sets as the Hausdorff measure of codimension one. The theory of functions of bounded variation plays an essential role in our arguments. The main tool is a modified version of the boxing inequality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambrosio L.: Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces. Adv. Math. 159(1), 51–67 (2001)
Ambrosio L.: Fine properties of sets of finite perimeter in doubling metric measure spaces. Calculus of variations, nonsmooth analysis and related topics. Set-Valued Anal. 10(2–3), 111–128 (2002)
Björn, A., Björn, J.: Nonlinear potential theory in metric spaces (to appear)
Björn, A., Björn, J., Parviainen, M.: Lebesgue points and the fundamental convergence theorem for superharmonic functions. Rev. Mat. Iberoam. (to appear)
Björn A., Björn J., Shanmugalingam N.: Quasicontinuity of Newton–Sobolev functions and density of Lipschitz functions on metric spaces. Houst. Math. J. 34(4), 1197–1211 (2008)
Choquet G.: Forme abstraite du téorème de capacitabilité (French). Ann. Inst. Fourier. (Grenoble) 9, 83–89 (1959)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag Inc., New York (1969)
Federer, H., Ziemer, W.P.: The Lebesgue set of a function whose distribution derivatives are p-th power summable. Indiana Univ. Math. J. 22, 139–158 (1972/73)
Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel (1984)
Gol’dshtein V., Troyanov M.: Capacities in metric spaces. Integral Equ. Oper. Theory 44(2), 212–242 (2002)
Gustin W.: Boxing inequalities. J. Math. Mech. 9, 229–239 (1960)
Hajłasz, P.: Sobolev spaces on metric-measure spaces. In: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002). Contemporary Mathematics, vol. 338, pp. 173–218. American Mathematical Society, Providence (2003)
Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688) (2000)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext, Springer-Verlag, New York (2001)
Kallunki S., Shanmugalingam N.: Modulus and continuous capacity. Ann. Acad. Sci. Fenn. Math. 26(2), 455–464 (2001)
Kilpeläinen T., Kinnunen J., Martio O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12(3), 233–247 (2000)
Kinnunen J., Korte R., Shanmugalingam N., Tuominen H.: Lebesgue points and capacities via the boxing inequality in metric spaces. Indiana Univ. Math. J. 57(1), 401–430 (2008)
Kinnunen J., Latvala V.: Lebesgue points for Sobolev functions on metric spaces. Rev. Math. Iberoam. 18, 685–700 (2002)
Kinnunen J., Martio O.: The Sobolev capacity on metric spaces. Ann. Acad. Sci. Fenn. Math. 21(2), 367–382 (1996)
Kinnunen, J., Martio, O.: Choquet property for the Sobolev capacity in metric spaces. In: Proceedings on Analysis and Geometry (Russian) (Novosibirsk Akademgorodok, 1999), pp. 285–290. Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Math., Novosibirsk (2000)
Kinnunen J., Martio O.: Sobolev space properties of superharmonic functions on metric spaces. Results Math. 44(1–2), 114–129 (2003)
Mäkäläinen T.: Adams inequality on metric measure spaces. Rev. Math. Iberoam. 25(2), 533–558 (2009)
Maz’ja, V.G.: Sobolev Spaces. Translated from the Russian by T. O. Shaposhnikova. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin (1985)
Miranda M.: Functions of bounded variation on “good” metric spaces. J. Math. Pures Appl. (9) 82(8), 975–1004 (2003)
Shanmugalingam N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Math. Iberoam. 16(2), 243–279 (2000)
Ziemer, W.P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, vol. 120. Springer-Verlag, New York (1989)