Abstract
We show a sharp relationship between the existence of space filling mappings with an upper gradient in a Lorentz space and the Poincaré inequality in a general metric setting. As key examples, we consider these phenomena in Cantor diamond spaces and the Heisenberg groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balogh Z.M., Rogovin K., Zürcher T.: The Stepanov differentiability theorem in metric measure spaces. J. Geom. Anal. 14(3), 405–422 (2004)
Bennett C., Sharpley R.: Interpolation of operators, volume 129 of Pure and Applied Mathematics. Academic Press Inc, Boston (1988)
Capogna L., Danielli D., Pauls S.D., Tyson J.T.: An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, volume 259 of Progress in Mathematics. Birkhäuser, Basel (2007)
Cheeger J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)
Costea Ş: Scaling invariant Sobolev-Lorentz capacity on \({\mathbb{R}^n}\). Indiana Univ. Math. J. 56(6), 2641–2669 (2007)
Hajłasz, P.: Sobolev spaces on metric-measure spaces. In Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), vol. 338 of Contemp. Math., pp. 173–218. Amer. Math. Soc., Providence (2003)
Hajłasz P., Koskela P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145(688), x101 (2000)
Hajłasz P., Tyson J.T.: Sobolev Peano cubes. Michigan Math. J. 56(3), 687–702 (2008)
Heinonen J.: Lectures on analysis on metric spaces. Universitext, Springer, New York (2001)
Heinonen J., Koskela P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181(1), 1–61 (1998)
Heinonen J., Koskela P., Shanmugalingam N., Tyson J.T.: Sobolev classes of Banach space-valued functions and quasiconformal mappings. J. Anal. Math. 85, 87–139 (2001)
Kaufman, R.: A singular map of a cube onto a square. J. Diff. Geom. 14(4), 593–594 (1981) 1979
Kauhanen J., Koskela P., Malý J.: On functions with derivatives in a Lorentz space. Manuscripta Math. 100(1), 87–101 (1999)
Keith S.: A differentiable structure for metric measure spaces. Adv. Math. 183(2), 271–315 (2004)
Keith S., Zhong X.: The Poincaré inequality is an open ended condition. Ann. Math. 167(2), 575–599 (2008)
Korte R.: Geometric implications of the Poincaré inequality. Results Math. 50(1–2), 93–107 (2007)
Koskela P., MacManus P.: Quasiconformal mappings and Sobolev spaces. Studia Math. 131(1), 1–17 (1998)
Malý J., Ziemer W.P.: Fine regularity of solutions of elliptic partial differential equations, volume 51 of Mathematical Surveys and Monographs. Amer. Math. Soc., Providence (1997)
Marola, N., Ziemer, W.P.: The coarea formula, condition (N) and rectifiable sets for Sobolev functions on metric spaces (preprint). http://arxiv.org/pdf/0807.2233v1
Rado, T., Reichelderfer, P.V.: Continuous transformations in analysis. With an introduction to algebraic topology. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd. LXXV. Springer, Berlin (1955)
Ranjbar-Motlagh A.: An embedding theorem for Sobolev type functions with gradients in a Lorentz space. Studia Math. 191(1), 1–9 (2009)
Romanov A.S.: On the absolute continuity of Sobolev-type functions on metric spaces. Sibirsk. Mat. Zh. 49(5), 1147–1156 (2008)
Semmes S.: Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities. Selecta Math. (N.S.) 2(2), 155–295 (1996)
Shanmugalingam N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16(2), 243–279 (2000)
Stein E.M.: Editor’s note: the differentiability of functions in R n. Ann. Math. 113(2), 383–385 (1981)
Troyanov, M.: Approximately Lipschitz mappings and Sobolev mappings between metric spaces. In: Proceedings on Analysis and Geometry (Russian) (Novosibirsk Akademgorodok, 1999), pp. 585–594. Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk (2000)
Tuominen, H.: Orlicz-Sobolev spaces on metric measure spaces. Ann. Acad. Sci. Fenn. Math. Diss. (135):86 (2004) (Dissertation, University of Jyväskylä, Jyväskylä (2004))
Wildrick K., Zürcher T.: Peano cubes with derivatives in a Lorentz space. Illinois J. Math. 53(2), 365–378 (2009)
Wildrick, K., Zürcher, T.: Mappings with an upper gradient in a Lorentz space. (preprint)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported by Academy of Finland grants 120972 and 128144.
The second author was partially supported by the Swiss National Science Foundation and Academy of Finland grant 120972.
Rights and permissions
About this article
Cite this article
Wildrick, K., Zürcher, T. Space filling with metric measure spaces. Math. Z. 270, 103–131 (2012). https://doi.org/10.1007/s00209-010-0787-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-010-0787-1