Revista Matemática Iberoamericana

Measure density and extendability of Sobolev functions

Piotr Hajłasz , Pekka Koskela , and Heli Tuominen

Full-text: Open access


We study necessary and sufficient conditions for a domain to be a Sobolev extension domain in the setting of metric measure spaces. In particular, we prove that extension domains must satisfy a measure density condition.

Article information

Rev. Mat. Iberoamericana, Volume 24, Number 2 (2008), 645-669.

First available in Project Euclid: 11 August 2008

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Sobolev extension doubling measure


Hajłasz, Piotr; Koskela, Pekka; Tuominen, Heli. Measure density and extendability of Sobolev functions. Rev. Mat. Iberoamericana 24 (2008), no. 2, 645--669.

Export citation


  • Björn, J. and Shanmugalingam, N.: Poincaré inequalities, uniform domains and extension properties for Newton-Sobolev functions in metric spaces. J. Math. Anal. Appl. 332 (2007), no. 1, 190-208.
  • Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9 (1999), 428-517.
  • Chua, S.-K.: Extension theorems on weighted Sobolev spaces. Indiana Univ. Math. J. 41 (1992), 1027-1076.
  • Chua, S.-K.: Some remarks on extension theorems for weighted Sobolev spaces. Illinois J. Math. 38 (1994), 95-126.
  • Coifman, R. R. and Weiss G.: Analyse harmonique non-commutative sur certains espaces homogènes. Lecture Notes in Mathematics 242. Springer-Verlag, Berlin-New York, 1971.
  • Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5 (1996), 403-415.
  • Hajłasz, P.: Sobolev spaces on metric-measure spaces. In Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 173-218. Contemp. Math. 338. Amer. Math. Soc. Providence, RI, 2003.
  • Hajłasz, P. and Kinnunen, J.: Hölder quasicontinuity of Sobolev functions on metric spaces. Rev. Mat. Iberoamericana 14 (1998), no. 3, 601-622.
  • Hajłasz, P. and Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688.
  • Hajłasz, P., Koskela, P. and Tuominen, T.: Sobolev embeddings, extensions and measure density condition. J. Funct. Anal. 254 (2008), no. 5, 1217-1234.
  • Harjulehto, P.: Sobolev extension domains on metric spaces of homogeneous type. Real Anal. Exchange 27 (2001/02), 583-597.
  • Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.
  • Heinonen, J. and Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181 (1998), 1-61.
  • Jones, P. W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 (1981), 71-88.
  • Keith, S. and Zhong, X.: The Poincaré inequality is an open ended condition. Ann. of Math. (2) 167 (2008), no. 2, 575-599.
  • Koskela, P. and MacManus, P.: Quasiconformal mappings and Sobolev spaces. Studia Math. 131 (1998), no. 1, 1-17.
  • Koskela, P. and Saksman, E.: Pointwise characterizations of Hardy-Sobolev functions. To appear in Math. Res. Lett.
  • Macías, R. A. and Segovia, C.: A decomposition into atoms of distributions on spaces of homogeneous type. Adv. in Math. 33 (1979), 271-309.
  • Mattila, P.: Geometry of sets and measures in Euclidean spaces. Cambridge Studies in Advanced Mathematics 44. Cambridge University Press, Cambridge, 1995.
  • Nhieu, D.-M.: Extension of Sobolev spaces on the Heisenberg group. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 1559-1564.
  • Nhieu, D.-M.: The Neumann problem for sub-Laplacians on Carnot groups and the extension theorem for Sobolev spaces. Ann. Mat. Pura Appl. (4) 180 (2001), 1-25.
  • Romanov, A. S.: On a generalization of Sobolev spaces. Siberian Math. J. 39 (1998), 821-824.
  • Semmes, S.: Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities. Selecta Math. (N.S.) 2 (1996), 155-295.
  • Shanmugalingam, N.: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16 (2000), 243-279.
  • Shanmugalingam, N.: Harmonic functions on metric spaces. Illinois J. Math. 45 (2001), 1021-1050.
  • Shvartsman, P.: On extensions of Sobolev functions defined on regular subsets of metric measure spaces. J. Approx. Theory 144 (2007), 139-161.