Measure density and extendability of Sobolev functions

Piotr Hajłasz , Pekka Koskela , and Heli Tuominen

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

Abstract

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.

Primary Subjects: 46E35

Keywords: Sobolev extension; doubling measure

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.rmi/1218475358

back to Table of Contents

References

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.

Mathematical Reviews (MathSciNet): MR2319654

Digital Object Identifier: doi:10.1016/j.jmaa.2006.09.064

Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9 (1999), 428-517.

Mathematical Reviews (MathSciNet): MR1708448

Digital Object Identifier: doi:10.1007/s000390050094

Chua, S.-K.: Extension theorems on weighted Sobolev spaces. Indiana Univ. Math. J. 41 (1992), 1027-1076.

Mathematical Reviews (MathSciNet): MR1206339

Digital Object Identifier: doi:10.1512/iumj.1992.41.41053

Chua, S.-K.: Some remarks on extension theorems for weighted Sobolev spaces. Illinois J. Math. 38 (1994), 95-126.

Mathematical Reviews (MathSciNet): MR1245837

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.

Mathematical Reviews (MathSciNet): MR499948

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.

Mathematical Reviews (MathSciNet): MR1922670

Project Euclid: euclid.rae/1212412857

Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.

Mathematical Reviews (MathSciNet): MR1800917

Heinonen, J. and Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181 (1998), 1-61.

Mathematical Reviews (MathSciNet): MR1654771

Digital Object Identifier: doi:10.1007/BF02392747

Jones, P. W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 (1981), 71-88.

Mathematical Reviews (MathSciNet): MR631089

Digital Object Identifier: doi:10.1007/BF02392869

Keith, S. and Zhong, X.: The Poincaré inequality is an open ended condition. Ann. of Math. (2) 167 (2008), no. 2, 575-599.

Mathematical Reviews (MathSciNet): MR2415381

Koskela, P. and MacManus, P.: Quasiconformal mappings and Sobolev spaces. Studia Math. 131 (1998), no. 1, 1-17.

Mathematical Reviews (MathSciNet): MR1628655

Koskela, P. and Saksman, E.: Pointwise characterizations of Hardy-Sobolev functions. To appear in Math. Res. Lett.

Mathematical Reviews (MathSciNet): MR2424909

Macías, R. A. and Segovia, C.: A decomposition into atoms of distributions on spaces of homogeneous type. Adv. in Math. 33 (1979), 271-309.

Mathematical Reviews (MathSciNet): MR546296

Digital Object Identifier: doi:10.1016/0001-8708(79)90013-6

Mattila, P.: Geometry of sets and measures in Euclidean spaces. Cambridge Studies in Advanced Mathematics 44. Cambridge University Press, Cambridge, 1995.

Mathematical Reviews (MathSciNet): MR1333890

Nhieu, D.-M.: Extension of Sobolev spaces on the Heisenberg group. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 1559-1564.

Mathematical Reviews (MathSciNet): MR1367807

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.

Mathematical Reviews (MathSciNet): MR1848049

Digital Object Identifier: doi:10.1007/s10231-001-8195-0

Romanov, A. S.: On a generalization of Sobolev spaces. Siberian Math. J. 39 (1998), 821-824.

Mathematical Reviews (MathSciNet): MR1654102

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.

Mathematical Reviews (MathSciNet): MR1414889

Digital Object Identifier: doi:10.1007/BF01587936

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

Mathematical Reviews (MathSciNet): MR1809341

Shanmugalingam, N.: Harmonic functions on metric spaces. Illinois J. Math. 45 (2001), 1021-1050.

Mathematical Reviews (MathSciNet): MR1879250

Shvartsman, P.: On extensions of Sobolev functions defined on regular subsets of metric measure spaces. J. Approx. Theory 144 (2007), 139-161.

Mathematical Reviews (MathSciNet): MR2293385

Digital Object Identifier: doi:10.1016/j.jat.2006.05.005

previous :: next