Scaled-oscillation and regularity
HTML articles powered by AMS MathViewer
- by Zoltán M. Balogh and Marianna Csörnyei PDF
- Proc. Amer. Math. Soc. 134 (2006), 2667-2675 Request permission
Abstract:
We give sufficient conditions for Sobolev and Lipschitz functions in terms of their lower scaled-oscillation. The sharpness of these conditions is shown by examples. Our examples also show that a Stepanov-type differentiability theorem does not hold under the boundedness assumption of the lower scaled-oscillation.References
- Zoltán M. Balogh and Pekka Koskela, Quasiconformality, quasisymmetry, and removability in Loewner spaces, Duke Math. J. 101 (2000), no. 3, 554–577. With an appendix by Jussi Väisälä. MR 1740689, DOI 10.1215/S0012-7094-00-10138-X
- Zoltán M. Balogh, Kevin Rogovin, and Thomas Zürcher, The Stepanov differentiability theorem in metric measure spaces, J. Geom. Anal. 14 (2004), no. 3, 405–422. MR 2077159, DOI 10.1007/BF02922098
- J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428–517. MR 1708448, DOI 10.1007/s000390050094
- Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
- Juha Heinonen and Pekka Koskela, Definitions of quasiconformality, Invent. Math. 120 (1995), no. 1, 61–79. MR 1323982, DOI 10.1007/BF01241122
- Sari Kallunki, Mappings of finite distortion: the metric definition, Ann. Acad. Sci. Fenn. Math. Diss. 131 (2002), 33. Dissertation, University of Jyväskylä, Jyväskylä, 2002. MR 1928755
- Stephen Keith, A differentiable structure for metric measure spaces, Adv. Math. 183 (2004), no. 2, 271–315. MR 2041901, DOI 10.1016/S0001-8708(03)00089-6
- Stephen Keith, Measurable differentiable structures and the Poincaré inequality, Indiana Univ. Math. J. 53 (2004), no. 4, 1127–1150. MR 2095451, DOI 10.1512/iumj.2004.53.2417
- Juha Kinnunen and Olli Martio, The Sobolev capacity on metric spaces, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 367–382. MR 1404091
- Sari Kallunki and Pekka Koskela, Exceptional sets for the definition of quasiconformality, Amer. J. Math. 122 (2000), no. 4, 735–743. MR 1771571, DOI 10.1353/ajm.2000.0028
- S. Kallunki and P. Koskela, Metric definition of $\mu$-homeomorphisms, Michigan Math. J. 51 (2003), no. 1, 141–151. MR 1960925, DOI 10.1307/mmj/1049832897
- Pekka Koskela, Removable sets for Sobolev spaces, Ark. Mat. 37 (1999), no. 2, 291–304. MR 1714767, DOI 10.1007/BF02412216
- P. Koskela, N. Shanmugalingam, and H. Tuominen, Removable sets for the Poincaré inequality on metric spaces, Indiana Univ. Math. J. 49 (2000), no. 1, 333–352. MR 1777027, DOI 10.1512/iumj.2000.49.1719
- O. Martio, V. Ryazanov, U. Srebro, E. Yabukov, Mappings of finite length distortion, to appear in J. d’Analyse Math.
- S. Saks, Theory of the Integral, Monografie Matematyczne, Warsaw, 1937.
- Nageswari Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), no. 2, 243–279. MR 1809341, DOI 10.4171/RMI/275
- Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009, DOI 10.1007/BFb0061216
- William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3
Additional Information
- Zoltán M. Balogh
- Affiliation: Mathematisches Institut, Universität Bern, CH–3012 Sidlerstrasse 5, Bern, Switzerland
- Email: zoltan.balogh@math-stat.unibe.ch
- Marianna Csörnyei
- Affiliation: Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom
- Email: mari@math.ucl.ac.uk
- Received by editor(s): May 27, 2004
- Received by editor(s) in revised form: April 3, 2005
- Published electronically: March 23, 2006
- Communicated by: Juha M. Heinonen
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 2667-2675
- MSC (2000): Primary 26B35; Secondary 26B05
- DOI: https://doi.org/10.1090/S0002-9939-06-08290-6
- MathSciNet review: 2213746