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1998 | 131 | 1 | 1-17
Tytuł artykułu

Quasiconformal mappings and Sobolev spaces

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EN
We examine how Poincaré change under quasiconformal maps between appropriate metric spaces having the same Hausdorff dimension. We also show that for many metric spaces the Sobolev functions can be identified with functions satisfying Poincaré, and this allows us to extend to the metric space setting the fact that quasiconformal maps from $ℝ^Q$ onto $ℝ^Q$ preserve the Sobolev space $L^{1,Q}(ℝ^Q)$.
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Twórcy
  • University of Jyväskylä, Department of Mathematics, P.O. Box 35, Fin-40351 Jyväskylä, Finland, pkoskela@math.jyu.fi
  • Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain, pmm@maths.ed.ac.uk
  • Permanent address: Department of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland
Bibliografia
  • [GR] V. M. Gol'dshteĭn and Yu. G. Reshetnyak, Quasiconformal Mappings and Sobolev Spaces, Kluwer, Dordrecht, 1990.
  • [Ha] P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403-415.
  • [HaK1] P. Hajłasz and P. Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris 320 (1995), 1211-1215.
  • [HaK2] P. Hajłasz and P. Koskela, Sobolev met Poincaré, preprint.
  • [HKM] J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Univ. Press, Oxford, 1993.
  • [HeK1] J. Heinonen and P. Koskela, From local to global in quasiconformal structures, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), 554-556.
  • [HeK2] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math., to appear.
  • [KS] J. N. Korevaar and R. M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), 561-659.
  • [K1] P. Koskela, Removable sets for Sobolev spaces, Ark. Mat., to appear.
  • [K2] P. Koskela, The degree of regularity of a quasiconformal mapping, Proc. Amer. Math. Soc. 122 (1994), 769-772.
  • [L] L. Lewis, Quasiconformal mapping and Royden algebras in space, Trans. Amer. Math. Soc. 158 (1971), 481-496.
  • [S] S. Semmes, Finding curves on general spaces through quantitative topology with applications for Sobolev and Poincaré inequalities, Selecta Math. (N.S.) 2 (1996), 155-295.
  • [ST] J. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, Berlin, 1989.
  • [T] J. Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Math., to appear.
  • [TV] P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, ibid. 5 (1980), 97-114.
  • [Z] W. P. Ziemer, Change of variables for absolutely continuous functions, Duke Math. J. 36 (1969), 171-178.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv131i1p1bwm
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