Quasiconformal mappings and Sobolev spaces

Pekka Koskela; Paul MacManus

Studia Mathematica (1998)

  • Volume: 131, Issue: 1, page 1-17
  • ISSN: 0039-3223

Abstract

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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 ) .

How to cite

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Koskela, Pekka, and MacManus, Paul. "Quasiconformal mappings and Sobolev spaces." Studia Mathematica 131.1 (1998): 1-17. <http://eudml.org/doc/216562>.

@article{Koskela1998,
abstract = {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 $L^\{1,Q\}(ℝ^Q)$.},
author = {Koskela, Pekka, MacManus, Paul},
journal = {Studia Mathematica},
keywords = {quasiconformal mappings in metric spaces},
language = {eng},
number = {1},
pages = {1-17},
title = {Quasiconformal mappings and Sobolev spaces},
url = {http://eudml.org/doc/216562},
volume = {131},
year = {1998},
}

TY - JOUR
AU - Koskela, Pekka
AU - MacManus, Paul
TI - Quasiconformal mappings and Sobolev spaces
JO - Studia Mathematica
PY - 1998
VL - 131
IS - 1
SP - 1
EP - 17
AB - 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 $L^{1,Q}(ℝ^Q)$.
LA - eng
KW - quasiconformal mappings in metric spaces
UR - http://eudml.org/doc/216562
ER -

References

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  1. [GR] V. M. Gol'dshteĭn and Yu. G. Reshetnyak, Quasiconformal Mappings and Sobolev Spaces, Kluwer, Dordrecht, 1990. 
  2. [Ha] P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403-415. Zbl0859.46022
  3. [HaK1] P. Hajłasz and P. Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris 320 (1995), 1211-1215. Zbl0837.46024
  4. [HaK2] P. Hajłasz and P. Koskela, Sobolev met Poincaré, preprint. Zbl0954.46022
  5. [HKM] J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Univ. Press, Oxford, 1993. 
  6. [HeK1] J. Heinonen and P. Koskela, From local to global in quasiconformal structures, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), 554-556. Zbl0842.30016
  7. [HeK2] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math., to appear. Zbl0915.30018
  8. [KS] J. N. Korevaar and R. M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), 561-659. Zbl0862.58004
  9. [K1] P. Koskela, Removable sets for Sobolev spaces, Ark. Mat., to appear. Zbl1070.46502
  10. [K2] P. Koskela, The degree of regularity of a quasiconformal mapping, Proc. Amer. Math. Soc. 122 (1994), 769-772. Zbl0814.30015
  11. [L] L. Lewis, Quasiconformal mapping and Royden algebras in space, Trans. Amer. Math. Soc. 158 (1971), 481-496. Zbl0214.38302
  12. [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. Zbl0870.54031
  13. [ST] J. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, Berlin, 1989. Zbl0676.42021
  14. [T] J. Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Math., to appear. Zbl0910.30022
  15. [TV] P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, ibid. 5 (1980), 97-114. Zbl0403.54005
  16. [Z] W. P. Ziemer, Change of variables for absolutely continuous functions, Duke Math. J. 36 (1969), 171-178. Zbl0177.08006

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