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2014 | 12 | 8 | 1229-1238
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Composition operators on W 1 X are necessarily induced by quasiconformal mappings

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EN
Let Ω ⊂ ℝn be an open set and X(Ω) be any rearrangement invariant function space close to L q(Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ↦ u ℴ f from W 1 X to W 1 X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.
Twórcy
Bibliografia
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  • [11] Hencl S., Kleprlík L., Composition of q-quasiconformal mappings and functions in Orlicz-Sobolev spaces, Illinois J. Math., 2012, 56(3), 661–1000
  • [12] Hencl S., Kleprlík L., Malý J., Composition operator and Sobolev-Lorentz spaces WL n.q, preprint available at http://msekce.karlin.mff.cuni.cz/ms-preprints/kma-preprints/2012-pap/2012-404.pdf
  • [13] Hencl S., Koskela P., Composition of quasiconformal mappings and functions in Triebel-Lizorkin spaces, Math. Nachr., 2013, 286(7), 669–678 http://dx.doi.org/10.1002/mana.201100130
  • [14] Hencl S., Malý J., Jacobians of Sobolev homeomorphisms, Calc. Var. Partial Differential Equations, 2010, 38(1–2), 233–242 http://dx.doi.org/10.1007/s00526-009-0284-8
  • [15] Iwaniec T., Martin G., Geometric Function Theory and Non-linear Analysis, Oxford Math. Monogr., Clarendon Press, Oxford University Press, New York, 2001
  • [16] Kauhanen J., Koskela P., Malý J., Mappings of finite distortion: condition N, Michigan Math. J., 2001, 49(1), 169–181 http://dx.doi.org/10.1307/mmj/1008719040
  • [17] Kleprlík L., Mappings of finite signed distortion: Sobolev spaces and composition of mappings, J. Math. Anal. Appl., 2012, 386(2), 870–881 http://dx.doi.org/10.1016/j.jmaa.2011.08.045
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