Composition operators on W 1 X are necessarily induced by quasiconformal mappings

Luděk Kleprlík

Open Mathematics (2014)

  • Volume: 12, Issue: 8, page 1229-1238
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Luděk Kleprlík. "Composition operators on W 1 X are necessarily induced by quasiconformal mappings." Open Mathematics 12.8 (2014): 1229-1238. <http://eudml.org/doc/269482>.

@article{LuděkKleprlík2014,
abstract = {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.},
author = {Luděk Kleprlík},
journal = {Open Mathematics},
keywords = {Sobolev space; Rearrangement invariant space; Quasiconformal mappings; Composition operator; rearrangement invariant space; quasiconformal mappings; composition operator},
language = {eng},
number = {8},
pages = {1229-1238},
title = {Composition operators on W 1 X are necessarily induced by quasiconformal mappings},
url = {http://eudml.org/doc/269482},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Luděk Kleprlík
TI - Composition operators on W 1 X are necessarily induced by quasiconformal mappings
JO - Open Mathematics
PY - 2014
VL - 12
IS - 8
SP - 1229
EP - 1238
AB - 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.
LA - eng
KW - Sobolev space; Rearrangement invariant space; Quasiconformal mappings; Composition operator; rearrangement invariant space; quasiconformal mappings; composition operator
UR - http://eudml.org/doc/269482
ER -

References

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  1. [1] Astala V., Iwaniec T., Martin G., Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Math. Ser., 48, Princeton University Press, Princeton, 2009 Zbl1182.30001
  2. [2] Bastero J., Milman M., Ruiz F.J., Rearrangement of Hardy-Littlewood maximal functions in Lorentz spaces, Proc. Amer. Math. Soc., 2000, 128(1), 65–74 http://dx.doi.org/10.1090/S0002-9939-99-05128-X Zbl0942.42011
  3. [3] Bennett C., Sharpley R., Interpolation of Operators, Pure Appl. Math., 129, Academic Press, Boston, 1988 Zbl0647.46057
  4. [4] Ciesielski M., Kaminska A., Lebesgue’s differentiation theorems in r.i. quasi-Banach spaces and Lorentz spaces Γp,w, J. Funct. Spaces Appl., 2012, #682960 Zbl1246.46026
  5. [5] Farroni F., Giova R., Quasiconformal mappings and exponentially integrable functions, Studia Math., 2011, 203(2), 195–203 http://dx.doi.org/10.4064/sm203-2-5 Zbl1221.30053
  6. [6] Farroni F., Giova R., Quasiconformal mappings and sharp estimates for the distance to L ∞ in some function spaces, J. Math. Anal. Appl., 2012, 395(2), 694–704 http://dx.doi.org/10.1016/j.jmaa.2012.05.057 Zbl1272.30036
  7. [7] Fiorenza A., Duality and reflexivity in grand Lebesgue spaces, Collect. Math., 2000, 51(2), 131–148 Zbl0960.46022
  8. [8] Gol’dshtein V., Gurov L., Romanov A., Homeomorphisms that induce monomorphisms of Sobolev spaces, Israel J. Math., 1995, 91(1–3), 31–60 http://dx.doi.org/10.1007/BF02761638 
  9. [9] HajŁasz P., Change of variables formula under minimal assumptions, Colloq. Math., 1993, 64(1), 93–101 Zbl0840.26009
  10. [10] Hencl S., Absolutely continuous functions of several variables and quasiconformal mappings, Z. Anal. Anwendungen, 2003, 22(4), 767–778 http://dx.doi.org/10.4171/ZAA/1172 Zbl1065.26017
  11. [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. [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 Zbl1293.30050
  13. [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 Zbl1271.46032
  14. [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 Zbl1198.26016
  15. [15] Iwaniec T., Martin G., Geometric Function Theory and Non-linear Analysis, Oxford Math. Monogr., Clarendon Press, Oxford University Press, New York, 2001 Zbl1045.30011
  16. [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 Zbl0997.30018
  17. [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 Zbl1236.46031
  18. [18] Koch H., Koskela P., Saksman E., Soto T., Bounded compositions on scaling invariant Besov spaces, preprint available at http://arxiv.org/abs/1209.6477 Zbl06320699
  19. [19] Koskela P., Lectures on quasiconformal and quasisymmetric mappings, Jyväskylä Lectures in Mathematics, 1, preprint available at http://users.jyu.fi/~pkoskela/quasifinal.pdf 
  20. [20] Koskela P., Yang D., Zhou Y., Pointwise characterization of Besov and Triebel-Lizorkin spaces and quasiconformal mappings, Adv. Math., 2011, 226(4), 3579–3621 http://dx.doi.org/10.1016/j.aim.2010.10.020 Zbl1217.46019
  21. [21] Pick L., Kufner A., John O., Fučík S., Function Spaces, I, 2nd ed., De Gruyter Ser. Nonlinear Anal. Appl., 14, Walter De Gruyter, Berlin, 2013 Zbl1275.46002
  22. [22] Reimann H.M., Functions of bounded mean oscillation and quasiconformal mappings, Comment. Math. Helv., 1974, 49, 260–276 http://dx.doi.org/10.1007/BF02566734 Zbl0289.30027
  23. [23] Rickman S., Quasiregular Mappings, Ergeb. Math. Grenzgeb., 26, Springer, Berlin, 1993 http://dx.doi.org/10.1007/978-3-642-78201-5 
  24. [24] Tukia P., Väisälä J., Quasiconformal extension from dimension n to n + 1, Ann. Math., 1982, 115(2), 331–348 http://dx.doi.org/10.2307/1971394 Zbl0484.30017
  25. [25] Ziemer W.P., Weakly Differentiable Functions, Grad. Texts in Math., 120, Springer, New York, 1989 Zbl0692.46022

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