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.
@article{bwmeta1.element.doi-10_2478_s11533-013-0392-8,
author = {Lud\v ek Kleprl\'\i k},
title = {Composition operators on W 1 X are necessarily induced by quasiconformal mappings},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {1229-1238},
zbl = {1306.46037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0392-8}
}
Luděk Kleprlík. Composition operators on W 1 X are necessarily induced by quasiconformal mappings. Open Mathematics, Tome 12 (2014) pp. 1229-1238. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0392-8/
[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 | Zbl 1182.30001
[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 | Zbl 0942.42011
[3] Bennett C., Sharpley R., Interpolation of Operators, Pure Appl. Math., 129, Academic Press, Boston, 1988 | Zbl 0647.46057
[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 | Zbl 1246.46026
[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 | Zbl 1221.30053
[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 | Zbl 1272.30036
[7] Fiorenza A., Duality and reflexivity in grand Lebesgue spaces, Collect. Math., 2000, 51(2), 131–148 | Zbl 0960.46022
[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] HajŁasz P., Change of variables formula under minimal assumptions, Colloq. Math., 1993, 64(1), 93–101 | Zbl 0840.26009
[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 | Zbl 1065.26017
[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 | Zbl 1293.30050
[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 | Zbl 1271.46032
[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 | Zbl 1198.26016
[15] Iwaniec T., Martin G., Geometric Function Theory and Non-linear Analysis, Oxford Math. Monogr., Clarendon Press, Oxford University Press, New York, 2001 | Zbl 1045.30011
[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 | Zbl 0997.30018
[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 | Zbl 1236.46031
[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 | Zbl 06320699
[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] 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 | Zbl 1217.46019
[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 | Zbl 1275.46002
[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 | Zbl 0289.30027
[23] Rickman S., Quasiregular Mappings, Ergeb. Math. Grenzgeb., 26, Springer, Berlin, 1993 http://dx.doi.org/10.1007/978-3-642-78201-5
[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 | Zbl 0484.30017
[25] Ziemer W.P., Weakly Differentiable Functions, Grad. Texts in Math., 120, Springer, New York, 1989 | Zbl 0692.46022