Composition of \(q\)-quasiconformal mappings and functions in Orlicz-Sobolev spaces. (English) Zbl 1291.30133

Summary: Let \(\Omega \subset \mathbb{R}^{n}, q\geq n\) and \({\alpha}\geq 0\) or \(1<q\leq n\) and \(\alpha\leq 0\). We prove that the composition of \(q\)-quasiconfomal mapping \(f\) and function \[ u\in WL^{q}\log^{\alpha}L_{\text{loc}}(f(\Omega)) \] satisfies \(u \circ f\in WL^{q}\log^{\alpha}L_{\text{loc}}(\Omega)\). Moreover, each homeomorphism \(f\) which introduces continuous composition operator from \(WL^{q}\log^{\alpha}L\) to \(WL^{q}\log^{\alpha}L\) is necessarily a \(q\)-quasiconformal mapping. As a new tool, we prove a Lebesgue density type theorem for Orlicz spaces.


30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: Euclid


[1] \beginbbook \bauthor\binitsK. \bsnmAstala, \bauthor\binitsT. \bsnmIwaniec and \bauthor\binitsG. \bsnmMartin, \bbtitleElliptic partial differential equations and quasiconformal mappings in the plane, \bsertitlePrinceton Mathematical Series, vol. \bseriesno48, \bpublisherPrinceton University Press, \blocationPrinceton, NJ, \byear2009. \endbbook \endbibitem
[2] \beginbbook \bauthor\binitsH. \bsnmFederer, \bbtitleGeometric measure theory, \bsertitleDie Grundlehren der mathematischen Wissenschaften, vol. \bseriesno153, \bpublisherSpringer-Verlag, \blocationNew York, \byear1969 \bcomment(Second edition 1996). \endbbook \endbibitem
[3] \beginbarticle \bauthor\binitsF. \bsnmFarroni and \bauthor\binitsR. \bsnmGiova, \batitleQuasiconformal mappings and exponentially integrable functions, \bjtitleStudia Math. \bvolume203 (\byear2011), page 195-\blpage203. \endbarticle \endbibitem · Zbl 1221.30053
[4] \beginbarticle \bauthor\binitsD. \bsnmGallardo, \batitleWeighted weak type integral inequalities for the Hardy-Littlewood maximal operator, \bjtitleIsrael J. Math. \bvolume67 (\byear1989), no. \bissue1, page 95-\blpage108. \endbarticle \endbibitem · Zbl 0683.42021
[5] \beginbarticle \bauthor\binitsV. \bsnmGold’stein, \bauthor\binitsL. \bsnmGurov and \bauthor\binitsA. \bsnmRomanov, \batitleHomeomorphisms that induce monomorphisms of Sobolev spaces, \bjtitleIsrael J. Math. \bvolume91 (\byear1995), page 31-\blpage60. \endbarticle \endbibitem · Zbl 0836.46021
[6] \beginbbook \bauthor\binitsV. \bsnmGold’stein and \bauthor\binitsY. G. \bsnmReshetnyak, \bbtitleQuasiconformal mappings and Sobolev spaces, \bpublisherKluwer Academic Publishers, \blocationDordrecht, \byear1990. \endbbook \endbibitem
[7] \beginbarticle \bauthor\binitsS. \bsnmHencl, \batitleAbsolutely continuous functions of several variables and quasiconformal mappings, \bjtitleZ. Anal. Anwendungen \bvolume22 (\byear2003), no. \bissue4, page 767-\blpage778. \endbarticle \endbibitem · Zbl 1065.26017
[8] \beginbotherref \oauthor\binitsS. \bsnmHencl, \oauthor\binitsL. and \oauthor\binitsJ. , Composition operator and Sobolev-Lorentz spaces \(WL^{n,q}\) , preprint MATH-KMA-2012/404, available at http://www.karlin.mff.cuni.cz/kma-preprints/. \endbotherref \endbibitem URL:
[9] \beginbarticle \bauthor\binitsS. \bsnmHencl and \bauthor\binitsP. \bsnmKoskela, \batitleMappings of finite distortion: Composition operator, \bjtitleAnn. Acad. Sci. Fenn. Math \bvolume33 (\byear2008), page 65-\blpage80. \endbarticle \endbibitem
[10] \beginbarticle \bauthor\binitsS. \bsnmHencl and \bauthor\binitsP. \bsnmKoskela, \batitleComposition of quasiconformal mappings and functions in fractional Triebel-Lizorkin spaces, \bjtitleMath. Nachr. \bvolume286 (\byear2013), page 669-\blpage678. \endbarticle \endbibitem · Zbl 1271.46032
[11] \beginbarticle \bauthor\binitsS. \bsnmHencl, \bauthor\binitsP. \bsnmKoskela and \bauthor\binitsJ. , \batitleRegularity of the inverse of a Sobolev homeomorphism in space, \bjtitleProc. Roy. Soc. Edinburgh Sect. A \bvolume136 (\byear2006), no. \bissue6, page 1267-\blpage1285. \endbarticle \endbibitem · Zbl 1122.30015
[12] \beginbarticle \bauthor\binitsS. \bsnmHencl and \bauthor\binitsJ. , \batitleJacobians of Sobolev homeomorphisms, \bjtitleCalc. Var. Partial Differential Equations \bvolume38 (\byear2010), page 233-\blpage242. \endbarticle \endbibitem · Zbl 1198.26016
[13] \beginbchapter \bauthor\binitsT. \bsnmIwaniec and \bauthor\binitsG. \bsnmMartin, \bctitleGeometric function theory and nonlinear analysis, \bbtitleOxford Mathematical Monographs, \bpublisherClarendon Press, \blocationOxford, \byear2001. \endbchapter \endbibitem
[14] \beginbotherref \bauthor\binitsL. , Composition operators on \(W^{1}X\) are necessarily induced by quasiconformal mappings , to appear in Cent. Eur. J. Math. \endbotherref \endbibitem
[15] \beginbarticle \bauthor\binitsL. , \batitleThe zero set of the Jacobian and composition of mappings, \bjtitleJ. Math. Anal. Appl. \bvolume386 (\byear2012), page 870-\blpage881. \endbarticle \endbibitem · Zbl 1236.46031
[16] \beginbarticle \bauthor\binitsP. \bsnmKoskela, \bauthor\binitsD. \bsnmYang and \bauthor\binitsY. \bsnmZhou, \batitlePointwise characterization of Besov and Triebel-Lizorkin spaces and quasiconformal mappings, \bjtitleAdv. Math. \bvolume226 (\byear2011), no. \bissue4, page 3579-\blpage3621. \endbarticle \endbibitem · Zbl 1217.46019
[17] \beginbotherref \oauthor\binitsP. \bsnmKoskela, Lectures on quasiconformal and quasisymmetric mappings , available at http://users.jyu.fi/ pkoskela/. \endbotherref \endbibitem URL:
[18] \beginbarticle \bauthor\binitsJ. \bsnmOnninen, \batitleDifferentiability of monotone Sobolev functions, \bjtitleReal Anal. Exchange \bvolume26 (\byear2000), no. \bissue2, page 761-\blpage772. \endbarticle \endbibitem
[19] \beginbbook \bauthor\binitsM. M. \bsnmRao and \bauthor\binitsZ. D. \bsnmRen, \bbtitleTheory of Orlicz spaces, \bsertitleMonographs and Textbooks in Pure and Applied Mathematics, vol. \bseriesno146, \bpublisherMarcel Dekker, \blocationNew York, \byear1991. \endbbook \endbibitem
[20] \beginbarticle \bauthor\binitsH. M. \bsnmReimann, \batitleFunctions of bounded mean oscillation and quasiconformal mappings, \bjtitleComment. Math. Helv. \bvolume49 (\byear1974), page 260-\blpage276. \endbarticle \endbibitem · Zbl 0289.30027
[21] \beginbbook \bauthor\binitsS. \bsnmRickman, \bbtitleQuasiregular mappings, \bsertitleErgebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. \bseriesno26, \bpublisherSpringer-Verlag, \blocationBerlin, \byear1993. \endbbook \endbibitem
[22] \beginbarticle \bauthor\binitsP. \bsnmTukia and \bauthor\binitsJ. , \batitleQuasiconformal extension from dimension \(n\) to \(n+1\), \bjtitleAnn. of Math. (2) \bvolume115 (\byear1982), no. \bissue2, page 331-\blpage348. \endbarticle \endbibitem · Zbl 0484.30017
[23] \beginbarticle \bauthor\binitsJ. , \batitleQuasi-symmetric embeddings in Euclidian spaces, \bjtitleTrans. Amer. Math. Soc. \bvolume264 (\byear1981), page 191-\blpage204. \endbarticle \endbibitem · Zbl 0456.30018
[24] \beginbbook \bauthor\binitsW. P. \bsnmZiemer, \bbtitleWeakly differentiable functions, \bsertitleGraduate Texts in Mathematics, vol. \bseriesno120, \bpublisherSpringer-Verlag, \blocationNew York, \byear1989. \endbbook \endbibitem · Zbl 0692.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.