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Bilipschitz embeddings of metric spaces into euclidean spaces.

Stephen Semmes (1999)

Publicacions Matemàtiques

When does a metric space admit a bilipschitz embedding into some finite-dimensional Euclidean space? There does not seem to be a simple answer to this question. Results of Assouad [A1], [A2], [A3] do provide a simple answer if one permits some small ("snowflake") deformations of the metric, but unfortunately these deformations immediately disrupt some basic aspects of geometry and analysis, like rectifiability, differentiability, and curves of finite length. Here we discuss a (somewhat technical)...

Bilipschitz extensions from smooth manifolds.

Taneli Huuskonen, Juha Partanen, Jussi Väisälä (1995)

Revista Matemática Iberoamericana

We prove that every compact C1-submanifold of Rn, with or without boundary, has the bilipschitz extension property in Rn.

Bilipschitz Extensions of Maps Having Quasiconformal Extensions.

Pekka Tukia, Jussi Väisälä (1984)

Mathematische Annalen

Bilipschitz maps and the modulus of rings.

Rohde, S. (1997)

Annales Academiae Scientiarum Fennicae. Mathematica

Bounded Turning and Quasiconformal Maps.

Jussi Väisälä (1991)

Monatshefte für Mathematik

Characterizations of snowflake metric spaces.

Tyson, Jeremy T., Wu, Jang-Mei (2005)

Annales Academiae Scientiarum Fennicae. Mathematica

Composition operator and Sobolev-Lorentz spaces $W L^{n, q}$

Stanislav Hencl, Luděk Kleprlík, Jan Malý (2014)

Studia Mathematica

Let Ω,Ω’ ⊂ ℝⁿ be domains and let f: Ω → Ω’ be a homeomorphism. We show that if the composition operator $T_{f} : u \mapsto u \circ f$ maps the Sobolev-Lorentz space $W L^{n, q} (Ω^{'})$ to $W L^{n, q} (Ω)$ for some q ≠ n then f must be a locally bilipschitz mapping.

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

Luděk Kleprlík (2014)

Open Mathematics

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.