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Quasicircles modulo bilipschitz maps.

Steffen Rohde (2001)

Revista Matemática Iberoamericana

We give an explicit construction of all quasicircles, modulo bilipschitz maps. More precisely, we construct a class S of planar Jordan curves, using a process similar to the construction of the van Koch snowflake curve. These snowflake-like curves are easily seen to be quasicircles. We prove that for every quasicircle Γ there is a bilipschitz homeomorphism f of the plane and a snowflake-like curve S ∈ S with Γ = f(S). In the same fashion we obtain a construction of all bilipschitz-homogeneous Jordan...

Quasiconformal mappings and exponentially integrable functions

Fernando Farroni, Raffaella Giova (2011)

Studia Mathematica

We prove that a K-quasiconformal mapping f:ℝ² → ℝ² which maps the unit disk onto itself preserves the space EXP() of exponentially integrable functions over , in the sense that u ∈ EXP() if and only if u ∘ f - 1 ∈ E X P ( ) . Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate 1 / ( 1 + K l o g K ) ≤ ( | | u ∘ f - 1 | | E X P ( ) ) / ( | | u | | E X P ( ) ) ≤ 1 + K l o g K for every u ∈ EXP(). Similarly, we consider the distance from L ∞ in EXP and we prove that if f: Ω → Ω’ is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then 1 / K ≤ ( d i s t E X P ( f ( G ) ) ( u ∘ f - 1 , L ∞ ( f ( G ) ) ) ) / ( d i s t E X P ( f ( G ) ) ( u , L ∞ ( G ) ) ) ≤ K for every u ∈ EXP(). We also prove that...

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