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Displaying 161 –
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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...
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 . Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate
for every u ∈ EXP(). Similarly, we consider the distance from in EXP and we prove that if f: Ω → Ω’ is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then
for every u ∈ EXP(). We also prove that...
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