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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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