Quasiconformal mappings and exponentially integrable functions

Fernando Farroni ; Raffaella Giova

Studia Mathematica, Tome 204 (2011), p. 195-203 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $u ∘ f^{-1} ∈ EXP()$. Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate $1/(1 + K logK) ≤ (||u ∘ f^{-1}||_{EXP()})/(||u||_{EXP()}) ≤ 1 + K log K$ 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 $1/K ≤ (dist_{EXP(f(G))}(u ∘ f^{-1},L^{∞}(f(G))))/(dist_{EXP(f(G))}(u,L^{∞}(G))) ≤ K$ for every u ∈ EXP(). We also prove that the last estimate is sharp, in the sense that there exist a quasiconformal mapping f: → , a domain G ⊂ ⊂ and a function u ∈ EXP(G) such that $dist_{EXP(f(G))}(u ∘ f^{-1},L^{∞}(f(G))) = K dist_{EXP(f(G))}(u,L^{∞}(G))$.

Détail

Publié le : 2011-01-01

Zbl 1221.30053

EUDML-ID : urn:eudml:doc:285676