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• # Artykuł - szczegóły

## Studia Mathematica

2011 | 203 | 2 | 195-203

## Quasiconformal mappings and exponentially integrable functions

EN

### Abstrakty

EN
We prove that a K-quasiconformal mapping f:ℝ² → ℝ² which maps the unit disk 𝔻 onto itself preserves the space EXP(𝔻) of exponentially integrable functions over 𝔻, in the sense that u ∈ EXP(𝔻) if and only if $u ∘ f^{-1} ∈ EXP(𝔻)$. Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate
$1/(1 + K logK) ≤ (||u ∘ f^{-1}||_{EXP(𝔻)})/(||u||_{EXP(𝔻)}) ≤ 1 + K log K$
for every u ∈ EXP(𝔻). 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(𝔾). We also prove that the last estimate is sharp, in the sense that there exist a quasiconformal mapping f: 𝔻 → 𝔻, a domain G ⊂ ⊂ 𝔻 and a function u ∈ EXP(G) such that
$dist_{EXP(f(G))}(u ∘ f^{-1},L^{∞}(f(G))) = K dist_{EXP(f(G))}(u,L^{∞}(G))$.

195-203

wydano
2011

### Twórcy

autor
• Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli Federico II, Via Cintia, 80126 Napoli, Italy
autor
• Dipartimento di Statistica e Matematica per la Ricerca Economica, Università degli Studi di Napoli Parthenope, Via Medina, 40, 80133 Napoli, Italy