Quasiconformal mappings and exponentially integrable functions

• Autorzy: Fernando Farroni , Raffaella Giova
• Języki publikacji: 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
$dist_{EXP(f(G))}(u ∘ f^{-1},L^{∞}(f(G))) = K dist_{EXP(f(G))}(u,L^{∞}(G))$.

Kategorie tematyczne

• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 30C62: Quasiconformal mappings in the plane
• 47B33: Composition operators

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2011
• Tom: 203
• Numer: 2
• Strony: 195-203

Daty

wydano

2011

Twórcy

autor

Fernando Farroni

• Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli Federico II, Via Cintia, 80126 Napoli, Italy

autor

Raffaella Giova

• Dipartimento di Statistica e Matematica per la Ricerca Economica, Università degli Studi di Napoli Parthenope, Via Medina, 40, 80133 Napoli, Italy

Identyfikatory

DOI

10.4064/sm203-2-5