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Quasiconformal mappings and exponentially integrable functions

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Farroni F., Giova R.

Studia Mathematica

2011

tom 203

nr 2

195-203

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 $dist_{EXP(f(G))}(u ∘ f^{-1},L^{∞}(f(G))) = K dist_{EXP(f(G))}(u,L^{∞}(G))$.

Ograniczanie wyników

1 Studia Mathematica

1 Farroni F.

1 Giova R.

1 2011

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Quasiconformal mappings and exponentially integrable functions