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Quasiconformal mappings and exponentially integrable functions. (English) Zbl 1221.30053

For a bounded domain \(G\subset\mathbb R^n\) with measure \(|G|\), the space \(\text{EXP}(G)\) consists of the exponentially integrable functions \(u: G\to\mathbb R\). \(\text{EXP}(G)\) is a Banach space equipped with the norm
\[ \|u\|^{}_{\text{EXP}(G)}=\sup_{0<t<|G|}\left(1+\log\frac{|G|}{t}\right)^{-1}u^*(t), \]
where \(u^*(t)=\sup\big\{\tau\geq0:\mu_u(\tau)>t\big\}\), \(t\in\big(0,|G|\big)\), \(\mu_u(\tau)=\big|\big\{x\in G:|u(x)|>\tau\big\}\big|\), and \(\tau\geq0\).
A quasiconformal mapping \(f:\mathbb R^2\to\mathbb R^2\) is principal if it is conformal outside the unit disk \(\mathbb D\) and satisfies the normalization condition
\[ \big|f(x)-x\big|=O\bigg(\frac{1}{|x|}\bigg)\quad\text{if}\quad |x|>1. \]
Theorem 1.2. Let \(f:\mathbb R^2\to\mathbb R^2\) be a \(K\)-quasiconformal principal mapping that is conformal outside \(\mathbb D\) and maps \(\mathbb D\) onto itself. Then, for every \(u\in\text{EXP}(\mathbb D)\),
\[ \frac{\|u\|_{\text{EXP}(\mathbb D)}}{1+K\log K}\leq\big\|u\circ f^{-1}\big\|^{}_{\text{EXP}(\mathbb D)}\leq(1+K\log K)\|u\|^{}_{\text{EXP}(\mathbb D)}. \]
The authors compare the distances in \(L^{\infty}\) of \(u\) and \(u\circ f^{-1}\) by the sharp estimates in the following theorem.
Theorem 1.3: Let \(\Omega\) and \(\Omega'\) be bounded domains in \(\mathbb R^2\), and let \(f:\Omega\to\Omega'\) be \(K\)-quasiconformal. Then, for every subdomain \(G\) of \(\Omega\) and for every \(u\in\text{EXP}(G)\),
\[ \frac{1}{K}\leq\frac{\text{dist}_{\text{EXP}(f(G))}\big(u\circ f^{-1},L^{\infty}(f(G))\big)}{\text{dist}_{\text{EXP}(G)}\big(u,L^{\infty}(G)\big)}\leq K. \]

MSC:

30C62 Quasiconformal mappings in the complex plane
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B33 Linear composition operators
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