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Asymptotic behavior of the coefficient of quasiconformality and boundary behavior of the mapping of the ball. (Russian) Zbl 0764.30016
For a homeomorphism $$f$$ let $$k(f,x)$$ denote its linear dilatation at a point of its domain of definition and let $$B^ n$$ stand for the unit ball in $$R^ n$$. The following theorem, related to earlier work of v.a.Zorich and others, is proved.
Theorem. Let $$f: B^ n\to R^ n$$ be a $$C^ 1-$$diffeomorphism such that $\int_{B^ n}{k(f,x) dx} < \infty.$ Then $$f$$ ha a radial limit at almost every point of the unit sphere.
It is not known whether an analog of his result holds for bounded quasiregular maps in dimensions $$n \geq 3$$ or whether even one radial limit exists in this case (cf. Problem (8) on p. 193 of the reviewer’s book “Conformal geometry and quasiregular mappings” (1988; Zbl 0646.30025)).

##### MSC:
 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations
##### Keywords:
diffeomorphism; quasiregular maps
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