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arXiv · 1005.0247

On boundary behavior of generalized quasi-isometries

D. Kovtonyuk·V. Ryazanov
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Abstract

It is established a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower $Q$-homeomorphisms $f$ between domains in $\overline{\Rn}=\Rn\cup\{\infty\}$, $n\geqslant2$, under integral constraints of the type $\intΦ(Q^{n-1}(x))\,dm(x)<\infty$ with a convex non-decreasing function $Φ:[0,\infty]\to[0,\infty]$. It is shown that integral conditions on the function $Φ$ found by us are not only sufficient but also necessary for a continuous extension of $f$ to the boundary. It is given also applications of the obtained results to the mappings with finite area distortion and, in particular, to finitely bi-Lipschitz mappings that are a far reaching generalization of isometries as well as quasi-isometries in $\Rn$. In particular, it is obtained a generalization and strengthening of the well-known theorem by Gehring--Martio on a homeomorphic extension to boundaries of quasiconformal mappings between QED (quasiextremal distance) domains.

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D. Kovtonyuk, V. Ryazanov. 2010-05-03. On boundary behavior of generalized quasi-isometries. https://arxiv.org/abs/1005.0247

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