Geodesic
On the Theory of Generalized Quasi-Isometries
Matematičeskie zametki, Tome 91 (2012) no. 4, pp. 571-577

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This paper is devoted to the study of so-called finitely bi-Lipschitz mappings, which are a far-reaching generalization of isometries and quasi-isometries. We obtain several criteria for the homeomorphic extension to the boundary of finitely bi-Lipschitz homeomorphisms $f$ between domains in $\mathbb{R}^n$, $n\geqslant2$, whose outer dilatations $K_O(x,f)$ satisfy the integral constraints $\int\Phi(K_O^{n-1}(x,f))\,dm(x)\infty$ with an increasing convex function $\Phi\colon[0,\infty]\to[0,\infty]$. Note that the integral conditions on the function $\Phi$ (obtained in the paper) are not only sufficient, but also necessary for the continuous extension of $f$ to the boundary.
Keywords: quasi-isometry, quasiconformal mapping, finitely bi-Lipschitz mapping, bi-Lipschitz homeomorphism, lower $Q$-homeomorphism
Mots-clés : Lebesgue integral. 
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     author = {D. A. Kovtonyuk and V. I. Ryazanov},
     title = {On the {Theory} of {Generalized} {Quasi-Isometries}},
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D. A. Kovtonyuk; V. I. Ryazanov. On the Theory of Generalized Quasi-Isometries. Matematičeskie zametki, Tome 91 (2012) no. 4, pp. 571-577. http://geodesic.mathdoc.fr/item/MZM_2012_91_4_a9/