On the Theory of Generalized Quasi-Isometries

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.