On the theory of generalized quasi-isometries

Abstract

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 ℝn, n ≥ 2, whose outer dilatations KO(x, f) satisfy the integral constraints $$\int {\Phi (K_O^{n - 1} (x,f))dm(x) < \infty } $$ with an increasing convex function Φ: [0,∞] → [0,∞]. Note that the integral conditions on the function Φ (obtained in the paper) are not only sufficient, but also necessary for the continuous extension of f to the boundary.