ON BOUNDARY BEHAVIOR OF GENERALIZED QUASI-ISOMETRIES

We establish a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower Q-homeomorphisms f between domains in (R-n) over bar = R-n boolean OR {infinity}, n >= 2, under integral constraints of the type integral Phi(Q(n-1)(x)) dm(x) < infinity with a convex non-decreasing function Phi : [0, infinity] -> [0, infinity]. Integral conditions on Phi are found that are necessary and sufficient for a continuous extension of f to the boundary. Our results are applied to finitely bi-Lipschitz mappings, which are a far-reaching generalization of isometries as well as quasi-isometries in R-n. In particular, a generalization and strengthening of the well-known theorem of Gehring-Martio on homeomorphic extension to boundaries of quasi-conformal mappings between QED (quasi-extremal distance) domains is obtained.