Homeomorphisms that induce monomorphisms of Sobolev spaces

Let G, G′ be domains in ℝⁿ. We obtain a geometrical description of the class of all homeomorphisms φ{symbol}:G→ G′ that induce bounded operators φ{symbol}^* from the seminormed Sobolev space L _p ¹ (G′) to L _p ¹ (G) by the rule φ{symbol}^* u =u o φ{symbol}. For p-Poincare domains the same classes of homeomorphisms induce bounded operators for classical Sobolev spaces W _p ¹ . These classes of homeomorphisms are natural generalizations of the class of quasiconformal homeomorphisms that correspond to the case p=n. We demonstrate some applications of our results for embedding theorems in domains with Hölder singularities.