Homeomorphisms that induce monomorphisms of Sobolev spaces

Composition operators on W 1 X are necessarily induced by quasiconformal mappings

“…Therefore, if T maps any L 1 into L 2 then is -quasiconformal. This generalizes the necessary condition for W 1 = WL from [8] and, moreover, we do not need to assume that is differentiable a.e. Our theorem also includes necessary condition from [12].…”

“…Moreover, we generalize results from [8,11], because we do not need to assume that T is continuous or that is differentiable a.e. We also allow T to be an operator between different function spaces.…”

“…the class of morphism is wider. If N is a set of zero measure then using (8) we can conclude that D = 0 a.e. on to L and again we can use the interpolation.…”

“…Moreover, each homeomorphism which maps W 1 loc (Ω 2 ) into W 1 loc (Ω 1 ) continuously is necessarily a quasiconformal mapping up to a reflection. Similarly, it is possible to characterize homeomorphisms for which the corresponding composition operator is continuous from W 1 loc to W 1 loc , these are -quasiconformal mappings [8] (see also [17]). Under a -quasiconformal mapping we understand a homeomorphism ∈ W 1 1 loc (Ω R ) such that for some constant K the distortion inequality |D ( )| ≤ K |J ( )| holds for a.e.…”

Distortion of mappings and L q, p -cohomology

“…The classes B D 1 s,∞ has been studied by different authors and under various names, see [3,15,[17][18][19]24,26,28,30]. The class B D n−1 s,∞ also appears in [26], where some obstructions to their existence are given.…”