Homeomorphisms of finite inner distortion: composition operators on Zygmund-Sobolev and Lorentz-Sobolev spaces

Letp > n-1 and α ∈ R and suppose that f: Ω →onto Ω′ is a homeomorphism in the Zygmund-Sobolev space WLp logα Lloc(Ω,Rn). Define r= p/p-n+1. Assume that u∈WLr log-α(r-1) Lloc(Ω). Then u ο f-1 ∈ BVloc(Ω′). We obtain a similar result whenever f is a homeomorphism in the Lorentz-Sobolev space WLloc p,q (Ω, Rn) with p, q > n-1 and u ∈ WLloc r,s (Ω) with r as before and s = q/q-n+1. Moreover, if we further assume that f has finite inner distortion we obtain in both cases u ο f-1 ∈ W loc 1,1 (Ω′).