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

Let p > n - 1 and a is an element of R and suppose that f: Omega ->(onto) Omega' is a homeomorphism in the Zygmund-Sobolev space WLp log(alpha) L-loc(Omega,R-n). Define r=p/p-n+1. Assume that u is an element of WLr log(-alpha(r-1)) L-loc(Omega). Then u o f(-1) is an element of BVloc(Omega'). We obtain a similar result whenever f is a homeomorphism in the Lorentz-Sobolev space WLlocp,q(Omega, R-n) with p,q > n - 1 and u is an element of WLlocr,s(Omega) 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 o f(-1) is an element of W-loc(1,1)(Omega').