Composition operator for functions of bounded variation

We study the optimal conditions on a homeomorphism $f:\Omega\subset \R^n\to \R^n$toguaranteethatthecomposition$u\circ f$ belongs to the space of functions of bounded variation for every function $u$ of bounded variation. We show that a sufficient and necessary condition is the existence of a constant $K$ such that $\mathrm{\mid }Df\mathrm{\mid }(f^{-1}(A))\le K\text{Ln}(A)$ for all Borel sets $A$. We also characterize homeomorphisms which maps sets of finite perimeter to sets of finite perimeter. Towards these results we study when $f^{-1}$ maps sets of measure zero onto sets of measure zero (i.e. $f$ satisfies the Lusin $(N^{-1})$ condition).