### abstract

- We study the invertibility of bounded composition operators of Sobolev spaces.We prove that if a homeomorphism φ of Euclidean domains D and D′ generates, by the composition rule \( \varphi ^* f - f \circ \varphi \), a bounded composition operator of the Sobolev spaces \( \varphi ^* :L_\infty ^1 \left( {D\prime} \right) \to L_p^1 \left( D \right) \), p > n - 1, has finite distortion and the Luzin N-property, then the inverse φ-1 generates the bounded composition operator from \( L_{p\prime}^1 \left( D \right),p\prime = p/\left( {p - n + 1} \right) \), into \( L_{1}^1 \left( D\prime \right)\).