Reshetnyak-class mappings and composition operators

For the Reshetnyak-class homeomorphisms $\phi :\mathrm{\Omega }\to Y$, where~$\mathrm{\Omega }$ is a~domain in some Carnot group and~$Y$ is a~metric space, we obtain an~equivalent description as the mappings which induce the bounded composition operator $${\phi }^{\ast }:\mathrm{L}\mathrm{i}\mathrm{p}(Y)\to L_q^1(\mathrm{\Omega }),$$ where $1\le q\le \mathrm{\infty }$, as ${\phi }^{\ast }u=u\circ \phi$ for $u\in \mathrm{L}\mathrm{i}\mathrm{p}(Y)$. We demonstrate the utility of our approach by characterizing the homeomorphisms $\phi :\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ of domains in some Carnot group~$\mathbb{G}$ which induce the bounded composition operator $${\phi }^{\ast }:L_p^1({\mathrm{\Omega }}^{\mathrm{\prime }})\cap {\mathrm{L}\mathrm{i}\mathrm{p}}_{\mathrm{l}\mathrm{o}\mathrm{c}}({\mathrm{\Omega }}^{\mathrm{\prime }})\to L_q^1(\mathrm{\Omega }),1\le q\le p\le \mathrm{\infty },$$ of homogeneous Sobolev spaces. The new proof is much shorter than the one already available, requires a~minimum of tools, and enables us to obtain new properties of the homeomorphisms in question.