Reshetnyak-class mappings and composition operators

Abstract

For the Reshetnyak-class homeomorphisms $\varphi:\Omega\to Y$, where~$\Omega$ 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 $$ \varphi^*:{\rm Lip}(Y)\to L_q^1(\Omega), $$ where $1\leq q\leq \infty$, as $\varphi^*u=u\circ\varphi$ for $u\in{\rm Lip}(Y)$. We demonstrate the utility of our approach by characterizing the homeomorphisms $\varphi:\Omega\to\Omega'$ of domains in some Carnot group~$\mathbb G$ which induce the bounded composition operator $$ \varphi^*: L^1_p(\Omega')\cap {\rm Lip}_{{\rm loc}}(\Omega')\to L^1_q (\Omega),\quad 1\leq q \leq p\leq \infty, $$ 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.