Reshetnyak-class mappings and composition operators

Abstract

For the Reshetnyak-class homeomorphisms $\ddot{I}†:\hat{I}\text{c◯}\to Y$, where~$\hat{I}\text{c◯}$ 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 $$\ddot{I}{†}^{\ast }:\mathrm{Lip}(Y)\to L_q^1(\hat{I}\text{c◯}),$$ where $1\le q\le \infty$, as $\ddot{I}{†}^{\ast }u=u\circ \ddot{I}†$ for $u\in \mathrm{Lip}(Y)$. We demonstrate the utility of our approach by characterizing the homeomorphisms $\ddot{I}†:\hat{I}\text{c◯}\to \hat{I}{\text{c◯}}^{\prime }$ of domains in some Carnot group~$\mathbb{G}$ which induce the bounded composition operator $$\ddot{I}{†}^{\ast }:L_p^1(\hat{I}{\text{c◯}}^{\prime })\cap {\mathrm{Lip}}_{\mathrm{loc}}(\hat{I}{\text{c◯}}^{\prime })\to L_q^1(\hat{I}\text{c◯}),1\le q\le p\le \infty ,$$ 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.

23 pages, 19 references